1 Introduction↩︎

Let \(({\mathfrak{g}},{\mathfrak l})\) be a finite-dimensional quadratic pair of complex Lie superalgebras. Thus \({\mathfrak{g}}\) is equipped with a non-degenerate even supersymmetric invariant bilinear form \(B\), and \({\mathfrak l}\subseteq{\mathfrak{g}}\) is a Lie subsuperalgebra for which \(B_{{\mathfrak l}} \mathrel{\vcenter{:}}= B\vert_{{\mathfrak l}}\) is non-degenerate. We write \[{\mathfrak{g}}={\mathfrak l}\oplus{\mathfrak{p}},\qquad {\mathfrak{p}}={\mathfrak l}^{\perp}.\] In this setting Chen and Kang [1] constructed a relative cubic Dirac operator in \(({\mathfrak U}({\mathfrak{g}})\otimes{\operatorname{Cl}}({\mathfrak{p}}))^{{\mathfrak l}},\) which is the \({\mathfrak l}\)-basic subalgebra \(W({\mathfrak{g}},{\mathfrak l})\) of the quantum Weil algebra \({\mathcal{W}}({\mathfrak{g}})\mathrel{\vcenter{:}}={\mathfrak U}({\mathfrak{g}})\otimes{\operatorname{Cl}}({\mathfrak{g}})\) of \({\mathfrak{g}}\) [2]. In terms of a homogeneous basis \(\{e_a\}\) of \({\mathfrak{p}}\) with \(B|_{{\mathfrak{p}}}\)-dual basis \(\{e^a\}\), this element is given by \[\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}} = \sum_a e^a\otimes e_a -\frac{1}{12} \sum_{a,b,c} (-1)^{p(e_{a})p(e_{b})+p(e_{c})}B(e_{a},[e_{b},e_{c}]) \bigl(1\otimes e^a\wedge e^b\wedge e^c\bigr).\] It is \({\mathfrak l}\)-invariant and satisfies the Parthasarathy-type formula \[\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}^{2} = \Omega_{{\mathfrak{g}}}\otimes 1 + \frac{1}{24}\operatorname{str}(\operatorname{ad}_{{\mathfrak{g}}}(\Omega_{{\mathfrak{g}}})) - j\!\left( \Omega_{{\mathfrak l}}\otimes 1 + \frac{1}{24}\operatorname{str}(\operatorname{ad}_{{\mathfrak l}}(\Omega_{{\mathfrak l}})) \right),\] where \(\Omega_{{\mathfrak{g}}}\) and \(\Omega_{{\mathfrak l}}\) denote the quadratic Casimir elements and \(j\) is the canonical embedding of \({\mathcal{W}}({\mathfrak l})\) into \({\mathcal{W}}({\mathfrak{g}})\). The associated supercommutator \(d=[\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}},\cdot]_{{\mathcal{W}}}\) defines a differential on \({\mathcal{W}}({\mathfrak{g}},{\mathfrak l})\). Its cohomology identifies with the center \(\mathfrak Z({\mathfrak l})\) of \({\mathfrak U}({\mathfrak l})\). This construction extends Kostant’s cubic Dirac operator [3] to quadratic Lie superalgebras and provides the algebraic framework underlying the super-analogue of Vogan’s conjecture. Representation-theoretic applications have been studied in [2], [4]–[6]. These works develop Dirac cohomology for Lie superalgebras, compute it in several classes of modules, and derive structural consequences for highest weight theory. In particular, the square formula gives rise to Dirac inequalities, which play a role in the classification of unitarizable highest weight supermodules, including the case of \({\mathfrak{sl}}(m\vert n)\) [5].

The super setting admits a direct extension to color Lie algebras, introduced by Rittenberg and Wyler [7] and systematically studied by Scheunert [8]. These are common generalizations of Lie algebras and Lie superalgebras, obtained by replacing the \({\mathbb{Z}}_2\)-grading by a grading over an arbitrary abelian group \(\Gamma\) and the Koszul sign rule by a commutation factor \(\varepsilon:\Gamma\times\Gamma\to{\mathbb{C}}^\times\). The first genuinely color examples occur for \(\Gamma={\mathbb{Z}}_2\times{\mathbb{Z}}_2\), for instance the \({\mathbb{Z}}_2\times{\mathbb{Z}}_2\)-graded general linear algebras \(\mathfrak{gl}(m_1,m_2\vert n_1,n_2)\) and their orthosymplectic analogues (see [7]). Recently, color Lie algebras have reappeared in mathematical physics as a natural language for symmetry structures beyond the ordinary boson–fermion paradigm. This renewed interest is particularly visible in \({\mathbb{Z}}_2\times{\mathbb{Z}}_2\)-graded models, including the graded symmetries of the Lévy–Leblond equation [9], \({\mathbb{Z}}_2\times{\mathbb{Z}}_2\)-graded mechanics and its quantization [10], double-graded supersymmetric quantum mechanics [11], and \({\mathbb{Z}}_2\times{\mathbb{Z}}_2\)-graded parastatistics in multiparticle quantum Hamiltonians [12]. Moreover, the first author showed in [2] that the general formulation of cubic Dirac operators for Lie superalgebras naturally leads to color Lie algebras, since the quantum Weil algebra carries an intrinsic color Lie algebra structure.

The finite-dimensional color analogue of the relative cubic Dirac operator was constructed by Meyer [13]. The properties of \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}\), the Parthasarathy-type square formula, and the algebraic form of Vogan’s conjecture extend to this setting after replacing the super sign rules by the corresponding \(\varepsilon\)-sign rules.

This article constructs relative cubic Dirac operators for infinite-dimensional \({\mathbb{Z}}\)-graded color Lie algebras \({\mathfrak{g}}=\bigoplus_{i\in{\mathbb{Z}}}{\mathfrak{g}}_i\) with finite-dimensional homogeneous components. This includes, in particular, Kac–Moody superalgebras. We study representation-theoretic consequences of the resulting operators, compute the kernel of the square for integrable highest weight modules over \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\), and derive consequences of unitarity. Our construction generalizes Meinrenken’s infinite-dimensional version of Kostant’s cubic Dirac operator [14] for ordinary Lie algebras. In this setting, formal infinite sums are replaced by normally ordered elements in suitable completions; the obstruction to the finite-dimensional identities is the Kac–Peterson cocycle, and triviality of its cohomology class yields a corrected cubic Dirac operator satisfying the expected square formula.

1.1 Main Results and Organization↩︎

We describe the main results and the organization of the paper. Throughout, \(\mathfrak g\) is a \(\mathbb{Z}\)-graded quadratic color Lie algebra with finite-dimensional homogeneous components; see Section 2.1. The commutation factor is denoted by \(\varepsilon\), and the non-degenerate invariant \(\varepsilon\)-symmetric bilinear form by \(B\). The \(\mathbb{Z}\)-grading determines the decomposition \[\label{eq::decomposition95intro} \mathfrak g=\mathfrak g_-\oplus\mathfrak g_+,\qquad \mathfrak g_+=\bigoplus_{i>0}\mathfrak g_i,\qquad \mathfrak g_-=\bigoplus_{i\leq0}\mathfrak g_i.\tag{1}\]

The starting point is the usual finite-dimensional formulas for the quadratic Casimir element and for the cubic Dirac operator. For a \(\mathbb{Z}\)-graded infinite-dimensional color Lie algebra these formulas become infinite sums. The decomposition \(\mathfrak g=\mathfrak g_-\oplus\mathfrak g_+\) fixes the normal-ordering convention and determines the completions in which the infinite-dimensional analogues of the Casimir element and the cubic Dirac operator are defined. Concretely, 1 is used to define the completions appearing below: the completed exterior and Clifford algebras \(\widehat{\bigwedge}_{\varepsilon}({\mathfrak{g}})\) in Section 2.2, the completed symmetric \(\widehat{S}_{\varepsilon}({\mathfrak{g}})\) and enveloping algebras \(\widehat{{\mathfrak U}}_{\varepsilon}({\mathfrak{g}})\) in Section 2.4, and the completed Weil \(\widehat{W}_{\varepsilon}({\mathfrak{g}})\) and quantum Weil algebras \(\widehat{{\mathcal{W}}}({\mathfrak{g}})\) in Section 2.5. In this context, the exterior-to-Clifford quantization has to be replaced by its normally ordered analogue \(\widehat q:\widehat{\bigwedge}_{\varepsilon}(\mathfrak g)\to\widehat{\operatorname{Cl}}_{\varepsilon}(\mathfrak g)\), constructed in Definition 2. Similarly, the PBW symmetrization is replaced by the normally ordered map \(\widehat Q:\widehat S_{\varepsilon}(\mathfrak g)\to\widehat U_{\varepsilon}(\mathfrak g)\), constructed in Definition 4. Combining both yields an isomorphism of graded vector spaces \(\widehat{{\mathcal{Q}}}: \widehat{W}_{\varepsilon}({\mathfrak{g}})\xrightarrow{\sim}\widehat{{\mathcal{W}}}_{\varepsilon}({\mathfrak{g}})\).

The obstruction to the finite-dimensional formulas is the color analogue of the Kac–Peterson class. The chosen decomposition defines a color Lie algebra \(2\)-cocycle \({\psi_{\operatorname{KP}}}\in \widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak{g}}^{\ast})\) by \[{\psi_{\operatorname{KP}}}(x,y) \mathrel{\vcenter{:}}= \tfrac12{\operatorname{tr}_{\varepsilon}}(\operatorname{ad}_x\pi_{-}\operatorname{ad}_y\pi_{+}) -\tfrac12\varepsilon(x,y){\operatorname{tr}_{\varepsilon}}(\operatorname{ad}_y\pi_{-}\operatorname{ad}_x\pi_{+})\] for homogeneous \(x,y \in {\mathfrak{g}}\), where \(\pi_{\pm}\) denote the projections onto \({\mathfrak{g}}_{\pm}\). Its cohomology class is the Kac–Peterson class. It measures the anomaly introduced by normal ordering: the normally ordered Casimir and the cubic Dirac operator have the expected finite-dimensional properties precisely after this class becomes trivial and a corresponding linear correction is chosen. More precisely, via \(B\) the cocycle \({\psi_{\operatorname{KP}}}\) corresponds to an endomorphism \(\Psi_{{\psi_{\operatorname{KP}}}}\) defined by \({\psi_{\operatorname{KP}}}(x,y)=B(\Psi_{\operatorname{KP}}(x),y)\), and Lemma 10 shows that \({\psi_{\operatorname{KP}}}\) is a coboundary, that is, \({\psi_{\operatorname{KP}}}=d\rho\), if and only if \(\Psi_{\operatorname{KP}}(x)=[\rho^\sharp,x]\) for all \(x\in{\mathfrak{g}}\). In this case the anomaly is inner.

This gives the corrected Casimir and Dirac operators. The normal-ordered Casimir is the quantization \(\Omega'_{{\mathfrak{g}}}\mathrel{\vcenter{:}}= \widehat Q(p)\) of the invariant quadratic element \[p=\sum_a\varepsilon(e_a)e_ae^a=\sum_a e^ae_a\in\widehat S_{\varepsilon}^{2}({\mathfrak{g}}).\] Theorem 12 gives \(L_x\Omega'_{{\mathfrak{g}}}=2\Psi_{{\psi_{\operatorname{KP}}}}(x)\). Hence, if \({\psi_{\operatorname{KP}}}=d\rho\), then \(\Omega_{{\mathfrak{g}}}\mathrel{\vcenter{:}}=\Omega'_{{\mathfrak{g}}}+2\rho^\sharp\) is central.

The classical precursor of the cubic Dirac operator is the element \[{\mathcal{D}}\mathrel{\vcenter{:}}= \sum_a e^a\otimes e_a+1\otimes\phi \in \widehat{W}_{\varepsilon}({\mathfrak{g}})\] introduced in Section 3.3. Here \(\phi\in \widehat{\bigwedge}_{\varepsilon}^{3}({\mathfrak{g}})\) is the structure constants tensor defined in Section 3.2, that is, the invariant cubic tensor encoding the Lie bracket of \({\mathfrak{g}}\) through the bilinear form \(B\).

Before quantizing, we explain the origin of the classical element \({\mathcal{D}}\). We first extend the notion of a \({\mathfrak{g}}\)-differential algebra in [15] to color Lie algebras and to the completed setting, and define connections \(A\) together with their curvatures \(F^{A}\). The completed Weil algebra satisfies the expected universal property: by Theorem 16, every connection in a completed \({\mathfrak{g}}\)-differential algebra \(\widehat{{\mathcal{A}}}\) induces a unique homomorphism of completed \({\mathfrak{g}}\)-differential algebras \(\widehat{W}_{\varepsilon}({\mathfrak{g}})\to \widehat{{\mathcal{A}}}\), namely the Chern–Weil homomorphism. In this setting, a Chern–Simons element in a completed \({\mathfrak{g}}\)-differential algebra for an invariant polynomial \(P\) and a connection \(A\) is an odd element \({\mathrm{CS}}_P(A)\) satisfying \(d\,{\mathrm{CS}}_P(A)=P(F^A)\). Applying this to the quadratic invariant polynomial defined by \(B\) gives a canonical Chern–Simons element in \(\widehat{W}_{\varepsilon}({\mathfrak{g}})\). Proposition 19 identifies this element with \({\mathcal{D}}\).

The uncorrected cubic Dirac operator is obtained by normal-ordered quantization, \(\operatorname{D}'\mathrel{\vcenter{:}}= {\mathcal{Q}}({\mathcal{D}}).\) It is not \({\mathfrak{g}}\)-invariant, and its square contains an obstruction term; see Theorem 21. This obstruction is again determined by \({\psi_{\operatorname{KP}}}\). If \({\psi_{\operatorname{KP}}}=d\rho\), the corrected cubic Dirac operator is \[\operatorname{D}\mathrel{\vcenter{:}}= \operatorname{D}'+1\otimes\rho^\sharp .\] We prove that \(\operatorname{D}\) is \({\mathfrak{g}}\)-invariant and satisfies the square formula \[\operatorname{D}^{2} = \Omega_{{\mathfrak{g}}}\otimes 1 + \tfrac{1}{24}{\operatorname{tr}_{\varepsilon}}(\operatorname{ad}_{{\mathfrak{g}}_{0}}(\Omega_{{\mathfrak{g}}_{0}}))(1\otimes 1) + B(\rho^{\sharp},\rho^{\sharp})(1\otimes 1);\] see Corollary 3.

The relative construction is developed in Section 3.5. Let \((\mathfrak g,\mathfrak l)\) be a quadratic pair satisfying the assumptions of that section. Then one has two Dirac operators \(\operatorname{D}_{{\mathfrak{g}}}\in \widehat{{\mathcal{W}}}_{\varepsilon}({\mathfrak{g}})\) and \(\operatorname{D}_{{\mathfrak l}}\in \widehat{{\mathcal{W}}}_{\varepsilon}({\mathfrak l})\). In order to compare these two absolute Dirac operators, one first has to regard the quantum Weil algebra of \({\mathfrak l}\) inside that of \({\mathfrak{g}}\). This is done by a homomorphism \(j:\widehat{\mathcal{W}}_{\varepsilon}({\mathfrak l})\longrightarrow \widehat{\mathcal{W}}_{\varepsilon}({\mathfrak{g}})\), constructed in Section 3.5. Conceptually, \(j\) is the diagonal embedding: it sends the enveloping part of \({\mathfrak l}\) to its image in \({\mathfrak{g}}\), and the Clifford part is embedded using the action of \({\mathfrak l}\) on the \(B\)-orthogonal complement \({\mathfrak s}\) in \({\mathfrak{g}}={\mathfrak l}\oplus{\mathfrak s}\). With this embedding, the absolute Dirac operator of \({\mathfrak l}\) can be subtracted from the absolute Dirac operator of \({\mathfrak{g}}\) and one defines the relative cubic Dirac operator by \[\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}\mathrel{\vcenter{:}}= \operatorname{D}_{{\mathfrak{g}}}-j(\operatorname{D}_{{\mathfrak l}}).\]

Theorem 24 shows that \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}\) belongs to the completion of the \({\mathfrak l}\)-basic subalgebra of \({\mathcal{W}}_{\varepsilon}({\mathfrak{g}})\); equivalently, \(\iota_x\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}=0\) and \(L_x\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}=0\) for all \(x \in {\mathfrak l}\). Moreover, \[\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}^{2} = \Omega_{{\mathfrak{g}}}-j(\Omega_{{\mathfrak l}}) +\frac{1}{24}\operatorname{tr}_{\varepsilon} \bigl(\operatorname{ad}_{{\mathfrak{g}}_0}(\Omega_{{\mathfrak{g}}_0})\bigr) -\frac{1}{24}\operatorname{tr}_{\varepsilon} \bigl(\operatorname{ad}_{{\mathfrak l}_0}(\Omega_{{\mathfrak l}_0})\bigr) +B(\rho^\sharp,\rho^\sharp) -B(\rho_{{\mathfrak l}}^\sharp,\rho_{{\mathfrak l}}^\sharp).\] If \([{\mathfrak s},{\mathfrak s}]\subset{\mathfrak l}\), then the cubic term vanishes and \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}\) is quadratic.

As an application, we consider Kac–Moody superalgebras in Section 4.1. For every symmetrizable Kac–Moody superalgebra, Proposition 27 gives \(\psi_{\mathrm{KP}}=d\rho\), where \(\rho\) is the Weyl vector. Hence the Kac–Peterson class is trivial in this setting. In particular, every symmetrizable Kac–Moody superalgebra admits a corrected cubic Dirac operator \(\operatorname{D}\in\widehat{\mathcal{W}}({\mathfrak{g}})\), which we simply call the Dirac operator of \({\mathfrak{g}}\).

Sections 4.2 and 4.3 study the action of the relative Dirac operator for the quadratic pair \(({\mathfrak{g}},{\mathfrak{g}_{\bar{0}}})\). For a highest weight \({\mathfrak{g}}\)-supermodule \(M\), the operator acts on the completed endomorphism space of \(M\otimes M({\mathfrak{g}_{\bar{1}}})\), \[\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\in \widehat{\operatorname{End}}(M\otimes M({\mathfrak{g}_{\bar{1}}})),\] where \(M({\mathfrak{g}_{\bar{1}}})\) is the oscillator module constructed in Section 4.2. We prove that \(\ker\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) is an \({\mathfrak{g}_{\bar{0}}}\)-module. Moreover, if \(M\) has highest weight \(\Lambda\) and \(V(\mu)\) is a highest weight \({\mathfrak{g}_{\bar{0}}}\)-submodule of \(M\otimes M({\mathfrak{g}_{\bar{1}}})\), then \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\) acts on \(V(\mu)\) by the scalar \[B(\Lambda+\rho,\Lambda+\rho)-B(\mu+\rho_{\bar0},\mu+\rho_{\bar0});\] see Proposition 29. Thus the square measures the difference between the \({\mathfrak{g}}\)- and \({\mathfrak{g}_{\bar{0}}}\)-Casimir eigenvalues. Proposition 30 gives the corresponding non-vanishing criterion and shows that \(\ker\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\neq0\) for every highest weight supermodule.

Section 4.4 specializes the preceding considerations to the Kac–Moody superalgebras associated to \(\mathfrak{osp}(1|2n)\). For integrable highest weight supermodules, Theorem 33 gives an explicit computation of \(\ker\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\). Here we refer to Section 4.1 for the definition of \(\rho_{\bar1}\) in the Kac–Moody setting.

We finally study the implications of \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) for unitarizable highest weight supermodules in Section 4.5. We recall the notion of unitarity following Jakobsen in [16]. Unitarity is defined relative to a conjugate-linear anti-involution \(\omega\) of \({\mathfrak{g}}\): a highest weight module is \(\omega\)-unitarizable if its contravariant Hermitian form is positive definite. The oscillator module \(M({\mathfrak{g}_{\bar{1}}})\) is constructed from a polarization of \({\mathfrak{g}_{\bar{1}}}\). To combine it with an \(\omega\)-unitarizable module \(M\), this polarization must be compatible with the anti-involution \(\omega\). This is the \(\omega\)-adaptedness condition. Under this assumption, Proposition 35 shows that \(M\otimes M({\mathfrak{g}_{\bar{1}}})\) is a \(\omega|_{{\mathfrak{g}_{\bar{0}}}}\)-unitarizable \({\mathfrak{g}_{\bar{0}}}\)-module. If the polarization is \(\omega\)-adapted, then \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) is self-adjoint or skew-adjoint. Combined with the square formula, this yields the Dirac inequality in Proposition 36: if \(M\) is \(\omega\)-unitarizable of highest weight \(\Lambda\), if \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) is self-adjoint, and if \(V(\mu)\subset M\otimes M({\mathfrak{g}_{\bar{1}}})\) is a highest weight \({\mathfrak{g}_{\bar{0}}}\)-submodule, then \[B(\Lambda+\rho,\Lambda+\rho)-B(\mu+\rho_{\bar0},\mu+\rho_{\bar0})\leq0.\]

The section concludes by relating the Dirac kernel to Lie superalgebra cohomology. Under assumptions on the polarization \({\mathfrak{g}_{\bar{1}}}= {\mathfrak{g}}_{+1}\oplus {\mathfrak{g}}_{-1}\) satisfied, for example, by Kac–Moody superalgebras such as \(\widehat{{\mathfrak{sl}}}(m\vert n)\), Theorem 37 gives an isomorphism of \({\mathfrak{g}_{\bar{0}}}\)-modules \[\ker \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}} \cong H^\ast({\mathfrak{g}}_{+1};M)\otimes\mathbb{C}_{-\rho_{\bar1}}.\]

1.2 Conventions↩︎

We write \({\mathbb{Z}}_{+}\) for the set of positive integers and \({\mathbb{Z}}_{2}\mathrel{\vcenter{:}}={\mathbb{Z}}/2{\mathbb{Z}}\), with elements \(\bar 0\) and \(\bar 1\). The ground field is \({\mathbb{C}}\).

Unless stated otherwise, \({\mathfrak{g}}\) denotes a \({\mathbb{Z}}\)-graded quadratic color Lie algebra. All color objects are \(\Gamma\)-graded. The additional \({\mathbb{Z}}\)-grading is the grading used for completions. Thus, by a \({\mathbb{Z}}\)-graded color Lie algebra, we mean a color Lie algebra endowed with a compatible \({\mathbb{Z}}\)-grading. If, in addition, there is a compatible \({\mathbb{Z}}_{2}\)-grading, we speak of a \({\mathbb{Z}}\)-graded color superalgebra in order to emphasize the role of this additional \({\mathbb{Z}}_{2}\)-grading.

If an object carries both a \({\mathbb{Z}}\)-grading and a \(\Gamma\)-grading, integer subscripts refer to the \({\mathbb{Z}}\)-grading and Greek subscripts to the \(\Gamma\)-grading.

For homogeneous \(x\in{\mathfrak{g}}\), the symbol \(L_x\) denotes the induced Lie derivative on the algebra under consideration. Thus the same notation is used on \(\bigwedge_{\varepsilon}({\mathfrak{g}})\), \(S_{\varepsilon}({\mathfrak{g}})\), \(U_{\varepsilon}({\mathfrak{g}})\), \(W_{\varepsilon}({\mathfrak{g}})\), and \({\mathcal{W}}_{\varepsilon}({\mathfrak{g}})\). The ambient algebra determines the meaning. If several such operators occur simultaneously, we write \(L_x^{\wedge}\), \(L_x^S\), \(L_x^U\), \(L_x^W\), and \(L_x^{{\mathcal{W}}}\).

2 Preliminaries↩︎

This section introduces the conventions and notation underlying this work, as well as the basic definitions employed throughout.

2.1 Color Lie Algebras↩︎

Color Lie algebras arise as the natural generalization of Lie superalgebras obtained by replacing the parity group \({\mathbb{Z}}_{2}\mathrel{\vcenter{:}}={\mathbb{Z}}/2{\mathbb{Z}}\) and the super sign by an arbitrary abelian grading group \(\Gamma\) and a commutation factor \(\varepsilon\colon \Gamma\times\Gamma\to {\mathbb{C}}^{\times}\). Historically, such structures already appeared in the work of Ree [17], were developed further in the physics literature by Rittenberg and Wyler [7] in connection with generalized statistics, and were subsequently given a systematic algebraic treatment by Scheunert [8]. Thus color Lie algebras provide a framework for graded Lie theory beyond the ordinary and super cases.

Let \(\Gamma\) be an abelian group, written additively, with neutral element \({0_{\scriptscriptstyle \Gamma}}\). A \(\Gamma\)-graded vector space is a vector space \(V\) together with a decomposition \(V=\bigoplus_{\gamma\in\Gamma}V_{\gamma}\). An element \(v\in V_{\gamma}\) is called homogeneous of degree \(\gamma\), and we write \(\vert v\vert =\gamma\).

A morphism \(f\colon V\to W\) of \(\Gamma\)-graded vector spaces is a linear map preserving the grading, that is, \(f(V_{\gamma})\subseteq W_{\gamma}\) for all \(\gamma\in\Gamma\). For fixed \(\Gamma\), we denote by \(\operatorname{Hom}_{{\mathrm{gr}}}(V,W)\) the space of morphisms of \(\Gamma\)-graded vector spaces from \(V\) to \(W\), and we set \(\operatorname{End}_{{\mathrm{gr}}}(V)\mathrel{\vcenter{:}}= \operatorname{Hom}_{{\mathrm{gr}}}(V,V)\). We write \(\operatorname{Hom}(V,W)\) for the graded vector space of finite sums of homogeneous linear maps from \(V\) to \(W\), and set \(\operatorname{End}(V)\mathrel{\vcenter{:}}=\operatorname{Hom}(V,V)\). This is again a \(\Gamma\)-graded vector space \(\operatorname{Hom}(V,W) = \bigoplus_{\gamma \in \Gamma} \operatorname{Hom}(V,W)_{\gamma}\), where \[\operatorname{Hom}(V,W)_{\gamma} \mathrel{\vcenter{:}}= \{T \in \operatorname{Hom}(V,W) : T(V_{\alpha}) \subseteq W_{\alpha+\gamma} \;\text{for all } \alpha \in \Gamma\}.\]

The \(\Gamma\)-graded tensor product of two \(\Gamma\)-graded vector spaces \(V=\bigoplus_{\alpha\in\Gamma}V_{\alpha}\) and \(W=\bigoplus_{\beta\in\Gamma}W_{\beta}\) is the \(\Gamma\)-graded vector space \[V\otimes W=\bigoplus_{\gamma\in\Gamma}(V\otimes W)_{\gamma},\qquad (V\otimes W)_{\gamma}\mathrel{\vcenter{:}}=\bigoplus_{\alpha+\beta=\gamma}V_{\alpha}\otimes W_{\beta}.\] In particular, if \(v\in V_{\alpha}\) and \(w\in W_{\beta}\) are homogeneous, then \(\vert v\otimes w\vert =\alpha+\beta\).

A commutation factor is a map \(\varepsilon\colon\Gamma\times\Gamma\to{\mathbb{C}}^{\times}\) such that, for all \(\alpha,\beta,\gamma\in\Gamma\), one has (see [18]) \[\label{eq::commutation95factor}\varepsilon(\alpha+\beta,\gamma)=\varepsilon(\alpha,\gamma)\varepsilon(\beta,\gamma),\qquad \varepsilon(\alpha,\beta+\gamma)=\varepsilon(\alpha,\beta)\varepsilon(\alpha,\gamma),\qquad \varepsilon(\alpha,\beta)\varepsilon(\beta,\alpha)=1.\tag{2}\] In particular, \(\varepsilon(\gamma,\gamma)^{2}=1\) for all \(\gamma\in\Gamma\). For \(\Gamma\)-graded vector spaces, a commutation factor \(\varepsilon\) defines a symmetry \[\tau_{V,W}\colon V\otimes W\to W\otimes V,\qquad v\otimes w\mapsto \varepsilon(\vert v\vert ,\vert w\vert)w\otimes v\] for homogeneous \(v\in V\) and \(w\in W\). It satisfies \(\tau_{W,V}\circ\tau_{V,W}=\operatorname{id}_{V\otimes W}\). The symmetry induced by \(\varepsilon\) yields canonical actions of symmetric groups on tensor powers \(V^{\otimes n}\). Concretely, for each \(n\geq 1\), there is a unique right action \(\pi\colon S_{n}\to\operatorname{GL}(V^{\otimes n})\) such that, for every adjacent transposition \(s_{i}=(i,i+1)\), \[\pi(s_{i})(v_{1}\otimes\cdots\otimes v_{n})=\varepsilon(\vert v_{i}\vert,\vert v_{i+1}\vert)v_{1}\otimes\cdots\otimes v_{i+1}\otimes v_{i}\otimes\cdots\otimes v_{n}\] for homogeneous \(v_{1},\dots,v_{n}\in V\). In general, for \(\sigma\in S_{n}\), \[\pi(\sigma)(v_{1}\otimes\cdots\otimes v_{n})=p'(\sigma;v_{1},\dots,v_{n})v_{\sigma(1)}\otimes\cdots\otimes v_{\sigma(n)},\] where \[p'(\sigma;v_{1},\dots,v_{n})\mathrel{\vcenter{:}}= \prod_{\substack{1\leq i<j\leq n\\ \sigma^{-1}(i)>\sigma^{-1}(j)}}\varepsilon(\vert v_{i}\vert, \vert v_{j}\vert).\] Analogously, there is a unique signed right action of \(S_{n}\) given by \[\pi_{\mathrm{sgn}}(\sigma)(v_1\otimes\cdots\otimes v_n) = p(\sigma;v_1,\dots,v_n)v_{\sigma(1)}\otimes\cdots\otimes v_{\sigma(n)},\] where \(p(\sigma;v_{1},\ldots,v_{n})\mathrel{\vcenter{:}}=\operatorname{sgn}(\sigma)p'(\sigma;v_{1},\ldots,v_{n})\). Moreover, \[p(\sigma\sigma';v_{1},\dots,v_{n})=p(\sigma';v_{\sigma(1)},\dots,v_{\sigma(n)})p(\sigma;v_{1},\dots,v_{n}),\qquad p(\operatorname{id};v_{1},\dots,v_{n})=1.\] These are the color analogues of the permutation action and signed permutation action, respectively.

In what follows, we fix the pair \((\Gamma,\varepsilon)\). For homogeneous elements \(v\) and \(w\), we write \(\varepsilon(v,w)\mathrel{\vcenter{:}}=\varepsilon(\vert v\vert ,\vert w\vert)\) and \(\varepsilon(v)\mathrel{\vcenter{:}}=\varepsilon(\vert v\vert ,\vert v\vert)\). Then \({\boldsymbol{Vec}_{\Gamma}^{\varepsilon}}\) denotes the symmetric monoidal category whose objects are \(\Gamma\)-graded vector spaces, whose morphisms are grading-preserving linear maps, whose tensor product is the \(\Gamma\)-graded tensor product, and whose symmetry is induced by the commutation factor \(\varepsilon\).

The category \({\boldsymbol{Vec}_{\Gamma}^{\varepsilon}}\) is equipped with the \(\varepsilon\)-trace. Concretely, let \(V\in{\boldsymbol{Vec}_{\Gamma}^{\varepsilon}}\) be finite-dimensional, and define \(\mathcal{E}\in\operatorname{End}(V)\) by \(\mathcal{E}(v)\mathrel{\vcenter{:}}=\varepsilon(v)v\) for homogeneous \(v\in V\), extended linearly. The \(\varepsilon\)-trace is the linear map \(\operatorname{tr}_{\varepsilon}\colon\operatorname{End}(V)\to{\mathbb{C}}\) defined by \[\operatorname{tr}_{\varepsilon}(T)\mathrel{\vcenter{:}}=\operatorname{tr}(\mathcal{E}\circ T).\] Its basic property is the \(\varepsilon\)-cyclicity \[\operatorname{tr}_{\varepsilon}(ST)=\varepsilon(S,T)\operatorname{tr}_{\varepsilon}(TS)\] for homogeneous \(S,T\in\operatorname{End}(V)\). Of particular interest are \(\varepsilon\)-quadratic spaces in \({\boldsymbol{Vec}_{\Gamma}^{\varepsilon}}\). A \(\Gamma\)-graded vector space \(V\) is called \(\varepsilon\)-quadratic if it is endowed with an even, nondegenerate, \(\varepsilon\)-symmetric bilinear form \(\langle\cdot,\cdot\rangle\colon V\times V\to{\mathbb{C}}\). Here, even means that \(\langle v,w\rangle=0\) for homogeneous \(v,w\in V\) unless \(\vert v\vert +\vert w\vert = {0_{\scriptscriptstyle \Gamma}}\), and \(\varepsilon\)-symmetric means that \(\langle v,w\rangle=\varepsilon(v,w)\langle w,v\rangle\) for all homogeneous \(v,w\in V\). This notion is particularly relevant for the study of color Lie algebras endowed with invariant bilinear forms, to which we now turn.

A color algebra \({\mathcal{A}}\) is an algebra object in the symmetric monoidal category of \(\Gamma\)-graded vector spaces determined by the commutation factor \(\varepsilon\). If \({\mathcal{A}}\) and \(\mathcal{B}\) are color algebras, then \({\mathcal{A}}\otimes \mathcal{B}\) is again a color algebra, with multiplication \[(a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=\varepsilon(b_{1},a_{2})a_{1}a_{2}\otimes b_{1}b_{2}\] for homogeneous \(a_{1},a_{2}\in {\mathcal{A}}\) and \(b_{1},b_{2}\in \mathcal{B}\). Moreover, a homogeneous map \(D\in\operatorname{End}({\mathcal{A}})\) of degree \(\vert D\vert \in\Gamma\) is called an \(\varepsilon\)-derivation if \[\label{def::epsilon95derivation} D(ab)=D(a)b+\varepsilon(D,a)\,aD(b)\tag{3}\] for all homogeneous \(a,b\in{\mathcal{A}}\). We denote the \(\Gamma\)-graded vector space of all \(\varepsilon\)-derivations of \({\mathcal{A}}\) by \({\mathrm{Der}}_{(\Gamma,\varepsilon)}({\mathcal{A}})\). The two main examples of color algebras considered below also carry a compatible \({\mathbb{Z}}_2\)-grading with parity function \(p(\cdot)\). In this case we use the combined commutation factor \(\varepsilon'(D,a)\mathrel{\vcenter{:}}= (-1)^{p(D)p(a)}\varepsilon(D,a).\) A homogeneous \(\varepsilon'\)-derivation \(D\in{\mathrm{Der}}_{(\Gamma,\varepsilon')}({\mathcal{A}})\) is called even or odd according as \(p(D)=\bar0\) or \(p(D)=\bar1\), and then satisfies \[\label{eq::parity95epsilon95derivations} D(ab)=D(a)b+(-1)^{p(D)p(a)}\varepsilon(D,a)\,aD(b)\tag{4}\] for homogeneous \(a,b\in{\mathcal{A}}\). We denote the spaces of even and odd \(\varepsilon'\)-derivations by \({\mathrm{Der}}_{(\Gamma,\varepsilon')}^{\bar0}({\mathcal{A}})\) and \({\mathrm{Der}}^{\bar{1}}_{(\Gamma,\varepsilon')}({\mathcal{A}})\), respectively.

A color Lie algebra is a Lie algebra object in this symmetric monoidal category. Concretely:

A color Lie algebra \({\mathfrak{g}}\) is called quadratic if it is endowed with an invariant bilinear form \(B\) such that \(({\mathfrak{g}},B)\) is an \(\varepsilon\)-quadratic \(\Gamma\)-graded vector space. Here, invariance means that \[B([x,y],z)=B(x,[y,z])\] for all homogeneous \(x,y,z\in{\mathfrak{g}}\).

By a color (Lie) superalgebra we mean a color (Lie) algebra \({\mathfrak{g}}=\bigoplus_{\gamma\in\Gamma}{\mathfrak{g}}_{\gamma}\) endowed with a compatible \({\mathbb{Z}}_{2}\)-grading. We denote the associated parity of a homogeneous element \(x\in{\mathfrak{g}}\) by \(p(x)\in{\mathbb{Z}}_{2}\). Equivalently, after replacing \(\Gamma\) by \({\mathbb{Z}}_{2}\times\Gamma\), this is a color (Lie) algebra with commutation factor \(\varepsilon'(x,y)=(-1)^{p(x)p(y)}\varepsilon(x,y)\) for all homogeneous \(x,y\in {\mathfrak{g}}\).

A representation of a color Lie algebra \({\mathfrak{g}}\) is a \(\Gamma\)-graded vector space \(V\) together with a homomorphism of color Lie algebras \(\rho:{\mathfrak{g}}\to\mathfrak{gl}_{\varepsilon}(V)\). Equivalently, a \({\mathfrak{g}}\)-module is a \(\Gamma\)-graded vector space \(V=\bigoplus_{\gamma\in\Gamma}V_\gamma\) together with a bilinear map \({\mathfrak{g}}\times V\to V\), \((x,v)\mapsto x\cdot v\), such that for all homogeneous \(x,y\in{\mathfrak{g}}\) and \(v\in V\) one has \(x\cdot v\in V_{\vert x\vert +\vert v\vert }\) and \[[x,y]\cdot v=x\cdot(y\cdot v)-\varepsilon(x,y)y\cdot(x\cdot v).\] A basic example is the adjoint module, where \({\mathfrak{g}}\) acts on itself by \(\operatorname{ad}_x(y)\mathrel{\vcenter{:}}=[x,y]\). Standard notions such as submodules and simple modules are defined in the usual graded sense.

We now focus on color Lie algebras equipped with an additional \({\mathbb{Z}}\)-grading. This includes important infinite-dimensional examples, in particular Kac–Moody superalgebras.

2.1.1 \({\mathbb{Z}}\)-graded Color Lie Algebras↩︎

Let \({\mathfrak{g}}=\bigoplus_{i\in{\mathbb{Z}}}{\mathfrak{g}}_{i}\) be a \({\mathbb{Z}}\)-graded color Lie algebra, that is, a color Lie algebra with homogeneous components \({\mathfrak{g}}_{i}=\bigoplus_{\gamma\in\Gamma}{\mathfrak{g}}_{i,\gamma}\) satisfying \([{\mathfrak{g}}_{i},{\mathfrak{g}}_{j}]\subseteq{\mathfrak{g}}_{i+j}\). We assume that each \({\mathfrak{g}}_{i}\) is finite-dimensional. In particular, \({\mathfrak{g}}_{0}\) is a color Lie algebra. Thus \({\mathfrak{g}}\) is naturally \({\mathbb{Z}}\times\Gamma\)-graded, and every homogeneous element \(x\in{\mathfrak{g}}\) has a bidegree \((i,\vert x\vert)\in{\mathbb{Z}}\times\Gamma\). For simplicity, we write \({\operatorname{deg}}(x)\mathrel{\vcenter{:}}= i\) and call it the \({\mathbb{Z}}\)-degree of \(x\). Thus \({\mathfrak{g}}_{i}\) always refers to the \({\mathbb{Z}}\)-grading, whereas \({\mathfrak{g}}_{\gamma}\) refers to the \(\Gamma\)-grading. More generally, whenever an object carries both a \({\mathbb{Z}}\)-grading and a \(\Gamma\)-grading, integer subscripts refer to the \({\mathbb{Z}}\)-grading, whereas Greek subscripts refer to the \(\Gamma\)-grading. In this setting, a \({\mathbb{Z}}\)-graded \({\mathfrak{g}}\)-module is a \({\mathfrak{g}}\)-module \(V=\bigoplus_{j\in{\mathbb{Z}}}V_{j}\) such that \({\mathfrak{g}}_{i}\cdot V_{j}\subseteq V_{i+j}\) for all \(i,j\in{\mathbb{Z}}\).

If \({\mathfrak{g}}\) is quadratic with non-degenerate \(\varepsilon\)-symmetric invariant bilinear form \(B\), we assume in addition that \(B(x,y)=0\) for homogeneous \(x,y\in{\mathfrak{g}}\) unless \(\vert x\vert +\vert y\vert = {0_{\scriptscriptstyle \Gamma}}\) and \({\operatorname{deg}}(x)+{\operatorname{deg}}(y)=0\).

Set \({\mathfrak{g}}_{+} \mathrel{\vcenter{:}}= \bigoplus_{i >0} {\mathfrak{g}}_{i}\) and \({\mathfrak{g}}_{-} \mathrel{\vcenter{:}}= \bigoplus_{i \leq 0} {\mathfrak{g}}_{i}\), and let \(\pi_{\pm} : {\mathfrak{g}}\to {\mathfrak{g}}_{\pm}\) be the associated even projection maps, i.e., they preserve the \({\mathbb{Z}}\times\Gamma\)-grading. By construction, both are color Lie subalgebras.

The \({\mathbb{Z}}\times\Gamma\)-graded restricted dual \({\mathfrak{g}}^{\ast}\) of \({\mathfrak{g}}\) is the direct sum over the duals of the \(\Gamma\)-graded vector spaces \({\mathfrak{g}}_{i}\) with \({\mathbb{Z}}\times\Gamma\)-grading \(({\mathfrak{g}}^{\ast})_{(i,\gamma)} = ({\mathfrak{g}}_{(-i, -\gamma)})^{\ast}\).

We denote by \(\operatorname{End}({\mathfrak{g}})\) and \(\operatorname{End}_{{\mathrm{gr}}}({\mathfrak{g}})\) the direct sum of the spaces \(\operatorname{End}({\mathfrak{g}})_{i} \mathrel{\vcenter{:}}= \bigoplus_{r} \operatorname{Hom}({\mathfrak{g}}_{r}, {\mathfrak{g}}_{r+i})\) and \(\operatorname{End}_{{\mathrm{gr}}}({\mathfrak{g}})_{i} \mathrel{\vcenter{:}}= \bigoplus_{r} \operatorname{Hom}_{{\mathrm{gr}}}({\mathfrak{g}}_{r}, {\mathfrak{g}}_{r+i})\), respectively. The spaces \(\operatorname{End}({\mathfrak{g}})_{i}\) and \(\operatorname{End}_{{\mathrm{gr}}}({\mathfrak{g}})_{i}\) consist of linear maps of \({\mathbb{Z}}\)-degree \(i\) with only finitely many nonzero \(r\)-components. Such maps are called finitary. We set \[\label{def::completion95End} \widehat{\operatorname{End}}({\mathfrak{g}})_{i} \mathrel{\vcenter{:}}= \prod_{r} \operatorname{Hom}({\mathfrak{g}}_{r}, {\mathfrak{g}}_{r+i}), \qquad \widehat{\operatorname{End}}_{{\mathrm{gr}}}({\mathfrak{g}})_{i} \mathrel{\vcenter{:}}= \prod_{r} \operatorname{Hom}_{{\mathrm{gr}}}({\mathfrak{g}}_{r}, {\mathfrak{g}}_{r+i}).\tag{6}\] Then the completions \(\widehat{\operatorname{End}}({\mathfrak{g}})\) and \(\widehat{\operatorname{End}}_{{\mathrm{gr}}}({\mathfrak{g}})\) are the direct sums of these spaces, respectively.

Since we consider the restricted dual \({\mathfrak{g}}^{\ast}\) and fix a nondegenerate bilinear form \(B\), we have a natural identification of \({\mathfrak{g}}\) and \({\mathfrak{g}}^{\ast}\), given by the usual musical isomorphisms \[\label{eq::musical95isomorphisms} \begin{align} &\flat \colon {\mathfrak{g}}\xrightarrow{\;\sim\;} {\mathfrak{g}}^{\ast}, \qquad x \longmapsto x^{\flat}, \quad x^{\flat}(y) \mathrel{\vcenter{:}}= B(x,y),\\[2pt] &\sharp \colon {\mathfrak{g}}^{\ast} \xrightarrow{\;\sim\;} {\mathfrak{g}}, \qquad \alpha \longmapsto \alpha^{\sharp}, \quad B(\alpha^{\sharp},y)=\alpha(y), \end{align}\tag{7}\] for all \(y \in {\mathfrak{g}}\). In what follows, we identify \({\mathfrak{g}}\) and \({\mathfrak{g}}^{\ast}\).

Let \((e_{a})\) be a homogeneous basis of \({\mathfrak{g}}\) with \(B\)-dual basis \((e^{a})\), that is, \(B(e_{a},e^{b})=\delta_{ab}\). Then \(\vert e_{a}\vert =-\vert e^{a}\vert\) and \({\operatorname{deg}}(e^{a})=-{\operatorname{deg}}(e_{a})\). Moreover, \(\varepsilon(e_{a})=\varepsilon(e_{a},e^{a})\). Indeed, \[1=\varepsilon(e_{a},{0_{\scriptscriptstyle \Gamma}})=\varepsilon(e_{a},e_{a}+(-e_{a}))=\varepsilon(e_{a},e_{a})\varepsilon(e_{a},-e_{a}),\] hence \(\varepsilon(e_{a},e^{a})=\varepsilon(e_{a})^{-1}\). Since \(\varepsilon(e_{a})^{2}=1\), it follows that \(\varepsilon(e_{a})=\varepsilon(e_{a},e^{a})\).

If \((e_{a}^{\ast})\) denotes a homogeneous basis of the restricted dual such that \(e_{a}^{\ast}(e_{b})=\delta_{ab}\), then the identification 7 gives \[e_{a}^{\ast}=\varepsilon(e_{a})B(e^{a},\cdot) \quad\Leftrightarrow\quad (e_{a}^{\ast})^{\sharp}=\varepsilon(e_{a})e^{a}.\] In particular, we have a natural identification \[\label{eq::identification95End40g41} \operatorname{End}({\mathfrak{g}}) \cong {\mathfrak{g}}\otimes {\mathfrak{g}}^{\ast} \cong {\mathfrak{g}}\otimes {\mathfrak{g}}, \qquad f \mapsto \sum_{a}f(e_{a})\otimes e_{a}^{\ast} \mapsto \sum_{a}\varepsilon(e_{a})f(e_{a}) \otimes e^{a}.\tag{8}\] Note that the map is well-defined since \(f \in \operatorname{End}({\mathfrak{g}})\) is finitary.

The following lemma is immediate and will be useful in what follows.

In what follows, \(({\mathfrak{g}},B)\), or simply \({\mathfrak{g}}\), denotes a \({\mathbb{Z}}\)-graded quadratic color Lie algebra, unless otherwise stated.

2.2 Completion of \(\varepsilon\)-Exterior and \(\varepsilon\)-Clifford Algebras of \({\mathfrak{g}}\)↩︎

Chen–Kang [19] and Meyer [13] introduced \(\varepsilon\)-exterior algebras and \(\varepsilon\)-Clifford algebras for finite-dimensional color Lie algebras. In this section, we adopt their notation and conventions in order to define these objects for the \({\mathbb{Z}}\)-graded color Lie algebras \({\mathfrak{g}}\) introduced above. In particular, we also formulate the corresponding completions.

Consider the tensor algebra \(\mathrm{T}({\mathfrak{g}})\), endowed with the \({\mathbb{Z}}\times \Gamma\)-grading induced from \({\mathfrak{g}}\), where each generator in \({\mathfrak{g}}_{(i,\gamma)}\) is assigned bidegree \((i,\gamma)\). It is a unital associative algebra with identity element \(1_{\mathrm{T}({\mathfrak{g}})}\), and moreover it is naturally equipped with an additional canonical \({\mathbb{Z}}\)-grading: for each integer \(k \geq 0\), the homogeneous component of tensor length \(k\), denoted \(\mathrm{T}^{k}({\mathfrak{g}})\), is the subspace spanned by all elements \(x_{1}\otimes \cdots \otimes x_{k}\) with \(x_{i} \in {\mathfrak{g}}\); by convention one sets \(\mathrm{T}^{0}({\mathfrak{g}}) = {\mathbb{C}}\). This tensor-length grading is compatible with the \({\mathbb{Z}}\times\Gamma\)-grading in the evident sense, that is, \(\mathrm{T}({\mathfrak{g}})=\bigoplus_{k\geq 0}\;\bigoplus_{(i,\gamma)\in{\mathbb{Z}}\times\Gamma}\mathrm{T}^{k}({\mathfrak{g}})_{(i,\gamma)}\). Moreover, reducing this grading modulo \(2\) endows \(\mathrm{T}({\mathfrak{g}})\) with the structure of a superalgebra, with even and odd parts \[\mathrm{T}({\mathfrak{g}})_{\bar{0}} \mathrel{\vcenter{:}}= \bigoplus_{k \in 2{\mathbb{Z}}_{+}} \mathrm{T}^{k}({\mathfrak{g}}), \qquad \mathrm{T}({\mathfrak{g}})_{\bar{1}} \mathrel{\vcenter{:}}= \bigoplus_{k \in 2{\mathbb{Z}}_{+}+1} \mathrm{T}^{k}({\mathfrak{g}}).\] In what follows, we consider \(\mathrm{T}({\mathfrak{g}})\) naturally as a \(({\mathbb{Z}}\times{\mathbb{Z}})\)-graded color algebra.

2.2.1 \(\varepsilon\)-Exterior Algebra and Completion↩︎

Let \(I_{\wedge}\) denote the (\({\mathbb{Z}}\times \Gamma\)-graded) homogeneous ideal of \(\mathrm{T}({\mathfrak{g}})\) generated by the elements \[x\otimes y+\varepsilon(x,y)y\otimes x,\qquad x,y\in{\mathfrak{g}}\;\text{homogeneous}.\] The quotient \(\bigwedge_{\varepsilon}({\mathfrak{g}})\mathrel{\vcenter{:}}= \mathrm{T}({\mathfrak{g}})/I_{\wedge}\) is the \(\varepsilon\)-exterior algebra of \({\mathfrak{g}}\), that is, the unital associative \({\mathbb{Z}}\)-graded color algebra generated by \({\mathfrak{g}}\) with defining relations \(x\wedge y+\varepsilon(x,y)y\wedge x=0\) for homogeneous \(x,y\in{\mathfrak{g}}\). We call \(\wedge\) the exterior multiplication and we write \(\bigwedge_{\varepsilon}({\mathfrak{g}})_{(i,\gamma)}\) for its homogeneous component of bidegree \((i,\gamma)\in{\mathbb{Z}}\times\Gamma\).

The canonical tensor-length \({\mathbb{Z}}\)-grading of \(\mathrm{T}({\mathfrak{g}})\) induces an additional \({\mathbb{Z}}\)-grading on \(\bigwedge_{\varepsilon}({\mathfrak{g}})\), giving the decomposition \(\bigwedge_{\varepsilon}({\mathfrak{g}}) = \bigoplus_{k \geq 0} \bigwedge^{k}_{\varepsilon}({\mathfrak{g}})\), where \(\bigwedge^{k}_{\varepsilon}({\mathfrak{g}})\) is the subspace spanned by exterior products of \(k\) elements of \({\mathfrak{g}}\). We denote the associated degree by \(\vert \cdot\vert_{\wedge}\). By declaring \(\bigwedge_{\varepsilon}({\mathfrak{g}})_{\bar{0}} \mathrel{\vcenter{:}}= \bigoplus_{k \in 2{\mathbb{Z}}_{+}} \bigwedge^{k}_{\varepsilon}({\mathfrak{g}})\) and \(\bigwedge_{\varepsilon}({\mathfrak{g}})_{\bar{1}} \mathrel{\vcenter{:}}= \bigoplus_{k \in 2{\mathbb{Z}}_{+}+1} \bigwedge^{k}_{\varepsilon}({\mathfrak{g}})\), the algebra \(\bigwedge_{\varepsilon}({\mathfrak{g}})\) is endowed with its natural structure of a superalgebra, compatible with its \({\mathbb{Z}}\times \Gamma\)-grading. In what follows, we consider \(\bigwedge_{\varepsilon}({\mathfrak{g}})\) as a \({\mathbb{Z}}\times {\mathbb{Z}}\)-graded color algebra. Moreover, we write \(\bigwedge_{\varepsilon}({\mathfrak{g}}_{\pm})\) for the \(\varepsilon\)-exterior algebra of the \({\mathbb{Z}}\)-graded color subalgebras \({\mathfrak{g}}_{\pm}\), and \(\bigwedge_{\varepsilon}({\mathfrak{g}}_{\pm})_{(i,\gamma)}\) for their homogeneous components of bidegree \((i,\gamma)\). Both are \({\mathbb{Z}}\times {\mathbb{Z}}\)-graded color subalgebras of \(\bigwedge_{\varepsilon}({\mathfrak{g}})\) such that \(\bigwedge_{\varepsilon}({\mathfrak{g}})\cong \bigwedge_{\varepsilon}({\mathfrak{g}}_{-})\otimes \bigwedge_{\varepsilon}({\mathfrak{g}}_{+})\) as \({\mathbb{Z}}\times {\mathbb{Z}}\)-graded objects in \({\boldsymbol{Vec}_{\Gamma}^{\varepsilon}}\).

On \(\bigwedge_{\varepsilon}({\mathfrak{g}})\), there are three natural operations. For \(v\in{\mathfrak{g}}\), let \(\ell(v)\) denote left exterior multiplication on \(\bigwedge_{\varepsilon}({\mathfrak{g}})\). Moreover, the operation \[\label{eq::contraction} \iota_{v}(x_{1}\otimes\cdots\otimes x_{l})\mathrel{\vcenter{:}}=\sum_{k=1}^{l}(-1)^{k-1}\varepsilon(v,x_{1}+\cdots+x_{k-1})B(v,x_{k})x_{1}\otimes\cdots\otimes\widehat{x}_{k}\otimes\cdots\otimes x_{l}\tag{9}\] on \(\mathrm{T}({\mathfrak{g}})\) leaves \(I_{\wedge}\) invariant (cf. [13]) and therefore descends to an operator on \(\bigwedge_{\varepsilon}({\mathfrak{g}})\), called contraction. It satisfies the Leibniz rule \[\label{eq::Leibniz95rule95eins}\iota_{v}(\omega\wedge\eta)=\iota_{v}(\omega)\wedge\eta+(-1)^{\vert \omega\vert_{\wedge}}\varepsilon(v,\omega)\omega\wedge\iota_{v}(\eta)\tag{10}\] for homogeneous \(\omega,\eta\in\bigwedge_{\varepsilon}({\mathfrak{g}})\). Finally, for homogeneous \(T\in\operatorname{End}({\mathfrak{g}})\), define \[\label{eq::Lie95derivative} L_{T}(x_{1}\otimes\cdots\otimes x_{l})\mathrel{\vcenter{:}}=\sum_{k=1}^{l}\varepsilon(T,x_{1}+\cdots+x_{k-1})x_{1}\otimes\cdots\otimes T(x_{k})\otimes\cdots\otimes x_{l}\tag{11}\] for \(x_{1},\dots,x_{l}\in{\mathfrak{g}}\). This operator preserves the ideal \(I_{\wedge}\) and hence descends to an operator on \(\bigwedge_{\varepsilon}({\mathfrak{g}})\), called the Lie derivative. It is an \(\varepsilon\)-derivation \(L_{T}\in {\mathrm{Der}}_{(\Gamma,\varepsilon)}(\bigwedge_{\varepsilon}({\mathfrak{g}}))\), that is, \[\label{eq::Leibniz95rule95zwei} L_{T}(\omega\wedge\eta)=L_{T}(\omega)\wedge\eta+\varepsilon(T,\omega)\omega\wedge L_{T}(\eta)\tag{12}\] for homogeneous \(\omega,\eta\in\bigwedge_{\varepsilon}({\mathfrak{g}})\).

To describe the relation between the Lie derivative and the contraction, let \(T\in\operatorname{End}({\mathfrak{g}})\) be homogeneous. Since \(B\) is nondegenerate, we can define its \(B\)-adjoint \(T^{\ast}\) by \[B(T^{\ast}x,y)=\varepsilon(T,x)B(x,Ty)\] for homogeneous \(x,y\in{\mathfrak{g}}\). For example, by invariance of \(B\), the adjoint of \(\operatorname{ad}_{x}\) is \(-\operatorname{ad}_{x}\) for every homogeneous \(x\in{\mathfrak{g}}\), whereas \(T \in \mathfrak{so}_{\varepsilon}({\mathfrak{g}})\), then \(T^{\ast}=-T\).

In addition, \(\bigwedge_{\varepsilon}({\mathfrak{g}})\) carries a bilinear form induced by \(B\). For \(k\geq 0\), define on \(\mathrm{T}^{k}({\mathfrak{g}})\) \[\langle x_{1}\otimes\cdots\otimes x_{k},y_{1}\otimes\cdots\otimes y_{k}\rangle_{\mathrm{T}^{k}}\mathrel{\vcenter{:}}=\prod_{i=0}^{k-1}B(x_{k-i},y_{1+i}),\] and then set \[\langle x_{1}\otimes\cdots\otimes x_{k},y_{1}\otimes\cdots\otimes y_{k}\rangle_{\wedge^{k}}\mathrel{\vcenter{:}}=\sum_{\sigma\in S_{k}}p(\sigma;x_{1},\ldots,x_{k})\langle x_{\sigma(1)}\otimes\cdots\otimes x_{\sigma(k)},y_{1}\otimes\cdots\otimes y_{k}\rangle_{\mathrm{T}^{k}},\] where \(p(\sigma;x_{1},\ldots,x_{k})\) is the signed color permutation factor from Section 2.1. This form descends to \(\bigwedge^{k}_{\varepsilon}({\mathfrak{g}})\), and hence defines a bilinear form \(\langle\cdot,\cdot\rangle_{\wedge}\) on \(\bigwedge_{\varepsilon}({\mathfrak{g}})\). As in [19], one proves the following lemma.

Recall that \({\mathfrak{g}}^{\ast}\) denotes the restricted dual of \({\mathfrak{g}}\). The natural pairing between \(\bigwedge_{\varepsilon}({\mathfrak{g}})\) and \(\bigwedge_{\varepsilon}({\mathfrak{g}}^{\ast})\) identifies \(\bigwedge_{\varepsilon}({\mathfrak{g}})_{(i,\gamma)}\) with a subspace of the linear maps \(\bigwedge_{\varepsilon}({\mathfrak{g}}^{\ast})_{(-i,-\gamma)} \to {\mathbb{C}}\). The completion of \(\bigwedge_{\varepsilon}({\mathfrak{g}})\) is defined as the space of all such linear maps. Equivalently, using \(\bigwedge_{\varepsilon}({\mathfrak{g}}) \cong \bigwedge_{\varepsilon}({\mathfrak{g}}_{-})\otimes \bigwedge_{\varepsilon}({\mathfrak{g}}_{+})\) and the identification \({\mathfrak{g}}\cong {\mathfrak{g}}^{\ast}\), one has the decomposition \(\widehat{\bigwedge}_{\varepsilon}({\mathfrak{g}}) = \bigoplus_{(i,\gamma)\in {\mathbb{Z}}\times \Gamma} \widehat{\bigwedge}_{\varepsilon}({\mathfrak{g}})_{(i,\gamma)},\) where \[\widehat{\bigwedge}_{\varepsilon}({\mathfrak{g}})_{(i,\gamma)} \mathrel{\vcenter{:}}= \prod_{r\geq 0}\;\bigoplus_{\substack{\alpha,\beta\in\Gamma\\ \alpha+\beta=\gamma}} \bigwedge\nolimits_{\varepsilon}({\mathfrak{g}}_{-})_{(i-r,\alpha)}\otimes \bigwedge\nolimits_{\varepsilon}({\mathfrak{g}}_{+})_{(r,\beta)}.\] The exterior multiplication extends componentwise to \(\widehat{\bigwedge}_{\varepsilon}({\mathfrak{g}})\). Thus \(\widehat{\bigwedge}_{\varepsilon}({\mathfrak{g}})\) is a \({\mathbb{Z}}\)-graded color algebra. The contraction, the Lie derivative, and \(\langle\cdot, \cdot\rangle_{\wedge}\) extend to the completion, denoted by the same symbol. Likewise, for each \(k\geq 0\) one defines the completed exterior powers \(\widehat{\bigwedge}_{\varepsilon}^{k}({\mathfrak{g}})\) such that \(\widehat{\bigwedge}_{\varepsilon}({\mathfrak{g}})=\bigoplus_{k\geq 0}\widehat{\bigwedge}_{\varepsilon}^{k}({\mathfrak{g}})\). Thus \(\widehat{\bigwedge}_{\varepsilon}({\mathfrak{g}})\) carries, in addition, a compatible \({\mathbb{Z}}\)-grading induced by tensor length.

2.2.2 \(\varepsilon\)-Exterior Algebra and \(\mathfrak{so}_{\varepsilon}({\mathfrak{g}})\)↩︎

We consider the \(\varepsilon\)-orthogonal algebra \(\mathfrak{so}_{\varepsilon}({\mathfrak{g}})\) of \(({\mathfrak{g}},B)\) introduced in Example 1. It is a \({\mathbb{Z}}\)-graded quadratic color Lie subalgebra of \(\operatorname{End}({\mathfrak{g}})\), where the \({\mathbb{Z}}\)-grading is induced by that of \({\mathfrak{g}}\).

We define the completion \(\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\) of \(\mathfrak{so}_{\varepsilon}({\mathfrak{g}})\) as in 6 .

For finite-dimensional \({\mathfrak{g}}\), one has \(\operatorname{ad}_{x}\in\mathfrak{so}_{\varepsilon}({\mathfrak{g}})\) for every \(x\in{\mathfrak{g}}\). Relative to the map \(\operatorname{ad}\colon{\mathfrak{g}}\to\mathfrak{so}_{\varepsilon}({\mathfrak{g}}),\) the moment map \(\require{upgreek} \upmu\colon{\mathfrak{g}}\times{\mathfrak{g}}\to\mathfrak{so}_{\varepsilon}({\mathfrak{g}})\) is defined implicitly by \[\require{upgreek} \label{eq::implicit95def95moment95map} -\tfrac{1}{2}\operatorname{tr}_{\varepsilon}(\operatorname{ad}_{x}\circ\upmu(y,z))=B([x,y],z),\qquad x,y,z\in{\mathfrak{g}},\tag{13}\] which is well-defined since \(\mathfrak{so}_{\varepsilon}({\mathfrak{g}})\) is quadratic with respect to the bilinear form \((S,T)\mapsto-\tfrac{1}{2}\operatorname{tr}_{\varepsilon}(ST).\) The map \(\require{upgreek} \upmu\) is \(\varepsilon\)-skew-symmetric, homogeneous of degree \((0,{0_{\scriptscriptstyle \Gamma}})\), and is given by [13] \[\require{upgreek} \label{eq::explicit95def95moment95map} \upmu(x,y)(z)=\varepsilon(y,z)B(x,z)y-B(y,z)x,\qquad x,y,z\in{\mathfrak{g}}.\tag{14}\] In the present setting, \(\operatorname{ad}_{x}\) need only lie in \(\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\), so 13 is no longer available. We therefore take 14 as the definition of the moment map.

Under the identification \({\mathfrak{g}}\cong {\mathfrak{g}}^{\ast}\) via \(B\), we can describe \(\require{upgreek} \uplambda\) as follows. For any \(T \in \mathfrak{so}_{\varepsilon}({\mathfrak{g}})\), define \(\omega_{T}(x,y) \mathrel{\vcenter{:}}= B(Tx,y)\) for all \(x,y \in {\mathfrak{g}}\). A direct calculation shows that this form is skew-\(\varepsilon\)-symmetric.

Since \(\require{upgreek} \upmu\) is linear and preserves the \({\mathbb{Z}}\times\Gamma\)-grading, it extends componentwise to the completions. The inverse \(\require{upgreek} \uplambda\) extends correspondingly, and both are \({\mathbb{Z}}\times \Gamma\)-graded isomorphisms in \({\boldsymbol{Vec}_{\Gamma}^{\varepsilon}}\). We denote the extensions by the same symbols.

2.2.3 \(\varepsilon\)-Clifford Algebra and Completion↩︎

We now introduce the \(\varepsilon\)-Clifford algebra associated with the quadratic color Lie algebra \(({\mathfrak{g}},B)\). Let \(I_{{\operatorname{Cl}}}({\mathfrak{g}})\) be the two-sided (\({\mathbb{Z}}\times \Gamma\)-graded) homogeneous ideal of \(\mathrm{T}({\mathfrak{g}})\) generated by all elements \[x\otimes y+\varepsilon(x,y)y\otimes x-2B(x,y)1_{\mathrm{T}({\mathfrak{g}})},\qquad x,y\in{\mathfrak{g}}\;\text{homogeneous}.\] The quotient \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\mathrel{\vcenter{:}}= \mathrm{T}({\mathfrak{g}})/I_{{\operatorname{Cl}}}({\mathfrak{g}})\) is a \({\mathbb{Z}}\)-graded associative color algebra, called the \(\varepsilon\)-Clifford algebra. Identifying \({\mathfrak{g}}\) with its image in \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\), it is generated by \({\mathfrak{g}}\) subject to the relations \[\label{eq::Clifford95relation} xy+\varepsilon(x,y)yx=2B(x,y)1_{{\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})}\tag{15}\] for all \(x,y\in{\mathfrak{g}}\) homogeneous. The operators 9 and 11 preserve \(I_{{\operatorname{Cl}}}({\mathfrak{g}})\). Hence, for every \(x\in{\mathfrak{g}}\) and \(T\in\operatorname{End}({\mathfrak{g}})\), they descend to operators on \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\), again denoted by \(\iota_{x}\) and \(L_{T}\). They satisfy the Leibniz rules 10 and 12 with respect to the Clifford product.

The restriction of \(B\) to \({\mathfrak{g}}_{\pm}\) yields a natural construction of Clifford algebras \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}}_{\pm})\), where \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}}_{+}) = \bigwedge_{\varepsilon}({\mathfrak{g}}_{+})\) as \(B\) restricts to zero on \({\mathfrak{g}}_{+}\). These are \({\mathbb{Z}}\)-graded color subalgebras of \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\) and the Clifford multiplication defines an isomorphism of \({\mathbb{Z}}\times\Gamma\)-graded vector spaces \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}}) \cong {\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}}_{-}) \otimes {\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}}_{+})\). As for the \({\mathbb{Z}}\)-graded \(\varepsilon\)-exterior algebra, we define the completion \(\widehat{{\operatorname{Cl}}}_{\varepsilon}({\mathfrak{g}})\) of \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\) as the direct sum of all \[\widehat{{\operatorname{Cl}}}_{\varepsilon}({\mathfrak{g}})_{(i,\gamma)} \mathrel{\vcenter{:}}= \prod_{r\geq 0}\;\bigoplus_{\substack{\alpha,\beta\in\Gamma\\ \alpha+\beta=\gamma}} {\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}}_{-})_{(i-r,\alpha)} \otimes {\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}}_{+})_{(r,\beta)}.\]

2.2.4 \(\widehat{\bigwedge}_{\varepsilon}({\mathfrak{g}}), \widehat{{\operatorname{Cl}}}_{\varepsilon}({\mathfrak{g}})\) and Normal-Ordered Quantization↩︎

We now relate \(\widehat{\bigwedge}_{\varepsilon}({\mathfrak{g}})\) and \(\widehat{{\operatorname{Cl}}}_{\varepsilon}({\mathfrak{g}})\). We define \(\gamma : {\mathfrak{g}}\to \operatorname{End}(\bigwedge_{\varepsilon}({\mathfrak{g}}))\) by \(\gamma(x) \mathrel{\vcenter{:}}= \ell(x) + \iota_{x}\) where \(\ell(x)\) denotes exterior multiplication by \(x\) and \(\iota_{x}\) denotes contraction. It satisfies \(\gamma(x)\gamma(y) + \varepsilon(x,y)\gamma(y)\gamma(x) = 2B(x,y)\) for any homogeneous \(x,y \in {\mathfrak{g}}\). Thus, by the universal property of the \(\varepsilon\)-Clifford algebra (see [19]), this realizes \(\bigwedge_{\varepsilon}({\mathfrak{g}})\) as a \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\)-module, i.e., \(\gamma : {\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}}) \to \operatorname{End}(\bigwedge_{\varepsilon}({\mathfrak{g}}))\).

As in [1], the quantization map intertwines contraction.

The quantization map \(q : \bigwedge_{\varepsilon}({\mathfrak{g}}) \to {\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\) does not extend to completions, which leads to the normal-ordered quantization map.

The Lie derivative \(L_{T}\) for any \(T \in \widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\) on \(\widehat{{\operatorname{Cl}}}_{\varepsilon}({\mathfrak{g}})\) is naturally related to the quantization map. For any \(T \in \widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\), define \[\require{upgreek} \widehat{\gamma}(T) \mathrel{\vcenter{:}}= \widehat{q}\left(\uplambda(T)\right).\] Furthermore, let \([\cdot,\cdot]_{\widehat{{\operatorname{Cl}}}_{\varepsilon}}\) denote the natural Lie \(\varepsilon\)-bracket on \(\widehat{{\operatorname{Cl}}}_{\varepsilon}({\mathfrak{g}})\), that is, \([v,w]_{\widehat{{\operatorname{Cl}}}_{\varepsilon}}=vw-\varepsilon(v,w)wv\) for any homogeneous \(v,w \in \widehat{{\operatorname{Cl}}}_{\varepsilon}({\mathfrak{g}})\).

2.3 Generalized Kac–Peterson Class↩︎

We construct an \(\varepsilon\)-symmetric analogue of the Kac–Peterson cocycle for the \({\mathbb{Z}}\)-graded quadratic color Lie algebra \(({\mathfrak{g}},B)\). In the symmetrizable Kac–Moody setting, the Kac–Peterson cocycle was introduced in [20] in order to formulate the generalized Casimir operator and the corresponding \(\rho\)-correction.

In Section 2.1 we introduced the \(\varepsilon\)-trace \({\operatorname{tr}_{\varepsilon}}\) on \(\operatorname{End}(V)\) for finite-dimensional \(V\in{\boldsymbol{Vec}_{\Gamma}^{\varepsilon}}\). For homogeneous \(S,T,U\in\operatorname{End}(V)\), it satisfies \({\operatorname{tr}_{\varepsilon}}(ST)=\varepsilon(S,T){\operatorname{tr}_{\varepsilon}}(TS)\), \({\operatorname{tr}_{\varepsilon}}([S,T]_{\operatorname{End}}U)={\operatorname{tr}_{\varepsilon}}(S[T,U]_{\operatorname{End}})\), and \({\operatorname{tr}_{\varepsilon}}(ST)=0\) unless \(\vert S\vert +\vert T\vert = {0_{\scriptscriptstyle \Gamma}}\). If \((V,\langle\cdot,\cdot\rangle)\) is quadratic and \((e_{a})\) is a homogeneous basis of \(V\) with \(\langle\cdot,\cdot\rangle\)-dual basis \((e^{a})\), then for every \(T\in\operatorname{End}(V)\), \[\label{eq::epsilon95trace} {\operatorname{tr}_{\varepsilon}}(T)=\sum_{a}\varepsilon(e_{a})\langle T(e_{a}),e^{a}\rangle=\sum_{a}\langle e^{a},T(e_{a})\rangle.\tag{17}\]

In general, for \(x\in{\mathfrak{g}}\), one has \(\operatorname{ad}_{x}\in\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\subset\widehat{\operatorname{End}}({\mathfrak{g}})\) by Lemma 5. The operator \(\operatorname{ad}_x\) need not be finitary. However, \(\pi_{-}\operatorname{ad}_{x}\pi_{+}\) has finite rank. This makes the following definition well-defined.

By the properties of the \(\varepsilon\)-trace \({\operatorname{tr}_{\varepsilon}}\), the Kac–Peterson cocycle has the following properties.

In particular, \({\psi_{\operatorname{KP}}}\) defines a class in the color Lie algebra cohomology \(\operatorname{H}^2({\mathfrak{g}};{\mathbb{C}})\), called the Kac–Peterson class. For the definition of \(\operatorname{H}^{\bullet}({\mathfrak{g}};{\mathbb{C}})\), we refer to [22]. If \({\mathfrak{g}}\) is the loop superalgebra of a basic classical Lie superalgebra with non-vanishing super Killing form, then the class of \({\psi_{\operatorname{KP}}}\) spans \(\operatorname{H}^2({\mathfrak{g}};{\mathbb{C}})\) [23]. In this case, \({\psi_{\operatorname{KP}}}\) defines a central extension of \({\mathfrak{g}}\), as follows immediately from Lemma 9.

The Kac–Peterson class vanishes exactly if there exists some \(\rho \in ({\mathfrak{g}}^{\ast})_{(0,{0_{\scriptscriptstyle \Gamma}})}\) such that \({\psi_{\operatorname{KP}}}= \operatorname{d}\!\rho\), where \[\operatorname{d}\!\rho(x,y) \mathrel{\vcenter{:}}= \rho([x,y]) = B(\rho^{\sharp},[x,y]), \qquad x,y \in {\mathfrak{g}}.\] In Section 2.2.2 we identified \(\widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak{g}})\) and \(\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\) as \({\mathbb{Z}}\times \Gamma\)-graded vector spaces. As \({\psi_{\operatorname{KP}}}\in \widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak{g}}^{\ast})\), the Kac–Peterson \(2\)-cocycle corresponds to an element \(\Psi_{\operatorname{KP}} \in \widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\) via the relation \[\label{eq::relations95psi95Psi} {\psi_{\operatorname{KP}}}(x,y) = B(\Psi_{\operatorname{KP}}(x),y), \qquad x,y\in {\mathfrak{g}}.\tag{18}\]

The musical isomorphism \(\sharp\) in 7 allows us to identify \(\widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak{g}}^{\ast})\) with \(\widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak{g}})\). We denote the image of \({\psi_{\operatorname{KP}}}\) under \(\sharp\) by \({\psi^{\sharp}_{\operatorname{KP}}}\), that is, \({\psi^{\sharp}_{\operatorname{KP}}}\) is the unique element in \(\widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak{g}})\) defined by the property \[\label{eq::definition95psi95sharp} {\psi_{\operatorname{KP}}}(x,y)=B(\iota_{x}{\psi^{\sharp}_{\operatorname{KP}}},y), \qquad x,y \in {\mathfrak{g}}.\tag{19}\]

We denote again by \({\psi_{\operatorname{KP}}}\) the bilinear form on \(\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\) defined by \[{\psi_{\operatorname{KP}}}(S,T)\mathrel{\vcenter{:}}=\operatorname{tr}_{\varepsilon}(S\pi_{-}T\pi_{+})-\varepsilon(S,T)\operatorname{tr}_{\varepsilon}(T\pi_{-}S\pi_{+})\] for homogeneous \(S,T\in\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\). For homogeneous \(x,y\in{\mathfrak{g}}\), one has \({\psi_{\operatorname{KP}}}(\operatorname{ad}_{x},\operatorname{ad}_{y})={\psi_{\operatorname{KP}}}(x,y),\) so this extends the Kac–Peterson cocycle of \({\mathfrak{g}}\). Moreover, \({\psi_{\operatorname{KP}}}\) measures the failure of \(\widehat q\) to be \(\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\)-equivariant.

2.4 Completion of \(\varepsilon\)-Symmetric and Universal Enveloping Algebras of \({\mathfrak{g}}\)↩︎

In this section, we introduce the completions of the \(\varepsilon\)-symmetric algebra and the universal enveloping algebra of \({\mathfrak{g}}\) and relate both by the PBW \(\varepsilon\)-symmetrization.

2.4.1 \(\varepsilon\)-Symmetric Algebras and Completion↩︎

Let \(V \in \mathbf{Vec}_{\Gamma}^{\varepsilon}\). Assume in addition that \(V\) is \({\mathbb{Z}}\)-graded, that is, \(V=\bigoplus_{n\in{\mathbb{Z}}}V^n\) with \(V^n\in \mathbf{Vec}_{\Gamma}^{\varepsilon}\). Let \(I_{S}\) denote the \(({\mathbb{Z}}\times \Gamma)\)-graded homogeneous ideal of \(\mathrm{T}(V)\) generated by \[v\otimes w-\varepsilon(v,w)w\otimes v,\qquad v,w\in V\;\text{homogeneous}.\] The \(\varepsilon\)-symmetric algebra of \(V\) is the quotient \(S_{\varepsilon}(V)\mathrel{\vcenter{:}}= \mathrm{T}(V)/I_{S}.\) Equivalently, it is generated by \(1_{S_{\varepsilon}(V)}\) and \(V\), subject to \(vw-\varepsilon(v,w)wv=0\) for homogeneous \(v,w\in V.\) Since \(I_{S}\) is \({\mathbb{Z}}\times\Gamma\)-homogeneous, \(S_{\varepsilon}(V)\) inherits the \({\mathbb{Z}}\times\Gamma\)-grading of \(\mathrm{T}(V)\) induced by \(V\), and is a \({\mathbb{Z}}\)-graded color algebra. Moreover, \(S_{\varepsilon}(V)\) carries the canonical \(\varepsilon\)-symmetric power grading \(S_{\varepsilon}(V)=\bigoplus_{k\geq 0}S^{k}_{\varepsilon}(V)\), induced by the tensor length grading \(\mathrm{T}(V)=\bigoplus_{k\geq 0}\mathrm{T}^{k}(V)\) of the tensor algebra. It is compatible with the \({\mathbb{Z}}\times \Gamma\)-grading induced by \(V\). In what follows, we view \(S_{\varepsilon}(V)\) as a \({\mathbb{Z}}\times {\mathbb{Z}}\)-graded color algebra.

We are interested in the case when \(V\) is the \({\mathbb{Z}}\)-graded color Lie algebra \({\mathfrak{g}}\). Recall that \({\mathfrak{g}}^{\ast}\) denotes the restricted dual and \({\mathfrak{g}}= {\mathfrak{g}}_{+}\oplus {\mathfrak{g}}_{-}\). The natural pairing between \(S_{\varepsilon}({\mathfrak{g}})\) and \(S_{\varepsilon}({\mathfrak{g}}^{\ast})\) identifies \(S_{\varepsilon}({\mathfrak{g}})_{(i, \gamma)}\) with a subspace of the linear maps \(S_{\varepsilon}({\mathfrak{g}}^{\ast})_{(-i, -\gamma)}\;\longrightarrow\;{\mathbb{C}}.\) We define \(\widehat{S}_{\varepsilon}({\mathfrak{g}})_{(i, \gamma)}\mathrel{\vcenter{:}}= \operatorname{Hom}_{{\mathbb{C}}}\!\bigl(S_{\varepsilon}({\mathfrak{g}}^{\ast})_{(-i,-\gamma)},{\mathbb{C}}\bigr).\) Equivalently, using \(S_{\varepsilon}({\mathfrak{g}})\cong S_{\varepsilon}({\mathfrak{g}}_-)\otimes S_{\varepsilon}({\mathfrak{g}}_+)\), one has \[\widehat{S}_{\varepsilon}({\mathfrak{g}})_{(i,\gamma)} \cong \prod_{r\ge 0}\; \bigoplus_{\substack{\alpha,\beta\in\Gamma\\ \alpha+\beta=\gamma}} S_{\varepsilon}({\mathfrak{g}}_{-})_{(i-r,\alpha)} \otimes S_{\varepsilon}({\mathfrak{g}}_{+})_{(r,\beta)}.\] The completion \(\widehat{S}_{\varepsilon}({\mathfrak{g}})\) of \(S_{\varepsilon}({\mathfrak{g}})\) is the direct sum of all \(\widehat{S}_{\varepsilon}({\mathfrak{g}})_{(i, \gamma)}\). The multiplication on \(S_{\varepsilon}({\mathfrak{g}})\) extends to \(\widehat{S}_{\varepsilon}({\mathfrak{g}})\), which is thus a \({\mathbb{Z}}\)-graded color algebra.

For each \(k\geq 0\), the homogeneous component \(S^{k}_{\varepsilon}({\mathfrak{g}})\) admits an analogous completion \(\widehat{S}_{\varepsilon}^{k}({\mathfrak{g}})\subseteq\widehat{S}_{\varepsilon}({\mathfrak{g}})\). Thus \(\widehat{S}_{\varepsilon}({\mathfrak{g}})\) carries, in addition, a compatible \({\mathbb{Z}}\)-grading induced by tensor length.

2.4.2 \({\mathfrak U}_{\varepsilon}({\mathfrak{g}})\) and Completion↩︎

Let \(I_{U}\) be the two-sided homogeneous ideal of \(\mathrm{T}({\mathfrak{g}})\) generated by \[x\otimes y-\varepsilon(x,y)y\otimes x-[x,y], \qquad x,y\in {\mathfrak{g}}\;\text{homogeneous}.\] The universal enveloping algebra is the quotient \({\mathfrak U}_{\varepsilon}({\mathfrak{g}}) \mathrel{\vcenter{:}}= \mathrm{T}({\mathfrak{g}})/I_{U}.\) Equivalently, \({\mathfrak U}_{\varepsilon}({\mathfrak{g}})\) is the associative color algebra generated by \({\mathfrak{g}}\) subject to the relations \(xy-\varepsilon(x,y)yx=[x,y]\) for all homogeneous \(x,y\in {\mathfrak{g}}\).

In [8], Scheunert proves a Poincaré–Birkhoff–Witt theorem for \({\mathfrak U}_{\varepsilon}({\mathfrak{g}})\). The tensor-degree filtration on \(\mathrm{T}({\mathfrak{g}})\) defined by \(F^{k}\mathrm{T}({\mathfrak{g}})\mathrel{\vcenter{:}}=\bigoplus_{j\leq k}\mathrm{T}^{j}({\mathfrak{g}})\), induces a filtration on \({\mathfrak U}_{\varepsilon}({\mathfrak{g}})\). Let \(\operatorname{gr}{\mathfrak U}_{\varepsilon}({\mathfrak{g}})\) denote the associated graded color algebra. Then there is a canonical isomorphism of graded color algebras [8] \[\operatorname{gr}{\mathfrak U}_{\varepsilon}({\mathfrak{g}})\cong S_{\varepsilon}({\mathfrak{g}}).\] Since \(I_U\) is \({\mathbb{Z}}\times\Gamma\)-graded, this is in fact an isomorphism of \({\mathbb{Z}}\)-graded color algebras. Here the \({\mathbb{Z}}\)-grading refers to the one induced by \({\mathfrak{g}}\).

Let \({\mathfrak U}_{\varepsilon}({\mathfrak{g}}_{\pm})\) denote the universal enveloping algebra of the \({\mathbb{Z}}\)-graded color subalgebras \({\mathfrak{g}}_{\pm}\) of \({\mathfrak{g}}\). By the Poincaré–Birkhoff–Witt theorem, the multiplication map induces a \({\mathbb{Z}}\times \Gamma\)-graded vector space isomorphism \({\mathfrak U}_{\varepsilon}({\mathfrak{g}})\;\cong\;{\mathfrak U}_{\varepsilon}({\mathfrak{g}}_{-})\otimes {\mathfrak U}_{\varepsilon}({\mathfrak{g}}_{+}).\) We define a completion \(\widehat{{\mathfrak U}}_{\varepsilon}({\mathfrak{g}})\) by declaring its \((i,\gamma)\)-graded piece to be \[\widehat{{\mathfrak U}}_{\varepsilon}({\mathfrak{g}})_{(i,\gamma)} \mathrel{\vcenter{:}}= \prod_{r\ge 0}\; \bigoplus_{\substack{\alpha,\beta\in\Gamma\\ \alpha+\beta=\gamma}} {\mathfrak U}_{\varepsilon}({\mathfrak{g}}_{-})_{(i-r,\alpha)} \otimes {\mathfrak U}_{\varepsilon}({\mathfrak{g}}_{+})_{(r,\beta)},\] and setting \(\widehat{{\mathfrak U}}_{\varepsilon}({\mathfrak{g}})\mathrel{\vcenter{:}}= \bigoplus_{(i, \gamma)\in{\mathbb{Z}}\times\Gamma}\widehat{{\mathfrak U}}_{\varepsilon}({\mathfrak{g}})_{(i, \gamma)}.\) The multiplication on \({\mathfrak U}_{\varepsilon}({\mathfrak{g}})\) extends to \(\widehat{{\mathfrak U}}_{\varepsilon}({\mathfrak{g}})\), making \(\widehat{{\mathfrak U}}_{\varepsilon}({\mathfrak{g}})\) a \({\mathbb{Z}}\)-graded color algebra.

2.4.3 PBW \(\varepsilon\)-Symmetrization↩︎

The natural inclusion \({\mathfrak{g}}\hookrightarrow {\mathfrak U}_{\varepsilon}({\mathfrak{g}})\) is a homomorphism in \(\boldsymbol{Vec}^{\varepsilon}_{\Gamma}\) of \({\mathbb{Z}}\)-graded vector spaces. It extends canonically to a homomorphism of \({\mathbb{Z}}\)-graded vector spaces \(Q: S_{\varepsilon}({\mathfrak{g}}) \to {\mathfrak U}_{\varepsilon}({\mathfrak{g}})\) by PBW \(\varepsilon\)-symmetrization, that is, \[Q(x_{1}\ldots x_{k}) \mathrel{\vcenter{:}}= \tfrac{1}{k!}\sum_{\sigma \in S_{k}} p'(\sigma;x_{1}, \ldots, x_{k}) x_{\sigma(1)}\ldots x_{\sigma(k)},\] where \(p'(\sigma; x_{1},\ldots, x_{k})\) is given by (cf. Section 2.1) \[p'(\sigma; x_{1},\ldots, x_{k}) = \prod_{1 \leq i < j \leq k, \;\sigma^{-1}(i) > \sigma^{-1}(j)} \varepsilon(x_{i},x_{j}).\] This is the color analogue of the PBW symmetrization map, and it maps \(S_{\varepsilon}({\mathfrak{g}}_{\pm})\) to \({\mathfrak U}_{\varepsilon}({\mathfrak{g}}_{\pm})\).

The PBW \(\varepsilon\)-symmetrization does not admit a direct extension to completions. We therefore introduce the following normal-ordered analogue.

From the construction of the normal-ordered PBW \(\varepsilon\)-symmetrization, together with Theorem 8, one obtains the following proposition.

We rewrite Example 5 in an appropriate way. For this purpose, we introduce some notation. Fix a homogeneous basis \((e_{a})\) of \({\mathfrak{g}}\) with \(B\)-dual basis \((e^{a})\), and identify \({\mathfrak{g}}\) and \({\mathfrak{g}}^{\ast}\) using the musical isomorphisms 7 . Recall the identification of \(\operatorname{End}({\mathfrak{g}})\) with \({\mathfrak{g}}\otimes {\mathfrak{g}}\) given in 8 \[\operatorname{End}({\mathfrak{g}}) \cong {\mathfrak{g}}\otimes {\mathfrak{g}}^{\ast} \cong {\mathfrak{g}}\otimes {\mathfrak{g}}, \qquad T \mapsto \sum_{a}T(e_{a})\otimes e_{a}^{\ast} \mapsto \sum_{a}\varepsilon(e_{a})T(e_{a}) \otimes e^{a}.\] In what follows, we write \(\mathcal{I} : \operatorname{End}({\mathfrak{g}}) \to {\mathfrak{g}}\otimes {\mathfrak{g}}\) for this \({\mathbb{Z}}\)-graded isomorphism in \(\boldsymbol{Vec}^{\varepsilon}_{\Gamma}\). Its inverse is on homogeneous \(x,y,z\in {\mathfrak{g}}\) given by \[\mathcal{I}^{-1}(x\otimes y)(z) =\varepsilon(y)B(z,y)x\] and then extended linearly.

We view \(\operatorname{End}({\mathfrak{g}})\) as a \({\mathfrak{g}}\)-module via \(\operatorname{ad}\), and \({\mathfrak{g}}\otimes{\mathfrak{g}}\) as a \({\mathfrak{g}}\)-module via \[L_{X}(x\otimes y)\mathrel{\vcenter{:}}= [X,x]\otimes y+\varepsilon(X,x)x\otimes [X,y],\qquad X,x,y\in{\mathfrak{g}}.\] Then \(\mathcal{I}\) is an isomorphism of \({\mathfrak{g}}\)-modules, with inverse \(\mathcal{I}^{-1}\).

We define the braiding \({\operatorname{br}}: \operatorname{End}({\mathfrak{g}}) \to {\mathfrak{g}}\) to be the linear map given by the identification \(\operatorname{End}({\mathfrak{g}}) \cong {\mathfrak{g}}\otimes {\mathfrak{g}}\) followed by the \(\varepsilon\)-bracket, that is, \[{\operatorname{br}}(T) = \sum_{a}\varepsilon(e_{a})[T(e_{a}),e^{a}].\]

We define a natural linear map \(A : S^{2}_{\varepsilon}({\mathfrak{g}}) \to \operatorname{End}({\mathfrak{g}})\) by \[\label{eq::definition95A}A_{xy}\mathrel{\vcenter{:}}=\mathcal{I}^{-1}(x\otimes y+\varepsilon(x,y)y\otimes x), \qquad A_{xy}(z)=\varepsilon(y)B(z,y)x+\varepsilon(x)\varepsilon(x,y)B(z,x)y\tag{20}\] for homogeneous \(x,y\in {\mathfrak{g}}\). This is well-defined since it is compatible with the defining relation \(xy=\varepsilon(x,y)yx\) in \(S^2_{\varepsilon}({\mathfrak{g}})\). It extends degreewise to a linear map \(A : \widehat{S}_{\varepsilon}^{2}({\mathfrak{g}}) \to \widehat{\operatorname{End}}({\mathfrak{g}})\), denoted by the same symbol.

2.5 Completion of the Weil and Quantum Weil Algebra of \({\mathfrak{g}}\)↩︎

We introduce the Weil algebra and the quantum Weil algebra, describe their completions, and formulate their relation via quantization. Throughout, \({\mathfrak{g}}^{\ast}\) is identified with \({\mathfrak{g}}\) by means of the form \(B\).

2.5.1 Weil algebra of \({\mathfrak{g}}\) and Completion↩︎

The Weil algebra of \({\mathfrak{g}}\) is the \(\Gamma\)-graded tensor product \[W_{\varepsilon}({\mathfrak{g}})\mathrel{\vcenter{:}}= S_{\varepsilon}({\mathfrak{g}})\otimes\bigwedge\nolimits_{\varepsilon}({\mathfrak{g}}).\] It is a \({\mathbb{Z}}\)-graded associative color algebra, with \({\mathbb{Z}}\)-grading induced by that of \({\mathfrak{g}}\). We denote the \((i,\gamma)\)-graded component by \(W_{\varepsilon}({\mathfrak{g}})_{(i,\gamma)}\). It carries in addition the \({\mathbb{Z}}\)-grading \[\label{eq::Z95grading95Weil} W_{\varepsilon}^{k}({\mathfrak{g}})\mathrel{\vcenter{:}}=\bigoplus_{2r+s=k}S_{\varepsilon}^{r}({\mathfrak{g}})\otimes \bigwedge\nolimits_{\varepsilon}^{s}({\mathfrak{g}}).\tag{21}\] Thus the generators \(x\otimes 1\) have degree \(2\), whereas the generators \(1\otimes x\) have degree \(1\). Declaring \(x\otimes 1\) to be even and \(1\otimes x\) to be odd endows \(W_{\varepsilon}({\mathfrak{g}})\) with a compatible \({\mathbb{Z}}_{2}\)-grading, and we consider \(W_{\varepsilon}({\mathfrak{g}})\) with respect to this \({\mathbb{Z}}_{2}\)-grading as a superalgebra, that is, \(W_{\varepsilon}({\mathfrak{g}})=W_{\varepsilon}({\mathfrak{g}})_{\bar{0}}\oplus W_{\varepsilon}({\mathfrak{g}})_{\bar{1}}\) is viewed as a \({\mathbb{Z}}\)-graded color superalgebra. We emphasize here the additional compatible \({\mathbb{Z}}\)- and \({\mathbb{Z}}_{2}\)-gradings, since the former is relevant for the completion and the latter for the constructions below. We denote by \(p(\cdot)\) the parity with respect to this \({\mathbb{Z}}_{2}\)-grading so that \(p(x\otimes 1)=\bar{0}\) and \(p(1\otimes x)=\bar{1}\). This color superalgebra is \(\varepsilon\)-supercommutative, that is, \[AB = (-1)^{p(A)p(B)}\varepsilon(A,B)BA,\] with \(A,B\in W_{\varepsilon}({\mathfrak{g}})\) homogeneous; or equivalently, with \(\varepsilon'(A,B)=(-1)^{p(A)p(B)}\varepsilon(A,B)\), \(\varepsilon'\)-commutative. Note that \(W_{\varepsilon}({\mathfrak{g}})\) is the free graded \(\varepsilon\)-supersymmetric algebra freely generated by \(x\otimes 1\) and \(1\otimes x\) with \(x \in {\mathfrak{g}}\).

On \(W_{\varepsilon}({\mathfrak{g}})\) there are two natural operations, called contraction and Lie derivative. For homogeneous \(x\in{\mathfrak{g}}\), the contraction \(\iota_{x}\) is defined on generators by \[\iota_{x}(1\otimes y)=B(x,y)(1\otimes 1),\qquad \iota_{x}(y\otimes 1)=0.\] Further, \(x\) acts on \(S_{\varepsilon}({\mathfrak{g}})\) and \(\bigwedge_{\varepsilon}({\mathfrak{g}})\) by the Lie derivatives \(L_{x}^{S}\) and \(L_{x}^{\wedge}\), where the superscripts serve here only to indicate the ambient algebra. The Lie derivative on \(W_{\varepsilon}({\mathfrak{g}})\) is then defined by \[L_{x}(y\otimes z)\mathrel{\vcenter{:}}= L_{x}^{S}(y)\otimes z+\varepsilon(x,y)\,y\otimes L_{x}^{\wedge}(z)\] for any homogeneous \(x\in {\mathfrak{g}}\) and \(y\otimes z\in W_{\varepsilon}({\mathfrak{g}})\). The Lie derivative and contraction are even and odd \(\varepsilon'\)-derivations, respectively, and both are compatible:

Moreover, these operations are compatible with a natural differential on \(W_{\varepsilon}({\mathfrak{g}})\), which yields a universal property described in Section 3.3.

Similarly, we define the Weil algebras of the \({\mathbb{Z}}\)-graded color subalgebras \({\mathfrak{g}}_{\pm}\). Using \(W_{\varepsilon}({\mathfrak{g}})\cong W_{\varepsilon}({\mathfrak{g}}_{-})\otimes W_{\varepsilon}({\mathfrak{g}}_{+})\), we define the completion \(\widehat{W}_{\varepsilon}({\mathfrak{g}})\) as the sum over all \((i,\gamma)\in {\mathbb{Z}}\times\Gamma\) of \[\widehat{W}_{\varepsilon}({\mathfrak{g}})_{(i,\gamma)} \mathrel{\vcenter{:}}= \prod_{r\ge 0}\; \bigoplus_{\substack{\alpha,\beta\in\Gamma\\ \alpha+\beta=\gamma}} W_{\varepsilon}({\mathfrak{g}}_-)_{(i-r,\alpha)} \otimes W_{\varepsilon}({\mathfrak{g}}_+)_{(r,\beta)}.\] The operators \(\iota_x\) and \(L_x\) extend degreewise to \(\widehat{W}_{\varepsilon}({\mathfrak{g}})\), and we denote these extensions by the same symbols. Moreover, since multiplication extends degreewise, we view \(\widehat{W}_{\varepsilon}({\mathfrak{g}})\) as a \({\mathbb{Z}}\)-graded color superalgebra.

2.5.2 Quantum Weil Algebra of \({\mathfrak{g}}\) and Completion↩︎

The quantum Weil algebra, or noncommutative Weil algebra, is defined to be the quantized version of the Weil algebra, which replaces the \(\varepsilon\)-commutative color algebras \(S_{\varepsilon}({\mathfrak{g}})\) and \(\bigwedge_{\varepsilon}({\mathfrak{g}})\) by their noncommutative analogues \({\mathfrak U}_{\varepsilon}({\mathfrak{g}})\) and \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\), respectively. It was introduced by Alekseev and Meinrenken [15] for Lie algebras.

\({\mathcal{W}_{\varepsilon}({\mathfrak{g}})}\) is a \({\mathbb{Z}}\)-graded color algebra with \({\mathbb{Z}}\)-grading induced by that of \({\mathfrak{g}}\). We denote the \((i,\gamma)\)-graded component by \({\mathcal{W}_{\varepsilon}({\mathfrak{g}})}_{(i,\gamma)}\). We denote the generators of \({\mathcal{W}_{\varepsilon}({\mathfrak{g}})}\) by the elements \(x\otimes 1\) and \(1 \otimes x\) for \(x \in {\mathfrak{g}}\). By declaring that the generators \(1\otimes x\) are odd and the generators \(x\otimes 1\) are even, we endow \({\mathcal{W}_{\varepsilon}({\mathfrak{g}})}\) with an additional \({\mathbb{Z}}_{2}\)-grading so that \({\mathcal{W}_{\varepsilon}({\mathfrak{g}})}= {\mathcal{W}_{\varepsilon}({\mathfrak{g}})}_{\bar{0}} \oplus {\mathcal{W}_{\varepsilon}({\mathfrak{g}})}_{\bar{1}}\) is a \({\mathbb{Z}}\)-graded color superalgebra. We denote the associated parity by \(p(\cdot)\). This color superalgebra is non-\(\varepsilon\)-supercommutative, and becomes a color Lie superalgebra with bracket \[\label{eq::bracket95quantum95Weil} [A,B]_{{\mathcal{W}}}\mathrel{\vcenter{:}}= AB-\varepsilon'(A,B)\,BA,\qquad \varepsilon'(A,B)\mathrel{\vcenter{:}}=(-1)^{p(A)p(B)}\varepsilon(A,B)\tag{22}\] for any homogeneous \(A,B \in {\mathcal{W}_{\varepsilon}({\mathfrak{g}})}\).

On \({\mathcal{W}_{\varepsilon}({\mathfrak{g}})}\) there are two canonical operations. The contraction \(\iota_{x}\) is defined on generators by \[\iota_{x}(1\otimes y)=B(x,y)(1\otimes 1),\qquad \iota_{x}(y\otimes 1)=0,\] for \(x,y\in{\mathfrak{g}}\). The second is the Lie derivative. It is the \(\Gamma\)-grading-preserving combination of the Lie derivatives \(L^{U}\) on \({\mathfrak U}_{\varepsilon}({\mathfrak{g}})\) and \(L^{C}\) on \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\), and is given by \[L_{x}(y\otimes z)\mathrel{\vcenter{:}}= L_{x}^{U}(y)\otimes z+\varepsilon(x,y)y\otimes L_{x}^{C}(z)\] for homogeneous \(x\in{\mathfrak{g}}\), \(y\in{\mathfrak U}_{\varepsilon}({\mathfrak{g}})\), and \(z\in{\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\). As for \(W_{\varepsilon}({\mathfrak{g}})\), the Lie derivative and contraction are compatible:

Combining quantization \(q : \bigwedge_{\varepsilon}({\mathfrak{g}}) \to {\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\) and PBW \(\varepsilon\)-symmetrization \(Q: S_{\varepsilon}({\mathfrak{g}}) \to {\mathfrak U}_{\varepsilon}({\mathfrak{g}})\), we obtain the following proposition.

In what follows, we simply call \({\mathcal{Q}}\) the quantization map of \(W_{\varepsilon}({\mathfrak{g}})\).

Similarly, we define \(\mathcal{W}_{\varepsilon}({\mathfrak{g}}_{\pm})\) such that \({\mathcal{W}_{\varepsilon}({\mathfrak{g}})}\cong {\mathcal{W}}_{\varepsilon}({\mathfrak{g}}_{-})\otimes {\mathcal{W}}_{\varepsilon}({\mathfrak{g}}_{+})\). We define the completed quantum Weil algebra \(\widehat{\mathcal{W}}_{\varepsilon}({\mathfrak{g}})\) as the direct sum over all \((i, \gamma) \in {\mathbb{Z}}\times \Gamma\) of the components \[\widehat{\mathcal{W}}_{\varepsilon}({\mathfrak{g}})_{(i,\gamma)} \mathrel{\vcenter{:}}= \prod_{r\ge 0}\; \bigoplus_{\substack{\alpha,\beta\in\Gamma\\ \alpha+\beta=\gamma}} \mathcal{W}_{\varepsilon}({\mathfrak{g}}_-)_{(i-r,\alpha)}\otimes \mathcal{W}_{\varepsilon}({\mathfrak{g}}_+)_{(r,\beta)}.\] The Lie derivative \(L_{x}\) and the contraction \(\iota_{x}\) extend degreewise to \(\widehat{{\mathcal{W}}}_{\varepsilon}({\mathfrak{g}})\), and we denote these extensions by the same symbols. Moreover, we can view \(\widehat{{\mathcal{W}}}_{\varepsilon}({\mathfrak{g}})\) as a color Lie superalgebra with bracket \([\cdot,\cdot]_{{\widehat{{\mathcal{W}}}}}\) extending 22 .

In what follows, we define for any \(x \in {\mathfrak{g}}\) \[\gamma^{{\mathcal{W}}}(x) \mathrel{\vcenter{:}}= x\otimes 1 -\tfrac{1}{2}(1\otimes \widehat{\gamma}(x))\in {\mathcal{W}_{\varepsilon}({\mathfrak{g}})}, \qquad \widehat{\gamma}(x) := \widehat{\gamma}(\operatorname{ad}_{x}).\] The following lemma will be used later. It is proved by a direct computation, as in [2].

Moreover, we obtain a reformulation of the Lie derivative and contraction in \({\widehat{{\mathcal{W}}}_{\varepsilon}({\mathfrak{g}})}\).

Finally, we introduce the normal-ordered quantization, namely the map \(\widehat{\mathcal{Q}}:\widehat{W}_{\varepsilon}({\mathfrak{g}})\longrightarrow \widehat{\mathcal{W}}_{\varepsilon}({\mathfrak{g}}),\) which, on each homogeneous component \((i,\gamma)\in {\mathbb{Z}}\times \Gamma\), is defined by taking the direct product over \(r\geq 0\) and the direct sum over all \(\alpha,\beta\in \Gamma\) with \(\alpha+\beta=\gamma\) of the maps \[\mathcal{Q}\otimes \mathcal{Q}: W_{\varepsilon}({\mathfrak{g}}_{-})_{(i-r,\alpha)}\otimes W_{\varepsilon}({\mathfrak{g}}_{+})_{(r,\beta)} \longrightarrow \mathcal{W}_{\varepsilon}({\mathfrak{g}}_{-})_{(i-r,\alpha)}\otimes \mathcal{W}_{\varepsilon}({\mathfrak{g}}_{+})_{(r,\beta)}.\] Since \(\mathcal{Q}\) is an isomorphism of \({\mathbb{Z}}\times \Gamma\)-graded super vector spaces on each factor, \(\widehat{\mathcal{Q}}\) is an isomorphism of \({\mathbb{Z}}\times \Gamma\) super vector spaces.

3 Cubic Dirac Operators for Infinite-Dimensional Quadratic Color Lie Algebras↩︎

This section extends the cubic Dirac operator \(\operatorname{D}\) to quadratic (possibly infinite-dimensional) color Lie algebras with trivial Kac–Peterson class and studies the resulting action. As in the finite-dimensional quadratic Lie algebra case, the square of \(\operatorname{D}\) is expressed as a sum of normally ordered Casimir operators and a constant. Along the way, we extend the Chern–Weil map to general quadratic \({\mathfrak{g}}\)-differential algebras and identify the element whose quantization yields the cubic Dirac operator with the Chern–Simons element in the completed Weil algebra associated with the quadratic polynomial defined by \(B\) and the universal connection. We begin by recalling the necessary background.

3.1 Normal-ordered Casimir Element↩︎

In analogy with the finite-dimensional case, we define a normal-ordered Casimir element for general infinite-dimensional \({\mathbb{Z}}\)-graded color Lie algebras. In general, this element need not be central.

If \({\mathfrak{g}}\) is finite-dimensional, let \(\{e_{a}\}\) be a homogeneous basis with \(B\)-dual basis \(\{e^{a}\}\), and set \(p\mathrel{\vcenter{:}}=\sum_{a}\varepsilon(e_{a})\,e_{a}e^{a}\in S^{2}_{\varepsilon}({\mathfrak{g}}).\) Then \(p\) is \({\mathfrak{g}}\)-invariant, and the Casimir element is \(Q(p)\in{\mathfrak U}_{\varepsilon}({\mathfrak{g}})\) (cf. [19]). We now extend this construction to the infinite-dimensional \({\mathbb{Z}}\)-graded color Lie algebras.

We fix a homogeneous basis \((e_{a})\) of \({\mathfrak{g}}\) with \(B\)-dual basis \((e^{a})\), and define the analogue \[p \mathrel{\vcenter{:}}= \sum_{a}\varepsilon(e_{a})e_{a}e^{a} = \sum_{a}e^{a}e_{a} \in \widehat{S}_{\varepsilon}^{2}({\mathfrak{g}}).\]

This motivates the following definition.

However, \(\Omega_{{\mathfrak{g}}}'\) need not lie in the center of \(\widehat{{\mathfrak U}}_{\varepsilon}({\mathfrak{g}})\); its failure to be central is measured by \(\Psi_{\operatorname{KP}}\in\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\), the unique element corresponding to \({\psi_{\operatorname{KP}}}\in\widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak{g}}^{\ast})\) (cf. 18 ).

The theorem indicates that the normal-ordered Casimir element \(\Omega_{{\mathfrak{g}}}'\) admits a correction to a central element in \(\widehat{{\mathfrak U}}_{\varepsilon}({\mathfrak{g}})\) precisely when the Kac–Peterson class vanishes, that is, \({\psi_{\operatorname{KP}}}= \operatorname{d}\!\rho\) for some \(\rho \in ({\mathfrak{g}}^{\ast})_{(0,{0_{\scriptscriptstyle \Gamma}})}\). Recall the definition of the musical isomorphism \(\sharp \colon {\mathfrak{g}}^{\ast} \xrightarrow{\;\sim\;} {\mathfrak{g}}\) introduced in 7 , and denote by \(\rho^{\sharp}\) the image of any \(\rho \in {\mathfrak{g}}^{\ast}\) under this map.

3.2 Structure Constants Tensor↩︎

The structure constants tensor \(\phi\) is the tensorial incarnation of the Lie bracket after identification of \({\mathfrak{g}}\) with \({\mathfrak{g}}^{\ast}\) via \(B\). It therefore records the full color Lie algebra structure of \({\mathfrak{g}}\) as a canonical cubic element of \(\widehat{\bigwedge}^{3}_{\varepsilon}({\mathfrak{g}})\).

Concretely, the structure constants tensor is the unique cubic element \(\phi \in \widehat{\bigwedge}^{3}_{\varepsilon}({\mathfrak{g}})_{(0,{0_{\scriptscriptstyle \Gamma}})}\) such that \[\langle\phi,x\wedge y\wedge z\rangle_{\wedge}=-\tfrac{1}{2}B([x,y],z)\] for all homogeneous \(x,y,z\in {\mathfrak{g}}\). Its existence follows from the \(\varepsilon\)-skew-symmetry of the bracket together with the invariance of \(B\), which imply that \((x,y,z)\mapsto -\tfrac12 B([x,y],z)\) is an \(\varepsilon\)-alternating trilinear form. It packages the structure constants \(f_{abc}\mathrel{\vcenter{:}}= B([e_{a},e_{b}],e_{c})\) into a single canonical cubic tensor.

The normal-ordered quantization of \(\phi\) is \[\phi'\mathrel{\vcenter{:}}= \widehat{q}(\phi)\in\widehat{{\operatorname{Cl}}}_{\varepsilon}({\mathfrak{g}}).\] It is odd with respect to the canonical \({\mathbb{Z}}_{2}\)-grading. For \(r\in{\mathbb{Z}}\), let \(\phi'_{r}\) denote the component of \(\phi'\) in \(\bigwedge_{\varepsilon}({\mathfrak{g}}_{-})_{-r}\otimes \bigwedge_{\varepsilon}({\mathfrak{g}}_{+})_{r}\). Then \(\phi'_{0}\) is the structure constants tensor of the finite-dimensional color Lie algebra \({\mathfrak{g}}_{0}\).

In general, \(\phi'\) is not \({\mathfrak{g}}\)-invariant. Recall that \({\psi^{\sharp}_{\operatorname{KP}}}\in\widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak{g}})\) is the image of \({\psi_{\operatorname{KP}}}\in\widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak{g}}^{\ast})\) under the musical isomorphism \(\sharp\) in 7 .

This motivates the definition of a modified structure constants tensor, which is \({\mathfrak{g}}\)-invariant.

3.3 Chern–Simons Element Perspective↩︎

We introduced the \({\mathfrak{g}}\)-invariant elements \(p=\sum_{a}e^{a}\otimes e_{a}\) and \(1\otimes\phi\) in the \({\mathbb{Z}}\)-graded color superalgebra \(\widehat{W}_{\varepsilon}({\mathfrak{g}})\). Their sum \({\mathcal{D}}\) is central to this paper, since its quantization is the cubic Dirac operator of \({\mathfrak{g}}\). In this section, we study \({\mathcal{D}}\) and show that it is the canonical Chern–Simons element associated with the quadratic form \(B\) via the universal property of the completed Weil algebra \(\widehat{W}_{\varepsilon}({\mathfrak{g}})\). This also explains its appearance in physics as the boundary operator of one-dimensional Chern–Simons theory.

3.3.1 The Universal Property of the Weil Algebra↩︎

The Weil algebra \(W({\mathfrak{g}})\) of a Lie algebra \({\mathfrak{g}}\) is characterized by the following universal property [24]: it is the initial \({\mathfrak{g}}\)-differential algebra equipped with a connection. Accordingly, for every \({\mathfrak{g}}\)-differential algebra \(\mathcal{A}\) with connection \(A\), there exists a unique morphism of \({\mathfrak{g}}\)-differential algebras \(c_{A}\colon W({\mathfrak{g}})\to\mathcal{A}\) sending the universal connection \(A_{{\mathrm{univ}}}\) of \(W({\mathfrak{g}})\) to \(A\), and hence the universal curvature \(F_{{\mathrm{univ}}}\) to the curvature of \(A\). We study its completed version for general \({\mathbb{Z}}\)-graded color Lie algebras \({\mathfrak{g}}\).

The \({\mathbb{Z}}\)-graded color Lie superalgebra \((\widehat{W}_{\varepsilon}({\mathfrak{g}}),[\cdot,\cdot]_{\widehat{W}})\) is componentwise generated by the elements \(x\otimes 1\) and \(1\otimes x\), with \(x\in{\mathfrak{g}}\), and carries Lie derivative \(L_{x}\) and contraction \(\iota_{x}\). We regard these generators as \({\mathbb{Z}}\times\Gamma\)-homogeneous maps \(A_{{\mathrm{univ}}}\colon {\mathfrak{g}}\to \widehat{W}_{\varepsilon}({\mathfrak{g}})_{\bar1}\) and \(F_{{\mathrm{univ}}}\colon {\mathfrak{g}}\to \widehat{W}_{\varepsilon}({\mathfrak{g}})_{\bar0}\) defined by \[A_{{\mathrm{univ}}}(x)\mathrel{\vcenter{:}}= 1\otimes x,\qquad F_{{\mathrm{univ}}}(x)\mathrel{\vcenter{:}}= x\otimes 1.\] With respect to the \({\mathbb{Z}}\)-grading 21 , one has, in general, \(A_{{\mathrm{univ}}}(x)\in\widehat{W}^{1}_{\varepsilon}({\mathfrak{g}})\) and \(F_{{\mathrm{univ}}}(x)\in\widehat{W}^{2}_{\varepsilon}({\mathfrak{g}}).\)

To define a differential in terms of \({A_{\text{univ}}}\) and \(F_{{\mathrm{univ}}}\), we first introduce notation and define a bracket \([A,B]\) for \({\mathbb{Z}}\times \Gamma\)-homogeneous maps \(A,B\colon{\mathfrak{g}}\to\widehat{W}_{\varepsilon}({\mathfrak{g}})\). We identify \({\mathfrak{g}}\) with its restricted dual \({\mathfrak{g}}^{\ast}\) throughout. We define the completion \(\widehat{{\mathfrak{g}}\otimes {\mathfrak{g}}}\) as the direct sum of the spaces \((\widehat{{\mathfrak{g}}\otimes {\mathfrak{g}}})_{(i,\gamma)}\), where \[\widehat{{\mathfrak{g}}\otimes{\mathfrak{g}}}_{(i,\gamma)} = \prod_{r\geq 0} \bigoplus_{\alpha+\beta=\gamma} (({\mathfrak{g}}\otimes{\mathfrak{g}})_{-})_{(i-r,\alpha)} \otimes (({\mathfrak{g}}\otimes{\mathfrak{g}})_{+})_{(r,\beta)}.\] Here \(({\mathfrak{g}}\otimes{\mathfrak{g}})_{\pm}\) are defined with respect to the total \({\mathbb{Z}}\)-grading induced by \({\mathfrak{g}}\): the symbol \(+\) denotes the positive-degree part, and \(-\) denotes the non-positive-degree part. This contains \(\widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak{g}})\) canonically. The dual of the color Lie bracket, that is, \([\cdot,\cdot]^{\ast} : {\mathfrak{g}}\to \widehat{\bigwedge}^{2}_{\varepsilon}({\mathfrak{g}})\) so that \(\langle [\cdot,\cdot]^{\ast}(z),\,x\wedge y\rangle_{\wedge} = \langle z,\,[x,y]\rangle_{\wedge}\), is given by \[[\cdot,\cdot]^{\ast}(z) = -\tfrac{1}{2}\sum_{a,b}\varepsilon(e_{a})\varepsilon(e_{b})\varepsilon(e_{b},e_{a})B(z,[e_{a},e_{b}])e^{a}\wedge e^{b}.\] Likewise, using \(W_{\varepsilon}({\mathfrak{g}})\otimes W_{\varepsilon}({\mathfrak{g}})\cong (W_{\varepsilon}({\mathfrak{g}}_{-})\otimes W_{\varepsilon}({\mathfrak{g}}_{-})\otimes (W_{\varepsilon}({\mathfrak{g}}_{+})\otimes W_{\varepsilon}({\mathfrak{g}}_{+}))\) as \({\mathbb{Z}}\)-graded \(\varepsilon\)-supersymmetric color superalgebras, we define the completion \(\widehat{W_{\varepsilon}({\mathfrak{g}})\otimes W_{\varepsilon}({\mathfrak{g}})}\) as the direct sum of all spaces \(\widehat{W_{\varepsilon}({\mathfrak{g}})\otimes W_{\varepsilon}({\mathfrak{g}})}_{(i,\gamma)}\), where \[\widehat{W_{\varepsilon}({\mathfrak{g}})\otimes W_{\varepsilon}({\mathfrak{g}})}_{(i,\gamma)} = \prod_{r\geq 0} \bigoplus_{\alpha+\beta=\gamma} ((W_{\varepsilon}({\mathfrak{g}})\otimes W_{\varepsilon}({\mathfrak{g}}))_{-})_{(i-r,\alpha)} \otimes ((W_{\varepsilon}({\mathfrak{g}})\otimes W_{\varepsilon}({\mathfrak{g}}))_{+})_{(r,\beta)}.\] Here \(({\mathcal{W}_{\varepsilon}({\mathfrak{g}})}\otimes {\mathcal{W}_{\varepsilon}({\mathfrak{g}})})_{\pm}\) are defined with respect to the total \({\mathbb{Z}}\)-grading induced by \({\mathfrak{g}}\). For \({\mathbb{Z}}\times \Gamma\)-homogeneous maps \(A,B\colon{\mathfrak{g}}\to\widehat{W}_{\varepsilon}({\mathfrak{g}})\), the map \(A\otimes B\) extends componentwise to a linear map \(A\otimes B\colon \widehat{{\mathfrak{g}}\otimes{\mathfrak{g}}}\to\widehat{W_{\varepsilon}({\mathfrak{g}})\otimes W_{\varepsilon}({\mathfrak{g}})}.\) Indeed, since \(A\) and \(B\) have fixed bidegree, \(A\otimes B\) preserves the product decomposition up to a fixed shift. We then define \([A,B]\) as the composition \[\label{eq::def95bracket} {\mathfrak{g}}\xrightarrow{[\cdot,\cdot]^{\ast}}\widehat{\bigwedge}^{2}_{\varepsilon}({\mathfrak{g}})\hookrightarrow \widehat{{\mathfrak{g}}\otimes {\mathfrak{g}}}\xrightarrow{A \otimes B} \widehat{W_{\varepsilon}({\mathfrak{g}}) \otimes W_{\varepsilon}({\mathfrak{g}})}\xrightarrow{\text{mult}} \widehat{W}_{\varepsilon}({\mathfrak{g}}).\tag{23}\]

Altogether, we define the Weil differential on generators by \[\label{eq::def95Weil95differential} \begin{align} {{\operatorname{d}}^{W}}A_{{\mathrm{univ}}}(x) &\mathrel{\vcenter{:}}= F_{{\mathrm{univ}}}(x)-\tfrac12[A_{{\mathrm{univ}}},A_{{\mathrm{univ}}}](x)\\ &= x\otimes 1 +\tfrac14\sum_{a,b}\varepsilon(e_{a})\varepsilon(e_{b})\varepsilon(e_b,e_a)B(x,[e_a,e_b])\, 1\otimes e^{a}\wedge e^{b}, \\ {{\operatorname{d}}^{W}}F_{{\mathrm{univ}}}(x) &\mathrel{\vcenter{:}}= -[A_{{\mathrm{univ}}},F_{{\mathrm{univ}}}](x)\\ &=\tfrac12\sum_{a,b}\varepsilon(e_{a})\varepsilon(e_{b})\varepsilon(e_{b},e_{a})B(x,[e_a,e_b])\, e^{b}\otimes e^{a}, \end{align}\tag{24}\] and extend it uniquely to an odd \(\varepsilon\)-derivation (cf. 4 ). These formulas are the color analogues of the corresponding formulas in the finite-dimensional case, now in the infinite-dimensional setting and with infinite sums. The following lemma is proved analogously to the finite-dimensional case by direct computation.

This motivates the following notion of \({\mathfrak{g}}\)-differential algebra, in the spirit of Meinrenken and Alekseev [25]. Let \({\mathcal{A}}\) be a \({\mathbb{Z}}\)-graded color superalgebra. Assume \({\mathcal{A}}\) is \(\varepsilon'\)-commutative, that is, \[xy = \varepsilon'(x,y)yx, \qquad \varepsilon'(x,y) = (-1)^{p(x)p(y)}\varepsilon(x,y)\] for all homogeneous \(x,y\in {\mathcal{A}}\). Moreover, assume that \({\mathcal{A}}\) is quadratic, that is, equipped with a nondegenerate \(\varepsilon'\)-symmetric bilinear form \(\langle\cdot,\cdot\rangle\) such that \(\langle x,y\rangle=0\) unless \(\vert x\vert +\vert y\vert = {0_{\scriptscriptstyle \Gamma}}\), \(p(x)=p(y)\), and \({\operatorname{deg}}(x)+{\operatorname{deg}}(y)=0\).

We assume that \({\mathcal{A}}\) admits a polarization \({\mathcal{A}}\) with respect to the \({\mathbb{Z}}\)-grading such that as a \({\mathbb{Z}}\times\Gamma\times{\mathbb{Z}}_2\)-graded vector space \({\mathcal{A}}\cong{\mathcal{A}}_-\otimes{\mathcal{A}}_+\), where \[{\mathcal{A}}_+\mathrel{\vcenter{:}}=\bigoplus_{i>0}{\mathcal{A}}_i,\qquad {\mathcal{A}}_-\mathrel{\vcenter{:}}=\bigoplus_{i\leq 0}{\mathcal{A}}_i .\] Using the this decomposition and the bilinear form \(\langle\cdot,\cdot\rangle\), one defines \(\widehat{{\mathcal{A}}}\) as the direct sum of the spaces \(\widehat{{\mathcal{A}}}_{(i,\gamma)}\), where \[\widehat{{\mathcal{A}}}_{(i,\gamma)}\mathrel{\vcenter{:}}=\prod_{r\geq 0}\bigoplus_{\alpha+\beta=\gamma}({\mathcal{A}}_{-})_{(i-r,\alpha)}\otimes({\mathcal{A}}_{+})_{(r,\beta)}.\] The \({\mathbb{Z}}_2\)-grading of \({\mathcal{A}}\) induces a \({\mathbb{Z}}_2\)-grading on \(\widehat{{\mathcal{A}}}\).

Extracting the roles of \(A_{{\mathrm{univ}}}\) and \(F_{{\mathrm{univ}}}\), the definition of \({{\operatorname{d}}^{W}}\), and the identities \(\iota_{x}A_{{\mathrm{univ}}}(y)=B(x,y)\) and \(L_{x}A_{{\mathrm{univ}}}(y)=1\otimes [x,y]=A_{{\mathrm{univ}}}([x,y])\), one is led to the notions of connection and curvature. For \({\mathbb{Z}}\times\Gamma\)-homogeneous maps \(A,B\colon{\mathfrak{g}}\to\widehat{{\mathcal{A}}}\), we define the bracket \([A,B]\) exactly as in 23 . Concretely, if \((e_{a})\) is a homogeneous basis of \({\mathfrak{g}}\) with \(B\)-dual basis \((e^{a})\), then \[[A,B](x)\mathrel{\vcenter{:}}=-\tfrac12\sum_{a,b}\varepsilon(e_{a})\varepsilon(e_{b})\varepsilon(e_{b},e_{a})B(x,[e_{a},e_{b}])\,A(e^{a})B(e^{b}).\]

The following lemma collects useful identities.

The completed Weil algebra \(\widehat{W}_{\varepsilon}({\mathfrak{g}})\) is the universal completed \({\mathfrak{g}}\)-differential algebra equipped with a connection. Its universal connection is \({A_{\text{univ}}}\), with universal curvature \(F_{{\mathrm{univ}}}\), and every connection \(A\) in a completed \({\mathfrak{g}}\)-differential algebra \(\widehat{{\mathcal{A}}}\) arises uniquely as the image of \({A_{\text{univ}}}\) under a morphism of completed \({\mathfrak{g}}\)-differential algebras. Indeed, for any completed \({\mathfrak{g}}\)-differential algebra \(\widehat{{\mathcal{A}}}\) with connection \(A\) define on generators \[c_{A}(A_{{\mathrm{univ}}})\mathrel{\vcenter{:}}= A, \qquad c_{A}(F_{{\mathrm{univ}}})\mathrel{\vcenter{:}}= F^{A}.\] Since \(\widehat{{\mathcal{A}}}\) is \(\varepsilon'\)-commutative and \(A_{{\mathrm{univ}}},F_{{\mathrm{univ}}}\) generate \(\widehat{W}_{\varepsilon}({\mathfrak{g}})\) componentwise, the homogeneity of \(A\) and \(F^{A}\) implies that the assignments \(A_{{\mathrm{univ}}}\mapsto A\) and \(F_{{\mathrm{univ}}}\mapsto F^{A}\) define a unique (continuous) homomorphism \(c_A:\widehat{W}_{\varepsilon}({\mathfrak{g}})\to\widehat{{\mathcal{A}}}\). We call \(c_{A}\) the Chern–Weil map for completed \({\mathfrak{g}}\)-differential algebras of \({\mathbb{Z}}\)-graded color Lie algebras.

3.3.2 Chern–Simons Elements and the Quadratic Case↩︎

Let \(\widehat{{\mathcal{A}}}\) be a completed \({\mathfrak{g}}\)-differential algebra, let \(A\) be a connection on \(\widehat{{\mathcal{A}}}\), and let \(F^{A}\) be its curvature. For \(m\geq 1\), let \(\widehat S^{m}({\mathfrak{g}}^{\ast})^{{\mathfrak{g}}}\) denote the completed space of invariant symmetric \(m\)-linear forms on \({\mathfrak{g}}\). For \(P\in\widehat S^{m}({\mathfrak{g}}^{\ast})^{{\mathfrak{g}}}\), the element \(P(F^{A})\in\widehat{{\mathcal{A}}}_{\bar{0}}\) is called the characteristic element associated with \(P\) and \(A\). Explicitly, if \((e_{a})\) is a homogeneous basis of \({\mathfrak{g}}\) with \(B\)-dual basis \((e^{a})\), then \[\label{eq::characteristic95element} P(F^{A})=\sum_{a_{1},\ldots,a_{m}}P(e_{a_{1}},\ldots,e_{a_{m}})\,F^{A}(e^{a_{1}})\cdots F^{A}(e^{a_{m}}),\tag{25}\] where the sum is taken in the completed algebra \(\widehat{{\mathcal{A}}}\).

Since \({\mathfrak{g}}\) is quadratic the invariant \(\varepsilon\)-symmetric form \(B\) induces a polynomial \(P_{B}\in \widehat{S}^{2}_{\varepsilon}({\mathfrak{g}}^{\ast})\). We show that the Chern–Simons element for \(P_{B}\) and \({A_{\text{univ}}}\) is \[{\mathcal{D}}\mathrel{\vcenter{:}}= \sum_{a}e^{a}\otimes e_{a}+1\otimes \phi \in \widehat{W}_{\varepsilon}({\mathfrak{g}}).\] This element is central in our article since its quantization yields the cubic Dirac operator. The following properties are direct consequences of Lemma 19 and Lemma 20.

To prove that \({\mathcal{D}}\) is a Chern–Simons element, we need the following lemma which follows by definition and the fact that \(B\) is a nondegenerate \(\varepsilon\)-symmetric invariant form on \({\mathfrak{g}}\).

3.4 The Cubic Dirac Operator↩︎

The cubic Dirac operator is a distinguished element of the completed quantum Weil algebra \(\widehat{\mathcal{W}}_{\varepsilon}({\mathfrak{g}})\) characterized by a Parthasarathy–type formula [26]. We keep the conventions of Section 2.1. Let \((e_a)\) be a homogeneous basis of \({\mathfrak{g}}\) with \(B\)-dual basis \((e^a)\), so that \(B(e_a,e^b)=\delta_{ab}\). Following the ideas of Meinrenken [14], Kostant [3], and Kang–Chen [1], we define the cubic Dirac operator as the quantization (cf. Proposition 11) of \({\mathcal{D}}= \sum_{a}e^{a}\otimes e_{a}+1\otimes\phi\), where \(\phi\) is the structure constants tensor from Section 3.2.

Theorem 21 motivates the following definition.

The corrected cubic Dirac operator is \({\mathfrak{g}}\)-invariant and it has a nice square.

3.4.1 Some Comments on relations the BV-BFV formalism↩︎

The BV-BFV¹ formalism is a tool to treat gauge theories on manifolds with boundary, both at the classical [30], [31] and quantum [32] level. A central object in this formalism are the classical and quantum BFV charges associated with the boundary,² denoted \(S^\partial\) and \(\hat{S}^\partial\), respectively. I n the BV–BFV formulation of \(1d\) Chern–Simons theory with coefficient Lie algebra \(\mathfrak{g}\) (potentially with Wilson lines) these can be naturally identified with the classical and quantum cubic Dirac operators for \(\mathfrak{g}\), see [35], [36]. While the authors only treat finite-dimensional quadratic Lie algebras, our construction implies the formulation for general \({\mathbb{Z}}\)-graded (infinite-dimensional) color Lie algebras. We expect this to be relevant in the BV-BFV formulation of two-dimensional (super)conformal field theories. More generally, one can think of higher-dimensional Chern-Simons theories on cylinders as one-dimensional Chern-Simons theories with target in an infinite-dimensional differential graded lie algebra or \(L_\infty\)-algebra, see e.g. [37] for background on \(L_\infty\) Chern-Simons theory and [38], [39] for the case with boundary. Generalizing the notion of cubic Dirac operator further to such cases and investigating the relationship to the classical and quantum BFV charges are the subject of ongoing research.

3.5 Relative Cubic Dirac Operator↩︎

We modify the construction of the cubic Dirac operator to incorporate more color Lie subalgebras of \({\mathfrak{g}}\).

Let \({\mathfrak l}\) be a \({\mathbb{Z}}\)-graded color Lie subalgebra of \({\mathfrak{g}}\), that is, \({\mathfrak l}\) is a color Lie subalgebra satisfying \({\mathfrak l}_{i}\subset{\mathfrak{g}}_{i}\) for all \(i\in{\mathbb{Z}}\). Suppose that \(B\) restricts to an even, non-degenerate, \(\varepsilon\)-symmetric bilinear form on \({\mathfrak l}\), denoted by \(B_{{\mathfrak l}}\) in the following. Set \({\mathfrak s}\mathrel{\vcenter{:}}= {\mathfrak l}^{\perp}\) with respect to \(B\), and assume \([{\mathfrak l}, {\mathfrak s}] \subset {\mathfrak s}\). Then one has the orthogonal decomposition \[{\mathfrak{g}}={\mathfrak l}\oplus{\mathfrak s}.\] For example, if \({\mathfrak{g}}\) is a \({\mathbb{Z}}\)-graded quadratic Lie superalgebra, one can take \({\mathfrak l}\mathrel{\vcenter{:}}= {\mathfrak{g}_{\bar{0}}}\) and \({\mathfrak s}\mathrel{\vcenter{:}}= {\mathfrak{g}_{\bar{1}}}\). By definition of \({\mathfrak s}\), the bilinear form \(B\) restricts to a nondegenerate bilinear form on \({\mathfrak s}\), denoted by \(B_{{\mathfrak s}}\). In particular, we can define \(\varepsilon\)-orthogonal algebra \(\mathfrak{so}_{\varepsilon}({\mathfrak l}) \mathrel{\vcenter{:}}= \mathfrak{so}_{\varepsilon}({\mathfrak l}; B_{{\mathfrak l}})\) and \(\mathfrak{so}_{\varepsilon}({\mathfrak s})\mathrel{\vcenter{:}}= \mathfrak{so}_{\varepsilon}({\mathfrak s}; B_{{\mathfrak s}})\), as well as the \(\varepsilon\)-Clifford algebras \({\operatorname{Cl}}_{\varepsilon}({\mathfrak l}) \mathrel{\vcenter{:}}= {\operatorname{Cl}}_{\varepsilon}({\mathfrak l};B_{{\mathfrak l}})\) and \({\operatorname{Cl}}_{\varepsilon}({\mathfrak s}) \mathrel{\vcenter{:}}= {\operatorname{Cl}}_{\varepsilon}({\mathfrak s}; B_{{\mathfrak s}})\), respectively.

Each of the color Lie algebras \({\mathfrak{g}}\) and \({\mathfrak l}\) has a Kac–Peterson class, denoted by \(\psi_{\operatorname{KP}}\) and \(\psi_{\operatorname{KP}}^{{\mathfrak l}}\), respectively. We assume that both are trivial, that is, that there exist \(\rho\in({\mathfrak{g}}^{\ast})_{(0,{0_{\scriptscriptstyle \Gamma}})}\) and \(\rho_{{\mathfrak l}}\in({\mathfrak l}^{\ast})_{(0,{0_{\scriptscriptstyle \Gamma}})}\) such that \(d\rho=\psi_{\operatorname{KP}}\) and \(d\rho_{{\mathfrak l}}=\psi_{\operatorname{KP}}^{{\mathfrak l}}\). We denote the corresponding corrected Casimir elements by \(\Omega_{{\mathfrak{g}}}\) and \(\Omega_{{\mathfrak l}}\), and the associated corrected cubic Dirac operators by \(\operatorname{D}_{{\mathfrak{g}}}\in\widehat{\mathcal{W}}_{\varepsilon}({\mathfrak{g}})\) and \(\operatorname{D}_{{\mathfrak l}}\in\widehat{\mathcal{W}}_{\varepsilon}({\mathfrak l})\).

To define the relative cubic Dirac operator \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}\), we embed \(\widehat{\mathcal{W}}_{\varepsilon}({\mathfrak l})\) into \(\widehat{\mathcal{W}}_{\varepsilon}({\mathfrak{g}})\) via the adjoint action of \({\mathfrak l}\) on \({\mathfrak s}\). For \(x\in{\mathfrak l}\), the operator \(\operatorname{ad}_{x}\in\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak{g}})\) decomposes as \(\operatorname{ad}_{x}=\operatorname{ad}_{x}^{{\mathfrak l}}+\operatorname{ad}_{x}^{{\mathfrak s}}\) with \(\operatorname{ad}_{x}^{{\mathfrak l}}\in\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak l})\) and \(\operatorname{ad}_{x}^{{\mathfrak s}}\in\widehat{\mathfrak{so}}_{\varepsilon}({\mathfrak s})\). We set \(\require{upgreek} \widehat{\gamma}_{{\mathfrak s}}(x)\mathrel{\vcenter{:}}=\widehat q(\uplambda(\operatorname{ad}_{x}^{{\mathfrak s}}))\in\widehat{{\operatorname{Cl}}}_{\varepsilon}({\mathfrak s})\). Since \(B\) is invariant, \(\operatorname{ad}_{x}\) preserves \({\mathfrak s}\) for every \(x\in{\mathfrak l}\). By Proposition 7, the \(\varepsilon\)-commutator \([\widehat{\gamma}_{{\mathfrak s}}(x),\widehat{\gamma}_{{\mathfrak s}}(y)]_{\widehat{{\operatorname{Cl}}}_{\varepsilon}}\) defines a \(2\)-cocycle \(\psi_{\operatorname{KP}}^{{\mathfrak s}}\in\widehat{\bigwedge}_{\varepsilon}^{2}({\mathfrak l}^{\ast})\), and this yields a color Lie algebra homomorphism provided \(\psi_{\operatorname{KP}}^{{\mathfrak s}}=d\rho_{{\mathfrak s}}\), where \(\rho_{{\mathfrak s}}\mathrel{\vcenter{:}}= \rho_{{\mathfrak l}}-\rho|_{{\mathfrak l}}\in({\mathfrak l}^{\ast})_{(0,{0_{\scriptscriptstyle \Gamma}})}\).

In what follows, we assume \(\psi_{\operatorname{KP}}^{{\mathfrak s}} = \operatorname{d}\!\rho_{{\mathfrak s}}\) for some \(\rho_{{\mathfrak s}} \in ({\mathfrak l}^{\ast})_{(0,{0_{\scriptscriptstyle \Gamma}})}\). We define a map \(j : \mathcal{W}_{\varepsilon}({\mathfrak l}) \to \widehat{\mathcal{W}}_{\varepsilon}({\mathfrak{g}})\), given on generators by \[\begin{align} j(1\otimes x) = 1\otimes x, \qquad j(x\otimes 1) = x\otimes 1 + 1 \otimes \gamma'_{{\mathfrak s}}(x). \end{align}\] In particular, \(j\) defines a diagonal map of \({\mathfrak l}\) into \(\widehat{\mathcal{W}}_{\varepsilon}({\mathfrak{g}})\).

The following theorem collects the main properties of \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}\). We denote by \({\mathcal{W}}_{\varepsilon}({\mathfrak{g}},{\mathfrak l})\) the \({\mathbb{Z}}\)-graded color subsuperalgebra of \({\mathcal{W}_{\varepsilon}({\mathfrak{g}})}\) of elements annihilated by \(L_x\) and \(\iota_x\) for all \(x\in{\mathfrak l}\). Its completion is denoted by \(\widehat{{\mathcal{W}}}_{\varepsilon}({\mathfrak{g}},{\mathfrak l})\subset {\widehat{{\mathcal{W}}}_{\varepsilon}({\mathfrak{g}})}\).

Assume that \({\mathfrak l}\subset{\mathfrak{g}}\) is quadratic, with form \(B_{{\mathfrak l}}\), and that \({\mathfrak{g}}={\mathfrak l}\oplus{\mathfrak s}\) with \([{\mathfrak s},{\mathfrak s}]\subset{\mathfrak l}\). Then the cubic term in \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}\) vanishes. Thus \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak l}}\) is quadratic; we call it the quadratic Dirac operator attached to \(({\mathfrak{g}},{\mathfrak l})\). The Kac–Moody superalgebra case treated below is of this form.

4 Applications to Kac–Moody Superalgebras↩︎

In this section, we study the relative cubic Dirac operator for symmetrizable Kac–Moody superalgebras \({\mathfrak{g}}={\mathfrak{g}_{\bar{0}}}\oplus{\mathfrak{g}_{\bar{1}}}\), with \({\mathfrak l}={\mathfrak{g}_{\bar{0}}}\) and \({\mathfrak s}={\mathfrak{g}_{\bar{1}}}\).

We first recall basic facts on Kac–Moody superalgebras and highest weight supermodules \(M\), and describe the action of \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) on \(M\otimes M({\mathfrak{g}_{\bar{1}}})\), where \(M({\mathfrak{g}_{\bar{1}}})\) denotes the oscillator supermodule. We then determine \(\ker \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\) in the example \({\mathfrak{g}}=\widehat{{\mathfrak{osp}}}(1\vert 2n)\). Finally, we introduce unitarizable \({\mathfrak{g}}\)-supermodules, study the consequences of the properties of \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\), and relate the kernel of the corresponding Dirac operator \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) to a cohomology theory.

In what follows, we fix \(\Gamma={\mathbb{Z}}_{2}\) and let \(\varepsilon\) be the parity. Accordingly, we omit the index \(\varepsilon\) throughout and write, for example, \(\bigwedge({\mathfrak{g}})\), \({\operatorname{Cl}}({\mathfrak{g}})\), \(S({\mathfrak{g}})\), \({\mathcal{W}}({\mathfrak{g}})\), and \(\mathfrak{osp}({\mathfrak{g}})\) in place of \(\bigwedge_{\varepsilon}({\mathfrak{g}})\), \({\operatorname{Cl}}_{\varepsilon}({\mathfrak{g}})\), \(S_{\varepsilon}({\mathfrak{g}})\), \({\mathcal{W}}_{\varepsilon}({\mathfrak{g}})\), and \(\mathfrak{so}_{\varepsilon}({\mathfrak{g}})\). Likewise, we speak simply of Lie superalgebras, supermodules, superderivations, and supercommutators, and all signs are understood with respect to parity.

4.1 Generalities on Kac–Moody Superalgebras↩︎

This section introduces Kac–Moody superalgebras. We follow the standard references [40]–[42].

Let \(A \mathrel{\vcenter{:}}= (a_{ij})\) be an \(n\times n\) matrix. Set \(I=\{1,\ldots,n\}\), and let \(p:I\to{\mathbb{Z}}_{2}\) be a map, referred to as the parity function. Fix an even vector space \({\mathfrak h}\) of dimension \(2\left|{I}\right|-{\operatorname{rk}}(A)\). Then there exist linearly independent elements \(\alpha_{i}\in{\mathfrak h}^{\ast}\) and \(h_{i}\in{\mathfrak h}\) such that [43] \[\alpha_{j}(h_{i})=a_{ij},\qquad i,j\in I.\] Define the superalgebra \({\overline{{\mathfrak{g}}}}(A)\) with generators \(X_{i},Y_{i}\) (\(i\in I\)) of parities \(p(X_{i})=p(Y_{i})=p(i)\), together with \({\mathfrak h}\), and relations \[[h,X_{i}]=\alpha_{i}(h)X_{i},\qquad [h,Y_{i}]=-\alpha_{i}(h)Y_{i},\qquad [X_{i},Y_{j}]=\delta_{ij}h_{i},\qquad [{\mathfrak h},{\mathfrak h}]=0.\] There exists a maximal ideal \(\mathfrak{m}\) of \({\overline{{\mathfrak{g}}}}(A)\) intersecting \({\mathfrak h}\) trivially [43]. The contragredient Lie superalgebra associated to \(A\) is the quotient \({\mathfrak{g}}(A)\mathrel{\vcenter{:}}= {\overline{{\mathfrak{g}}}}(A)/\mathfrak{m}.\) We call \(A\) the Cartan matrix of the Lie superalgebra \({\mathfrak{g}}(A)\). Without loss of generality we assume \(a_{ii}\in\{2,0\}\), since \({\mathfrak{g}}(B)\cong{\mathfrak{g}}(A)\) whenever \(B=DA\) for some invertible diagonal \(n\times n\) matrix \(D\). Such a Cartan matrix \(A\) is called normalized.

A contragredient Lie superalgebra \({\mathfrak{g}}(A)\) is called quasisimple if for every ideal \({\mathfrak{j}}\) of \({\mathfrak{g}}(A)\) either \({\mathfrak{j}}\subset {\mathfrak h}\) or \({\mathfrak{j}}+ {\mathfrak h}= {\mathfrak{g}}(A)\). A Cartan matrix \(A\) is called admissible if it satisfies the following conditions [44]:

1. If \(a_{ii} = 0\) and \(p(i)=\bar{0}\), then \(a_{ij} = 0\) for every \(j \in I\).

2. If \(a_{ii} = 2\) and \(p(i)=\bar{0}\), then \(a_{ij} \in {\mathbb{Z}}_{\leq 0}\) for any \(j \in I\), \(i \neq j\) and \(a_{ij}=0\) implies \(a_{ji}=0\).

3. If \(a_{ii} = 2\) and \(p(i)=\bar{1}\), then \(a_{ij} \in 2{\mathbb{Z}}_{\leq 0}\) for any \(j \in I\), \(i \neq j\) and \(a_{ij}=0\) implies \(a_{ji}=0\).

If \(A\) is admissible, we call the associated contragredient Lie superalgebra \({\mathfrak{g}}(A)\) admissible. Note that \({\mathfrak{g}}(A)\) is admissible if and only if \(\operatorname{ad}_{X_{i}}\) and \(\operatorname{ad}_{Y_{i}}\) act locally nilpotently [45].

Any contragredient Lie superalgebra \({\mathfrak{g}}\mathrel{\vcenter{:}}= {\mathfrak{g}}(A)\) admits a root space decomposition, since the action of \({\mathfrak h}\) on \({\mathfrak{g}}(A)\) is diagonalizable: \[{\mathfrak{g}}= {\mathfrak h}\oplus \bigoplus_{\alpha \in \Delta} {\mathfrak{g}}^{\alpha}, \qquad {\mathfrak{g}}^{\alpha} \mathrel{\vcenter{:}}= \{x \in {\mathfrak{g}}: [h,x] = \alpha(h)x \;\text{for all} \;h \in {\mathfrak h}\},\] for some \(\Delta \subset {\mathfrak h}^{\ast} \setminus \{0\}\), called the set of roots. By definition, \(\alpha_{1}, \ldots, \alpha_{n}\) are roots, and every root \(\alpha \in \Delta\) can be uniquely expressed as \(\alpha = \sum_{i} n_{i}\alpha_{i}\), where all \(n_{i}\) are either nonnegative or nonpositive integers. The roots \(\alpha_{1}, \ldots, \alpha_{n}\) are called simple roots. The set \(\require{upgreek} \Uppi\) of all simple roots is called a simple system. Furthermore, by linear independence of the \(\alpha_{i}\), each root space \({\mathfrak{g}}^{\alpha}\) is either purely even or purely odd. We define the parity function \(p : \Delta \to {\mathbb{Z}}_{2}\) by setting \(p(\alpha) = 0\) or \(1\) whenever \({\mathfrak{g}}^{\alpha}\) is even or odd, respectively. Consequently, \(\require{upgreek} \Uppi=\Uppi_{\bar0}\sqcup\Uppi_{\bar1}\) and \(\Delta=\Delta_{\bar0}\sqcup\Delta_{\bar1}\), according to the decomposition into even and odd simple roots and roots, respectively.

A root \(\alpha \in \Delta\) is called positive if, in the expression \(\alpha = \sum_{i} n_{i}\alpha_{i}\), all coefficients \(n_{i}\) are non-negative and not all zero. Negative roots are defined analogously. We denote by \(\Delta^{+}\) and \(\Delta^{-}\) the sets of positive and negative roots, respectively, so that \(\Delta = \Delta^{+} \sqcup \Delta^{-}\). In particular, one obtains the triangular decomposition \[\label{eq::triangular95decomposition} {\mathfrak{g}}= {\mathfrak{n}}^{-} \oplus {\mathfrak h}\oplus {\mathfrak{n}}^{+}, \qquad {\mathfrak{n}}^{\pm} \mathrel{\vcenter{:}}= \bigoplus_{\alpha \in \Delta^{\pm}} {\mathfrak{g}}^{\alpha}.\tag{27}\]

For each simple root \(\require{upgreek} \alpha_i\in\Uppi\), set \(X_{\alpha_i}\mathrel{\vcenter{:}}= X_i,Y_{\alpha_i}\mathrel{\vcenter{:}}= Y_i\) and \(H_{\alpha_i}\mathrel{\vcenter{:}}= [X_{\alpha_i},Y_{\alpha_i}]=H_i.\) Then \[[h,X_i]=\alpha_{i}(h)X_{i},\quad [h,Y_i]=-\alpha_{i}(h)Y_i,\quad [X_i,Y_i]=h_{i}\] for all \(h\in {\mathfrak h}\). Define a \({\mathbb{Z}}\)-grading on \({\mathfrak{g}}\) by \[\label{eq::Z95grading95KM}{\operatorname{deg}}(X_{i})=1, \qquad {\operatorname{deg}}(Y_{i})=-1,\qquad {\operatorname{deg}}(h_{i})=0\tag{28}\] for all \(i\in I\). Moreover, there exists \(\rho\in{\mathfrak h}^{\ast}\) such that \[\require{upgreek} \label{eq::definition95Weyl95element} \rho(h_{\alpha})=\tfrac{1}{2}\alpha(h_{\alpha}),\qquad \forall\alpha\in\Uppi.\tag{29}\] We refer to \(\rho\) as the Weyl vector for \({\mathfrak{g}}\).

A simple root \(\alpha_{i}\) is called isotropic if \(a_{ii}=0\); otherwise it is called non-isotropic. Furthermore, \(\alpha_{i}\) is called regular if for any other simple root \(\alpha_{j}\), \(a_{ij}=0\) implies \(a_{ji}=0\); otherwise it is called singular. We call the Cartan matrix \(A\) regular if for any \(i,j \in I\), \(a_{ij}=0\) implies \(a_{ji}=0\).

We note that the definition adopted in this article differs from that in [44]. Moreover, if \(p(i) = \bar{0}\) for all \(i \in I\), then \(A\) is a generalized Cartan matrix, and \({\mathfrak{g}}(A)\) coincides with the Kac–Moody algebra associated with \(A\).

4.1.1 Symmetrizable Kac–Moody Superalgebras↩︎

We are concerned with a particular class of Kac–Moody superalgebras, namely the symmetrizable Kac–Moody superalgebras. A Cartan matrix \(A\) is said to be symmetrizable if there exists an invertible diagonal matrix \(D\) such that \(DA\) is symmetric. Symmetrizable Cartan matrices are regular. A contragredient Lie superalgebra \({\mathfrak{g}}(A)\) is called symmetrizable if its Cartan matrix \(A\) is symmetrizable.

This lemma justifies our interest in symmetrizable Kac–Moody superalgebras, for which a cubic Dirac operator can be constructed. In the sequel we write \({\mathfrak{g}}\mathrel{\vcenter{:}}= {\mathfrak{g}}(A)\) for a symmetrizable Kac–Moody superalgebra, where \(A\) is fixed. If \(D = \operatorname{diag}(d_{1}, \ldots, d_{n})\), we may normalize \(B(\cdot,\cdot)\) such that \[\label{eq::normalization95B} B(h,H_{j}) = d_{j}\alpha_{j}(h), \qquad h \in {\mathfrak h}.\tag{30}\] Such a bilinear form \(B\) is called the standard form, and we fix this normalization on \({\mathfrak{g}}\).

The restriction of \(B\) to \({\mathfrak h}\) is nondegenerate, and \(B({\mathfrak{g}}^{\alpha}, {\mathfrak{g}}^{\beta}) = 0\) for \(\alpha \ne -\beta\); hence \(B\) defines a nondegenerate pairing between \({\mathfrak{g}}^{\alpha}\) and \({\mathfrak{g}}^{-\alpha}\). Moreover, \(B\) induces an isomorphism \(\eta : {\mathfrak h}\to {\mathfrak h}^{\ast}\) and thereby a nondegenerate bilinear form on \({\mathfrak h}^{\ast}\), denoted by the same symbol by abuse of notation.

Examples of symmetrizable Kac–Moody superalgebras include the basic classical Lie superalgebras \[A(m\vert n)\;(m\neq n),\quad B(m\vert n),\quad C(n),\quad D(m\vert n),\quad D(2,1;\alpha)\;(\alpha\neq 0,-1),\quad F(4),\quad G(3)\] classified in [21], as well as the affine and twisted affine superalgebras. Our main interest lies in the latter.

4.1.2 Affine and Twisted Affine Lie Superalgebras↩︎

Let \({\mathfrak s}\) be a finite-dimensional contragredient simple Lie superalgebra, \({\mathfrak s}\neq {\mathfrak{psl}}(n\vert n)\), equipped with an even nondegenerate supersymmetric invariant form \(B\). We define the affine Lie superalgebra \(\widehat{{\mathfrak s}}\) to be the infinite-dimensional super vector space \[{\mathfrak s}\otimes {\mathbb{C}}[t,t^{-1}] \oplus {\mathbb{C}}D \oplus {\mathbb{C}}K,\] with even elements \(K,D\) and with the Lie bracket defined by \[\begin{align} \label{eq::commutation95relations95affine95Lie95superalgebra} [x \otimes t^{k}, y \otimes t^{l}] = [x,y] \otimes t^{k+l}+k\delta_{k,-l}B(x,y)K, \\ [K,D] = [K, x \otimes t^{l}]=0, \qquad [D, x \otimes t^{l}] = lx \otimes t^{l} \end{align}\tag{31}\] for any \(x,y \in {\mathfrak s}\) and \(k,l \in {\mathbb{Z}}\). Explicitly, if \(X_{1}, \ldots, X_{n},Y_{1},\ldots, Y_{n}\) are generators of \({\mathfrak s}\), there exist (unique) \(x_{0},y_{0}\in {\mathfrak s}\) such that \([Y_{i},x_{0}]=[X_{i},y_{0}]\) for all \(i > 0\), and if we set \(X_{0} \mathrel{\vcenter{:}}= x_{0}\otimes t^{-1}\) and \(Y_{0} \mathrel{\vcenter{:}}= y_{0} \otimes t\), then \(X_{0},X_{1},\ldots,X_{n}, Y_{0},Y_{1},\ldots,Y_{n}, K,D\) generate \(\widehat{{\mathfrak s}}\).

\(B\) extends to \(\widehat{{\mathfrak s}}\), again denoted by \(B\), by \(B(K,D)=1\), \(B(x\otimes t^{k},y\otimes t^{l})=\delta_{k,-l}B(x,y)\) and \(B(K,K)=B(D,D)=B(K,{\mathfrak s})=B(D,{\mathfrak s})=0\). By Lemma 29, this yields the following proposition.

The affinization \(\widehat{{\mathfrak{psl}}}(n\vert n)\) is the infinite-dimensional super vector space \[{\mathfrak s}\otimes {\mathbb{C}}[t,t^{-1}] \oplus {\mathbb{C}}D \oplus {\mathbb{C}}K \oplus {\mathbb{C}}D' \oplus {\mathbb{C}}K'\] with additional even elements \(D',K'\) and with Lie bracket \[\begin{align} [x \otimes t^{k}, y \otimes t^{l}] = [x,y] \otimes t^{k+l} + k\delta_{k,-l}B(x,y)K+\delta_{k,-l}\operatorname{tr}([x,y])K' \\ [D',x \otimes t^{k}] = \left(1-(-1)^{p(x)}\right)x \otimes t^{k}, \quad [K',x \otimes t^{k}] =0. \end{align}\] Here, \(\operatorname{tr}\) denotes the ordinary trace. \(\widehat{{\mathfrak{psl}}}(n\vert n)\) is a symmetrizable Kac–Moody superalgebra with two-dimensional center. In what follows, \({\mathfrak s}\) denotes a finite-dimensional quadratic contragredient simple Lie superalgebra, and \(\widehat{{\mathfrak s}}\) its affinization.

We describe the roots of \(\widehat{{\mathfrak s}}\). Let \({\mathfrak h}_{{\mathfrak s}}\) be a Cartan subalgebra of \({\mathfrak s}\), and set \({\mathfrak h}\mathrel{\vcenter{:}}={\mathfrak h}_{{\mathfrak s}}\oplus{\mathbb{C}}K\oplus{\mathbb{C}}D\). Let \(\delta,\kappa\in{\mathfrak h}^{\ast}\) be defined by \[\label{eq::definition95delta95kappa} \delta_{{\mathfrak h}_{{\mathfrak s}}\oplus{\mathbb{C}}K}=0, \quad \delta(D)=1, \qquad \kappa\vert_{{\mathfrak h}_{{\mathfrak s}}\oplus {\mathbb{C}}D}=0, \quad \kappa(K)=1,\tag{32}\] so that \({\mathfrak h}^{\ast}={\mathfrak h}^{\ast}_{{\mathfrak s}}\oplus{\mathbb{C}}\kappa\oplus{\mathbb{C}}\delta\). If \(\Delta_{{\mathfrak s}}\) denotes the root system of \({\mathfrak s}\), then the roots of \(\widehat{{\mathfrak s}}\) are the elements \(\alpha+k\delta\) with \(\alpha\in\Delta_{{\mathfrak s}}\) and \(k\in{\mathbb{Z}}\), together with the roots \(k\delta\) for \(k\in{\mathbb{Z}}\setminus\{0\}\). If \(\require{upgreek} \Uppi_{{\mathfrak s}}=\{\alpha_{1},\ldots,\alpha_{n}\}\) is the standard simple system of \({\mathfrak s}\), then the standard simple system of \(\widehat{{\mathfrak s}}\) is \(\require{upgreek} \Uppi \mathrel{\vcenter{:}}= \{\alpha_{0},\alpha_{1},\ldots,\alpha_{n}\}\), where \(\alpha_{0}\mathrel{\vcenter{:}}=\theta+\delta\) and \(\theta\) denotes the lowest root of \({\mathfrak s}\). We denote the set of roots of \(\widehat{{\mathfrak s}}\) by \(\Delta\). The even and odd roots are given by \[\Delta_{\bar{0}} =\{\alpha+n\delta : \alpha \in (\Delta_{{\mathfrak s}})_{\bar{0}}, \;n \in {\mathbb{Z}}\} \sqcup \{n\delta : n\in {\mathbb{Z}}\setminus\{0\}\}, \qquad \Delta_{\bar{1}} = \{\alpha + n \delta : \alpha \in (\Delta_{{\mathfrak s}})_{\bar{1}}, \;n \in {\mathbb{Z}}\}.\]

Finally, assume that \({\mathfrak s}_{\bar0}\) is reductive, say \({\mathfrak s}_{\bar0}=\bigoplus_{j=0}^{N}{\mathfrak s}_{\bar0,j}\), where \({\mathfrak s}_{\bar0,0}\) is abelian and \({\mathfrak s}_{\bar0,j}\) is a simple Lie algebra for \(j\geq 1\). Then, for each \(j=0,\ldots,N\), the affinization \(\widehat{{\mathfrak s}}\) contains the affine Kac–Moody algebra associated with \({\mathfrak s}_{\bar0,j}\), which we denote by \(\widehat{{\mathfrak s}}_{\bar0,j}\).

Next we introduce twisted affine Lie superalgebras. Consider an automorphism \(\phi\) of \({\mathfrak s}\) of finite order \(q\), which preserves the invariant form on \({\mathfrak s}\), that is, \(B(x,y)=B(\phi(x),\phi(y))\) for all \(x,y \in {\mathfrak s}\). In addition, let \(\epsilon\) be a \(q\)-th primitive root of \(1\). Extend \(\phi\) to \(\widehat{{\mathfrak s}}\) via \[\phi(x \otimes t^{k}) = \epsilon^{k} \phi(x)\otimes t^{k}, \quad \phi(D)=D, \quad \phi(K)=K.\] The twisted affine Lie superalgebra \({\mathfrak s}^{\phi}\) is the subsuperalgebra of the fixed points of \(\phi\). The construction does not depend on the choice of \(\epsilon\), and if \(\phi\) and \(\psi\) are two automorphisms in the same connected component of \(\operatorname{Aut}({\mathfrak s})\), then the corresponding twisted affine Lie superalgebras are isomorphic.

To describe the simple system of \({\mathfrak s}^{\phi}\), we introduce a \({\mathbb{Z}}_{q}\)-grading \[{\mathfrak s}= {\mathfrak s}^{0} \oplus {\mathfrak s}^{1} \oplus \cdots \oplus {\mathfrak s}^{q-1}, \qquad {\mathfrak s}^{k} \mathrel{\vcenter{:}}= \{ s \in {\mathfrak s}: \phi(s) = \epsilon^{k}s \}.\] It can be arranged that \(\phi\) satisfies the following properties: \({\mathfrak s}^{0}\) is a simple finite-dimensional Lie superalgebra, \({\mathfrak s}^{1}\) is a simple \({\mathfrak s}^{0}\)-module, and \({\mathfrak s}^{0} + {\mathfrak s}^{1}\) generates \({\mathfrak s}\) [44]. Let \(\Delta^{0}\) denote the set of roots of \({\mathfrak s}^{0}\), and \(\Delta^{j}\) the set of roots of \({\mathfrak s}^{j}\) with respect to a Cartan subalgebra of \({\mathfrak s}^{0}\). If \(\theta\) is the lowest weight of \({\mathfrak s}^{1}\) and \(\{\alpha_{1}, \ldots, \alpha_{n}\}\) is the standard simple system of \({\mathfrak s}^{0}\), then a simple system of \({\mathfrak s}^{\phi}\) is given by \(\{\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n}\}\), where \(\alpha_{0} = \theta + \delta\). The roots of \({\mathfrak s}^{\phi}\) are of the form \(\alpha + k\delta\) with \(\alpha \in \Delta^{j}\) and \(k \in j + q{\mathbb{Z}}\), or \(k\delta\) with \(k \in q{\mathbb{Z}}\setminus \{0\}\).

4.1.3 Kac–Peterson Class↩︎

Let \({\mathfrak{g}}\mathrel{\vcenter{:}}= {\mathfrak{g}}(A)\) be a symmetrizable Kac–Moody superalgebra, and let \(B\) denote the standard form on \({\mathfrak{g}}\). Let \((e_{a})\) be a homogeneous basis with \(B\)-dual basis \((e^{a})\). It admits a corrected Casimir element [41] \[\Omega_{{\mathfrak{g}}} = 2\rho^{\sharp} + \sum_{a} \varepsilon(e_{a}) e_{a} e^{a}.\] By Corollary 1, this implies that the Kac–Peterson class is trivial.

In what follows, we refer to the corrected cubic Dirac operator simply as the cubic Dirac operator.

4.2 Oscillator Module↩︎

Let \({\mathfrak{g}}\) be a symmetrizable Kac–Moody superalgebra with nondegenerate invariant supersymmetric bilinear form \(B\) in standard normalization. Assume that \(B(\alpha,\beta)\in{\mathbb{R}}\) for all simple roots \(\alpha,\beta\). This condition is satisfied for all twisted and untwisted affine Lie superalgebras constructed from basic classical Lie superalgebras, except for those associated with \(D(2,1;\alpha)\) when \(\alpha\notin{\mathbb{R}}\). For the rest of this article, we fix \({\mathfrak l}= {\mathfrak{g}_{\bar{0}}}\) and \({\mathfrak s}= {\mathfrak{g}_{\bar{1}}}\), and consider the relative Dirac operator \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\), which is an element of \(\widehat{{\mathcal{W}}}({\mathfrak{g}},{\mathfrak l})\) and quadratic since \([{\mathfrak{g}_{\bar{1}}},{\mathfrak{g}_{\bar{1}}}]\subset {\mathfrak{g}_{\bar{0}}}\). Moreover, following Section 3.5, we set \(\rho_{\bar1}\mathrel{\vcenter{:}}= \rho_{\bar0}-\rho\).

In what follows, we fix a simple system \(\require{upgreek} \Uppi\) of \({\mathfrak{g}}\), whose simple roots we denote by \(\alpha_{i}\). If \(M\) is a highest weight \({\mathfrak{g}}\)-module (cf. Section 4.3.1), we will see that \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) has a well-defined action on \(M\otimes M({\mathfrak{g}_{\bar{1}}})\), where \(M({\mathfrak{g}_{\bar{1}}})\) is the simple module over \({\operatorname{Cl}}({\mathfrak{g}_{\bar{1}}})\) induced by the triangular decomposition, called the oscillator module.

The restriction of \(B\) to \({\mathfrak{g}_{\bar{1}}}\) is nondegenerate and skew-supersymmetric. Hence \({\mathfrak{g}_{\bar{1}}}\) is symplectic, and \({\operatorname{Cl}}({\mathfrak{g}_{\bar{1}}})\) identifies with the Weyl algebra of \({\mathfrak{g}_{\bar{1}}}\), that is, the algebra generated by the elements of \({\mathfrak{g}_{\bar{1}}}\) with defining relations \(xy-yx=2B(x,y)\) for homogeneous \(x,y\in{\mathfrak{g}_{\bar{1}}}\). Let \(\widehat{{\operatorname{Cl}}}({\mathfrak{g}_{\bar{1}}})\) denote its completion.

Fix a triangular decomposition \({\mathfrak{g}}={\mathfrak{n}}^{-}\oplus{\mathfrak h}\oplus{\mathfrak{n}}^{+}\) (cf. 27 ). Then \[{\mathfrak{g}_{\bar{1}}}=({\mathfrak{g}_{\bar{1}}})_{-}\oplus({\mathfrak{g}_{\bar{1}}})_{+},\qquad ({\mathfrak{g}_{\bar{1}}})_{\pm}\mathrel{\vcenter{:}}= {\mathfrak{g}_{\bar{1}}}\cap{\mathfrak{n}}^{\pm}.\] The subspaces \(({\mathfrak{g}_{\bar{1}}})_{-}\) and \(({\mathfrak{g}_{\bar{1}}})_{+}\) are isotropic, and \(B\) identifies them in the restricted sense, that is, it induces a nondegenerate pairing \(({\mathfrak{g}_{\bar{1}}})_{+}\times({\mathfrak{g}_{\bar{1}}})_{-}\longrightarrow\mathbb{C}\) so that \(({\mathfrak{g}_{\bar{1}}})_{+} \cong ({\mathfrak{g}_{\bar{1}}})_{-}^{\ast}\) (restricted dual). Let \((x_{a})\) be a basis of \(({\mathfrak{g}_{\bar{1}}})_{-}\) and \((\partial_{a})\) a basis of \(({\mathfrak{g}_{\bar{1}}})_{+}\) such that \[B(x_{a},x_{b})=0,\qquad B(\partial_{a},\partial_{b})=0,\qquad B(\partial_{a},x_{b})=\tfrac{1}{2}\delta_{ab}.\] With this notation, the Weyl algebra of \({\mathfrak{g}_{\bar{1}}}\) is generated by the \(x_{a}\) and \(\partial_{a}\), and identifies with the algebra of polynomial differential operators in the variables \(x_{a}\), where \(\partial_{a}\) acts as \(\partial/\partial x_{a}\).

The simple \({\mathbb{Z}}\)-graded \({\operatorname{Cl}}({\mathfrak{g}_{\bar{1}}})\)-module is \[M({\mathfrak{g}_{\bar{1}}})\mathrel{\vcenter{:}}= S(({\mathfrak{g}_{\bar{1}}})_{-})=\bigoplus_{k\geq 0}S^{k}(({\mathfrak{g}_{\bar{1}}})_{-})\cong {\mathbb{C}}[x_{1},x_{2},\ldots],\] which is the polynomial algebra in countably many variables. Its action extends componentwise to \(\widehat{{\operatorname{Cl}}}({\mathfrak{g}_{\bar{1}}})\), although \(M({\mathfrak{g}_{\bar{1}}})\) is in general not a \(\widehat{{\operatorname{Cl}}}({\mathfrak{g}_{\bar{1}}})\)-module. In addition, the module \(M({\mathfrak{g}_{\bar{1}}})\) has a natural \({\mathbb{Z}}\)-grading coming from the \({\mathbb{Z}}\)-grading of \({\mathfrak{g}_{\bar{1}}}\). This \({\mathbb{Z}}\)-grading is by construction compatible with the action of \({\operatorname{Cl}}({\mathfrak{g}_{\bar{1}}})\).

Additionally, \(M({\mathfrak{g}_{\bar{1}}})\) carries a Hermitian form \(\langle\cdot,\cdot\rangle_{M({\mathfrak{g}_{\bar{1}}})}\) such that the operators \(x_{a}\) and \(\partial_{a}\) are adjoint to each other, that is, \(\langle x_{a}v,w\rangle_{M({\mathfrak{g}_{\bar{1}}})} = \langle v, \partial_{a}w\rangle_{M({\mathfrak{g}_{\bar{1}}})}\) for any \(v,w \in M({\mathfrak{g}_{\bar{1}}})\). Concretely, let \(\alpha = (\alpha_1,\alpha_2,\dots)\) be a multi-index with finite support. We write \[x^\alpha \mathrel{\vcenter{:}}= \prod_{i\ge 1} x_i^{\alpha_i}, \qquad \alpha! \mathrel{\vcenter{:}}= \prod_{i\ge 1} \alpha_i!.\] Then the Hermitian inner product on \(\mathbb{C}[x_1,x_2,\dots]\) is given by \[\langle x^\alpha, x^\beta \rangle_{M({\mathfrak{g}_{\bar{1}}})} = \delta_{\alpha,\beta}\alpha!.\] Here we use the convention that Hermitian forms are conjugate-linear in the first argument and linear in the second.

In what follows, we regard \(M({\mathfrak{g}_{\bar{1}}})\) as a module over \({\operatorname{Cl}}({\mathfrak{g}_{\bar{1}}})\) equipped with the Hermitian form \(\langle\cdot,\cdot\rangle_{M({\mathfrak{g}_{\bar{1}}})}\).

The assignment \(x_{a} \mapsto \partial_{a}\) and \(\partial_{a} \mapsto x_{a}\) extends uniquely to a \(^{\ast}\)-operation \((\cdot)^{\ast}\) on \({\operatorname{Cl}}({\mathfrak{g}_{\bar{1}}})\), that is, a conjugate-linear map such that \((x^{\ast})^{\ast}=x\) and \((xy)^{\ast}=y^{\ast}x^{\ast}\) for all \(x,y \in {\operatorname{Cl}}({\mathfrak{g}_{\bar{1}}})\). This allows us to regard \(M({\mathfrak{g}_{\bar{1}}})\) as a \(^{\ast}\)-module over \({\operatorname{Cl}}({\mathfrak{g}_{\bar{1}}})\), that is, \[\langle xv,w \rangle_{M({\mathfrak{g}_{\bar{1}}})}= \langle v,x^{\ast}w \rangle_{M({\mathfrak{g}_{\bar{1}}})}\] for all \(x \in {\operatorname{Cl}}({\mathfrak{g}_{\bar{1}}})\) and \(v,w \in M({\mathfrak{g}_{\bar{1}}})\).

Recall the Lie superalgebra morphism \(\gamma_{{\mathfrak{g}_{\bar{1}}}}'\colon {\mathfrak{g}_{\bar{0}}}\to\widehat{{\operatorname{Cl}}}({\mathfrak{g}_{\bar{1}}})\) from Lemma 28, which defines a diagonal embedding of \({\mathfrak{g}_{\bar{0}}}\) into \(\widehat{{\mathcal{W}}}({\mathfrak{g}})\). Via this map, we regard \(M({\mathfrak{g}_{\bar{1}}})\) as an \({\mathfrak{g}_{\bar{0}}}\)-module and denote the corresponding representation by \(\pi_{M({\mathfrak{g}_{\bar{1}}})}\). Recall \(\rho_{\bar1}\mathrel{\vcenter{:}}= \rho_{\bar0}-\rho.\)

Let \(\omega\) be a conjugate-linear anti-involution of \({\mathfrak{g}}\) so that \(\omega(x)=x^{\ast}\) for all \(x \in {\mathfrak{g}_{\bar{1}}}\). We assume that \(\omega\) is consistent, that is, \(\omega({\mathfrak{g}}^{\alpha})={\mathfrak{g}}^{-\alpha}\) for all \(\alpha \in \Delta\). Then \(\pi_{M({\mathfrak{g}_{\bar{1}}})}\) is \(\omega\)-contravariant.

Combining Lemma 28 and Remark 22, one has the following.

The following corollary is immediate.

If \(M\) is a \({\mathfrak{g}}\)-supermodule we have a natural action of \({\mathcal{W}}({\mathfrak{g}},{\mathfrak{g}_{\bar{0}}})\) on \(M\otimes M({\mathfrak{g}_{\bar{1}}})\). This action extends componentwise to an action of the completion \(\widehat{{\mathcal{W}}}({\mathfrak{g}},{\mathfrak{g}_{\bar{0}}})\) on \(\widehat{\operatorname{End}}(M\otimes M({\mathfrak{g}_{\bar{1}}}))\). In this case, we consider the relative cubic Dirac operator as the image under this representation, that is, \[\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}} \in \widehat{\operatorname{End}}(M \otimes M({\mathfrak{g}_{\bar{1}}})).\]

We examine this action in the next section for highest weight supermodules.

4.3 Action of the Cubic Dirac Operator↩︎

In this section we derive an explicit formula for \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\), and prove that \(\ker \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\neq\{0\}\). We begin by introducing highest weight supermodules of \({\mathfrak{g}}\).

4.3.1 Highest Weight Supermodules↩︎

Fix a simple system \(\require{upgreek} \Uppi\) such that \({\mathfrak b}\mathrel{\vcenter{:}}={\mathfrak h}\oplus{\mathfrak{n}}^{+}\) and \({\mathfrak{g}}={\mathfrak{n}}^{-}\oplus{\mathfrak h}\oplus{\mathfrak{n}}^{+}\). Let \(\rho\) denote the corresponding Weyl vector.

A \({\mathfrak{g}}\)-supermodule \(M\) is called a weight supermodule if \({\mathfrak h}\) acts semisimply on \(M\), that is, \[M=\bigoplus_{\mu\in{\mathfrak h}^{\ast}}M^{\mu},\qquad M^{\mu}\mathrel{\vcenter{:}}=\{v\in M:hv=\mu(h)v\;\text{for all }h\in{\mathfrak h}\},\] with \(\dim M^{\mu}<\infty\) for all \(\mu\in{\mathfrak h}^{\ast}\). A weight supermodule \(M\) is called a highest weight supermodule if there exist \(\Lambda\in{\mathfrak h}^{\ast}\) and \(v_{\Lambda}\in M\) such that \[{\mathfrak{n}}^{+}v_{\Lambda}=0,\qquad hv_{\Lambda}=\Lambda(h)v_{\Lambda}\;\text{for all }h\in{\mathfrak h},\qquad {\mathfrak U}({\mathfrak{g}})v_{\Lambda}=M.\] \(M\) is called integrable if \(X_{\alpha}\) and \(Y_{\alpha}\) act locally nilpotently for all even real simple roots \(\alpha\). In particular, \(\Lambda(h_{\alpha})\in {\mathbb{Z}}_{\geq 0}\) for all even real simple roots \(\alpha\).

Any highest weight supermodule of \({\mathfrak{g}}\) has a canonical \({\mathbb{Z}}\)-grading induced by the principal \({\mathbb{Z}}\)-grading of \({\mathfrak{g}}\) (see 28 ). This \({\mathbb{Z}}\)-grading will be used without further mention.

On every highest weight \({\mathfrak{g}}\)-supermodule \(M\) the corrected quadratic Casimir \(\Omega_{{\mathfrak{g}}}\) is a well-defined operator and acts by a scalar.

Examples of highest weight supermodules are given by Verma supermodules. For each \(\Lambda \in {\mathfrak h}^{\ast}\), we define the Verma supermodule \(M_{{\mathfrak b}}(\Lambda)\) of highest weight \(\Lambda\) by the induced supermodule \[M_{{\mathfrak b}}(\Lambda) \mathrel{\vcenter{:}}= {\mathfrak U}({\mathfrak{g}}) \otimes_{{\mathfrak U}({\mathfrak b})} {\mathbb{C}}_{\Lambda},\] where \({\mathbb{C}}_{\Lambda}\) is the one-dimensional \({\mathfrak b}\)-module with an even generator \(v\) such that \({\mathfrak{n}}^{+}v=0\) and \(hv=\Lambda(h)v\) for all \(h \in {\mathfrak h}\). The vector \(v\) is called the highest weight vector of \(M_{{\mathfrak b}}(\Lambda)\). Since \({\mathfrak b}\) is fixed, we omit the corresponding subscript in what follows.

4.3.2 Action on \(M\otimes M({\mathfrak{g}_{\bar{1}}})\)↩︎

Let \(M\) be a highest weight \({\mathfrak{g}}\)-supermodule of highest weight \(\Lambda\), and consider \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\in\widehat{\operatorname{End}}(M\otimes M({\mathfrak{g}_{\bar{1}}}))\). Since \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\in\widehat{{\mathcal{W}}}({\mathfrak{g}},{\mathfrak{g}_{\bar{0}}})\), it is \({\mathfrak{g}_{\bar{0}}}\)-invariant. Hence one has the following.

Next, we apply Theorem 24 to obtain an explicit formula for \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\). Since \({\mathfrak{g}}_{0}={\mathfrak h}=({\mathfrak{g}}_{\bar0})_{0}\) by the \({\mathbb{Z}}\)-grading 28 , one has \(\operatorname{str}(\operatorname{ad}_{{\mathfrak{g}}_{0}}(\Omega_{{\mathfrak{g}}_{0}}))=0\) and \(\operatorname{str}(\operatorname{ad}_{({\mathfrak{g}_{\bar{0}}})_{0}}(\Omega_{({\mathfrak{g}_{\bar{0}}})_{0}}))=0\). Together with Lemma 32, this shows that \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\) is a well-defined operator on \(M\otimes M({\mathfrak{g}_{\bar{1}}})\) and yields the following proposition.

This shows that \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\) measures the difference between the Casimir actions of \({\mathfrak{g}}\) and \({\mathfrak{g}_{\bar{0}}}\).

These results imply that the relative cubic Dirac operator has nontrivial kernel on every highest weight supermodule.

In general, \(\ker \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\neq\ker \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\) and equality holds for example in the unitarizable case (see Section 4.5). Nevertheless, Proposition 29 allows explicit computations of \(\ker \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\). We perform these calculations in one example: integrable highest weight supermodules of \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\).

4.4 An Example: Integrable Highest Weight Supermodules of \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\)↩︎

We consider the affine Lie superalgebras \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\), which constitute the simplest examples of affine Kac–Moody superalgebras (see Section 4.1) with a nontrivial odd part.

4.4.1 The affine Lie superalgebra \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\)↩︎

The affine Lie superalgebra \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\) is the untwisted affinization of the basic classical Lie superalgebra \({\mathfrak{osp}}(1\vert 2n)\). Its structure and representation theory are described in detail in [40], [42], [44], [46]. We briefly recall its construction and structural properties.

Let \(V=V_{\bar0}\oplus V_{\bar1}\) be a \({\mathbb{Z}}_{2}\)-graded vector space with \(\dim V_{\bar0}=1\) and \(\dim V_{\bar1}=2n\), endowed with a nondegenerate even supersymmetric bilinear form. The corresponding orthosymplectic Lie superalgebra is denoted by \({\mathfrak{osp}}(1\vert 2n)\); in the notation of Example 1, it is the \(\varepsilon\)-orthogonal Lie algebra of \(V\) for \(\Gamma={\mathbb{Z}}_{2}\) and \(\varepsilon\) given by parity. In Kac’s notation, this is the simple Lie superalgebra of type \(B(0,n)\). Its even part identifies canonically with \(\mathfrak{sp}(V_{\bar{1}})\cong \mathfrak{sp}(2n)\) and its odd part is the standard simple \(2n\)-dimensional module of \(\mathfrak{sp}(2n)\). Moreover, \({\mathfrak{osp}}(1\vert 2n)\) is simple and admits a nondegenerate even supersymmetric invariant bilinear form \(B(\cdot,\cdot)\) which is unique up to an overall scalar, hence it is basic classical. A canonical choice is obtained from the defining representation via the supertrace (cf. Lemma 4): \[B(x,y) \mathrel{\vcenter{:}}= -\operatorname{str}(xy), \qquad x,y\in{\mathfrak{osp}}(1\vert 2n).\]

Let \({\mathfrak h}_{\mathfrak{sp}(2n)}=\left\{\operatorname{diag}(h_{1},\ldots,h_{n},-h_{1},\ldots,-h_{n})\right\}\subseteq\mathfrak{sp}(2n)\) be a Cartan subalgebra of \(\mathfrak{sp}(2n)\), viewed as a subalgebra of the even part of \(\mathfrak{osp}(1\vert 2n)\). It is then a Cartan subalgebra of \({\mathfrak{osp}}(1\vert 2n)\), denoted by \({\mathfrak h}_{{\mathfrak{osp}}}\). For \(1\leq i\leq n\), define \(\varepsilon_{i}\in{\mathfrak h}^{\ast}_{{\mathfrak{osp}}}\) by \(\varepsilon_{i}(h)=h_{i}\) for \(h\in{\mathfrak h}_{{\mathfrak{osp}}}\). Then the root system of \({\mathfrak{osp}}(1\vert 2n)\) is \(\Delta_{{\mathfrak{osp}}}=(\Delta_{{\mathfrak{osp}}})_{\bar{0}}\sqcup (\Delta_{{\mathfrak{osp}}})_{\bar{1}}\), where \[(\Delta_{{\mathfrak{osp}}})_{\bar0}=\{\pm\varepsilon_{i}\pm\varepsilon_{j},\;\pm2\varepsilon_{i}:i\neq j\},\qquad (\Delta_{{\mathfrak{osp}}})_{\bar1}=\{\pm\varepsilon_{i}\}.\] The form \(B\) restricts to a nondegenerate symmetric bilinear form on \({\mathfrak h}_{{\mathfrak{osp}}}\). Via this restriction, we identify \({\mathfrak h}_{{\mathfrak{osp}}}\) with \({\mathfrak h}^{\ast}_{{\mathfrak{osp}}}\). This induces a nondegenerate symmetric bilinear form on \({\mathfrak h}^{\ast}_{{\mathfrak{osp}}}\), again denoted by \(B\). With the above normalization one has \[B(\epsilon_i,\epsilon_j)=\tfrac12\delta_{ij}.\]

Importantly, every root has positive length \[\label{eq::positivity95roots95osp}B(\pm\epsilon_i,\pm\epsilon_i)=\tfrac12,\qquad B(\pm\epsilon_i\pm\epsilon_j,\pm\epsilon_i\pm\epsilon_j)=1,\qquad B(\pm2\epsilon_i,\pm2\epsilon_i)=2.\tag{33}\]

The affine Lie superalgebra associated with \({\mathfrak{osp}}(1\vert 2n)\) is \(\widehat{{\mathfrak{osp}}}(1\vert 2n) = ({\mathfrak{osp}}(1\vert 2n)\otimes_{{\mathbb{C}}} {\mathbb{C}}[t,t^{-1}]) \oplus {\mathbb{C}}K\oplus {\mathbb{C}}D\) with parity decomposition \[\widehat{{\mathfrak{osp}}}(1\vert 2n)_{\bar0} = ({\mathfrak{osp}}(1\vert 2n)_{\bar0}\otimes_{{\mathbb{C}}} {\mathbb{C}}[t,t^{-1}]) \oplus {\mathbb{C}}K\oplus {\mathbb{C}}D, \qquad \widehat{{\mathfrak{osp}}}(1\vert 2n)_{\bar1} = {\mathfrak{osp}}(1\vert 2n)_{\bar1}\otimes_{{\mathbb{C}}} {\mathbb{C}}[t,t^{-1}]\] and the commutation relations from 31 . Recall that \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\) is a symmetrizable Kac–Moody superalgebra, equipped with the nondegenerate invariant supersymmetric bilinear form induced by \(B\), again denoted by \(B\); see Section 4.1. As above, let \({\mathfrak h}\mathrel{\vcenter{:}}= {\mathfrak h}_{{\mathfrak{osp}}}\oplus{\mathbb{C}}K\oplus{\mathbb{C}}D\) be the affine Cartan subalgebra of \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\), with dual space \({\mathfrak h}^{\ast}={\mathfrak h}^{\ast}_{{\mathfrak{osp}}}\oplus{\mathbb{C}}\kappa\oplus{\mathbb{C}}\delta\) as in 32 . Then \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\) admits the root space decomposition with root system \[\Delta=\{\alpha+n\delta:\alpha\in\Delta_{{\mathfrak{osp}}},\;n\in{\mathbb{Z}}\}\cup\{n\delta:n\in{\mathbb{Z}}\setminus\{0\}\}.\]

The set of roots \(\Delta\) of \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\) decomposes into real and imaginary roots. The real roots are \[\pm\epsilon+n\delta,\qquad \pm2\epsilon+n\delta,\qquad n\in{\mathbb{Z}},\] where \(\epsilon+n\delta\) are odd roots and \(2\epsilon+n\delta\) are even roots. The imaginary roots are the nonzero multiples \[n\delta,\qquad n\in{\mathbb{Z}}\setminus\{0\},\] which are even and have multiplicity one.

The standard simple system is \[\require{upgreek} \Uppi_{{\text{st}}}=\{\delta-2\epsilon_{1},\epsilon_{1}-\epsilon_{2},\ldots,\epsilon_{n-1}-\epsilon_{n},\epsilon_{n}\}.\] Let \(\rho_{{\mathfrak{osp}}}\) denote the corresponding Weyl vector of \({\mathfrak{osp}}(1\vert 2n)\). Then the affine Weyl vector is \(\rho=\rho_{{\mathfrak{osp}}}+h^{\vee}\kappa\), where \(h^{\vee}=n+\tfrac12\) is the dual Coxeter number.

The Weyl group \(W\) of \({\widehat{{\mathfrak{osp}}}(1\vert 2)}\) is the subgroup of \(\operatorname{GL}({\mathfrak h}^{\ast})\) generated by the even reflections \[r_{\beta}(\alpha) = \alpha-B(\alpha,\beta)\beta, \qquad \beta\;\text{real even root}.\] The form \(B\) is invariant under the action of \(W\). Moreover, the simple systems of \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\) form a single orbit under the Weyl group. Consequently, \(\require{upgreek} \Uppi_{{\text{st}}}\) is the unique, up to \(W\) action, simple system of \({\mathfrak{g}}\).

4.4.2 Action on Integrable Highest Weight Supermodules↩︎

Fix a simple system \(\require{upgreek} \Uppi = \{\alpha_{1}, \ldots, \alpha_{n+1}\}\). Let \(\Delta^{+}\) be the associated positive system, and \({\mathfrak b}\) the Borel subalgebra.

Any element \(\Lambda\in{\mathfrak h}^{\ast}\) is uniquely of the form \[\Lambda=\Lambda\vert_{{\mathfrak h}_{{\mathfrak{osp}}}}+k\kappa+d_{0}\delta.\] We call \(\Lambda\vert_{{\mathfrak h}_{{\mathfrak{osp}}}}\) the \({\mathfrak h}\)-weight and \(k\) the level of \(\Lambda\). With this notation, Kac and Wakimoto classified the simple integrable highest weight \(\widehat{{\mathfrak{osp}}}(1\vert 2n)\)-supermodules in [40].

In what follows, let \(L_{{\mathfrak b}}(\Lambda)\) be a fixed integrable highest weight supermodule. We determine \(\ker \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\) by solving (Proposition 29) \[B(\Lambda+\rho,\Lambda+\rho)=B(\mu+\rho_{\bar{0}},\mu+\rho_{\bar{0}}),\] where \(\mu\) is a weight of \(L_{{\mathfrak b}}(\Lambda)\otimes M({\mathfrak{g}_{\bar{1}}})\). The weights of \(M\otimes M({\mathfrak{g}_{\bar{1}}})\) are parametrized as follows. Let \(Q\) be the root lattice generated by the simple roots \(\alpha_{i}\), and let \(Q_{+}\) be the submonoid generated by positive linear combinations of the \(\alpha_{i}\). Then the set of weights \(\mathcal{P}_{L_{{\mathfrak b}}(\Lambda)}\) of \(L_{{\mathfrak b}}(\Lambda)\) is contained in \(\{\Lambda-\gamma:\gamma\in Q_{+}\}\). On the other hand, by Corollary 5, the weights of \(M({\mathfrak{g}_{\bar{1}}})\) are contained in \(\{-\rho_{\bar1}-\sum_{\alpha\in\Delta_{\bar1}^{+}}n_{\alpha}\alpha:n_{\alpha}\in{\mathbb{Z}}_{+}\}\) with \(\rho_{\bar{1}}\mathrel{\vcenter{:}}=\rho_{\bar{0}}-\rho\). Hence the weights of \(M\otimes M({\mathfrak{g}_{\bar{1}}})\) are contained in \[\label{eq::weight95set} \Bigl\{\Lambda-\rho_{\bar1}-\gamma-\sum_{\alpha\in\Delta_{\bar1}^{+}}n_{\alpha}\alpha:\gamma\in Q_{+},\;n_{\alpha}\in{\mathbb{Z}}_{+}\Bigr\}.\tag{34}\]

The preceding argument relies on the positivity property \(B(\alpha,\alpha)\geq 0\) for all simple roots \(\alpha\). It is this property which makes it possible to determine the equation \(B(\Lambda+\rho,\Lambda+\rho)=B(\mu+\rho_{\bar0},\mu+\rho_{\bar0})\) uniquely. For general basic classical Lie superalgebras, this mechanism breaks down: isotropic odd positive roots occur, and the form \(B\) is in general highly indefinite. Consequently, the equation \(B(\Lambda+\rho,\Lambda+\rho)=B(\mu+\rho_{\bar0},\mu+\rho_{\bar0})\) typically admits many solutions. Indeed, if \(\mu=\nu-\rho_{\bar1}\) and \(\gamma\mathrel{\vcenter{:}}=\Lambda-\nu\), then the condition is equivalent to \[2B(\Lambda+\rho,\gamma)=B(\gamma,\gamma).\] If \(\gamma\) is odd isotropic, this reduces to the atypicality condition for \(\Lambda\). Thus the determination of \(\ker \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}^{2}\) requires new methods.

4.5 Unitarizable Highest Weight Supermodules↩︎

The relative cubic Dirac operator \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) for a symmetrizable Kac–Moody superalgebra \({\mathfrak{g}}\) is directly relevant to the study of unitarizable highest weight supermodules. It acts skew-adjointly on \(M\otimes M({\mathfrak{g}_{\bar{1}}})\) and yields a Dirac inequality, hence a necessary condition for unitarity. It also provides a relation between \(\ker \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) and Lie superalgebra cohomology.

In what follows, \({\mathfrak{g}}\) denotes a symmetrizable Kac–Moody superalgebra. We retain the notation of Section 4.1 and Section 4.2. We begin with unitarizable supermodules.

4.5.1 Unitarizable Supermodules of Kac–Moody Superalgebras↩︎

Unitarizable highest weight supermodules of affine Kac–Moody superalgebras were studied by Jakobsen in [16]. We follow his conventions.

Let \(M\) be a highest weight supermodule of \({\mathfrak{g}}\) for a fixed Borel subalgebra \({\mathfrak b}\subset {\mathfrak{g}}\). We equip \(M\) with a Hermitian form. A conjugate-linear map \(\omega : {\mathfrak{g}}\to {\mathfrak{g}}\) is called an anti-involution if \(\omega^{2} = \operatorname{id}_{{\mathfrak{g}}}\) and \(\omega([x,y]) = [\omega(y), \omega(x)]\) for all \(x,y \in {\mathfrak{g}}\). It is consistent if \(\omega({\mathfrak{g}}^{\alpha}) = {\mathfrak{g}}^{-\alpha}\) for all \(\alpha \in \Delta\). In what follows, \(\omega\) denotes a consistent conjugate-linear anti-involution. It extends to \({\mathfrak U}({\mathfrak{g}})\) in the obvious way.

Any real form of \({\mathfrak{g}}\) determines a conjugate-linear anti-involution. As in the case of Kac–Moody algebras, the following proposition holds (see for example [16], [43]).

We call a \(\Lambda \in {\mathfrak h}^{\ast}\) such that \(\Lambda(h)=\overline{\Lambda(\omega(h))}\) symmetric. In what follows, we assume that \(M\) is a highest weight \({\mathfrak{g}}\)-supermodule with symmetric highest weight \(\Lambda\). Let \(\langle\cdot, \cdot\rangle\) denote the associated unique contravariant Hermitian form on \(M\).

Explicit examples of unitarizable supermodules over affine Lie superalgebras are constructed in [16].

4.5.2 Dirac inequality↩︎

Let \(M\) be a \(\omega\)-unitarizable \({\mathfrak{g}}\)-supermodule, where \(\omega\) is a consistent conjugate-linear anti-involution of \({\mathfrak{g}}\), and let \(\langle\cdot,\cdot\rangle_{M}\) be the corresponding positive-definite Hermitian form. We let \({\mathfrak{g}_{\bar{0}}}\) act on \(M\otimes M({\mathfrak{g}_{\bar{1}}})\) by \[x(v\otimes P)\mathrel{\vcenter{:}}= xv\otimes P+v\otimes \pi_{M({\mathfrak{g}_{\bar{1}}})}(x)P,\qquad x\in{\mathfrak{g}_{\bar{0}}},\;v\in M,\;P\in M({\mathfrak{g}_{\bar{1}}}).\] Moreover, we equip \(M\otimes M({\mathfrak{g}_{\bar{1}}})\) with the Hermitian form \[\langle v\otimes P,w\otimes Q\rangle\mathrel{\vcenter{:}}=\langle v,w\rangle_{M}\langle P,Q\rangle_{M({\mathfrak{g}_{\bar{1}}})}.\] Lemma 30 then yields the following.

Since the definition of \(\operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) is independent of the choice of basis, the conventions of Section 4.2 give \[\label{eq::explicit95form95Dirac95g95even} \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}=2\sum_{a}(x_{a}\otimes \partial_{a}-\partial_{a}\otimes x_{a}) \in \widehat{\operatorname{End}}(M\otimes M({\mathfrak{g}_{\bar{1}}})).\tag{35}\]

A polarization \({\mathfrak{g}_{\bar{1}}}={\mathfrak{g}_{\bar{1}}}_{-}\oplus{\mathfrak{g}_{\bar{1}}}_{+}\) is called \(\omega\)-adapted of sign \(\sigma\in\{\pm 1\}\) if there exist \(B\)-dual homogeneous bases \((x_{a})\) of \(({\mathfrak{g}_{\bar{1}}})_{-}\) and \((\partial_{a})\) of \(({\mathfrak{g}_{\bar{1}}})_{+}\), normalized by \(B(\partial_{a},x_{b})=\tfrac12\delta_{ab}\), such that \(\omega(x_{a})=\sigma\partial_{a}\) and \(\omega(\partial_{a})=\sigma x_{a}\) for all \(a\). Such a basis is called an \(\omega\)-adapted basis of sign \(\sigma\). An explicit example is given in [6] for \({\mathfrak{sl}}(m\vert n)\), and extends directly to \(\widehat{{\mathfrak{sl}}}(m\vert n)\) via Example 6.

As a direct consequence of 35 and Lemma 30, one obtains the following.

The lemma yields the Dirac inequality.

4.5.3 Cohomology↩︎

We now relate \(\ker \operatorname{D}_{{\mathfrak{g}},{\mathfrak{g}_{\bar{0}}}}\) to the \({\mathfrak{g}}_{+1}\)-cohomology for a class of affine Lie superalgebras. Let \({\mathfrak{g}_{\bar{1}}}=({\mathfrak{g}}_{\bar1})_{+}\oplus({\mathfrak{g}}_{\bar1})_{-}\) be the decomposition from Section 4.2, where \(({\mathfrak{g}}_{\bar1})_{+}\) is spanned by the \(\partial_{a}\) and \(({\mathfrak{g}}_{\bar1})_{-}\) by the \(x_{a}\) such that \(({\mathfrak{g}}_{\bar{1}})_{\pm}\) are abelian. For brevity, write \({\mathfrak{g}}_{+1}\mathrel{\vcenter{:}}=({\mathfrak{g}}_{\bar1})_{+}\) and \({\mathfrak{g}}_{-1}\mathrel{\vcenter{:}}=({\mathfrak{g}}_{\bar1})_{-}\). Assume that the \({\mathfrak{g}_{\bar{0}}}\)-action is compatible, that is, \({\mathfrak{g}}_{+1}\cong{\mathfrak{g}}_{-1}\) as \({\mathfrak{g}_{\bar{0}}}\)-modules. The main example is the Kac–Moody superalgebra associated to \({\mathfrak{sl}}(m\vert n)\), with \(\widehat{{\mathfrak{sl}}}(m\vert n)_{0}\mathrel{\vcenter{:}}=\widehat{{\mathfrak{sl}}}(m\vert n)_{\bar0}\) and \(\widehat{{\mathfrak{sl}}}(m\vert n)_{\pm1}\mathrel{\vcenter{:}}= {\mathfrak{sl}}(m\vert n)_{\pm1}\otimes_{{\mathbb{C}}}{\mathbb{C}}[t,t^{-1}]\), where \({\mathfrak{sl}}(m\vert n)_{-1}\) and \({\mathfrak{sl}}(m\vert n)_{+1}\) denote the lower and upper odd block matrices in the standard realization.

Let \(M\) be a highest weight \({\mathfrak{g}}\)-supermodule. Assume that \(M\) is discretely decomposable as an \({\mathfrak{g}_{\bar{0}}}\)-module and \(\omega\)-unitarizable for a conjugate-linear anti-involution \(\omega\) satisfying \(\omega(x_{a})=\partial_{a}\) and \(\omega(\partial_{a})=x_{a}\), that is, such that the polarization is \(\omega\)-adapted of sign \(+1\). The case of sign \(-1\) is analogous. In particular, \(M\otimes M({\mathfrak{g}_{\bar{1}}})\) is a \(\omega|_{{\mathfrak{g}_{\bar{0}}}}\)-unitarizable \({\mathfrak{g}_{\bar{0}}}\)-module for the diagonal action of Section 4.2 and it is discretely decomposable as an \({\mathfrak{g}_{\bar{0}}}\)-module. Indeed, \(M\) is discretely decomposable by assumption, and \(M({\mathfrak{g}_{\bar{1}}})\) is a \(\omega\)-unitarizable \({\mathfrak{g}_{\bar{0}}}\)-weight module whose weights are bounded above (Corollary 5). Hence every nonzero \({\mathfrak{g}_{\bar{0}}}\)-submodule of \(M({\mathfrak{g}_{\bar{1}}})\) contains a highest weight vector, and the positive-definite contravariant form yields an orthogonal decomposition into highest weight submodules. Since \(M({\mathfrak{g}_{\bar{1}}})\) is a weight module, this decomposition is an algebraic direct sum.

We define the cochain complex \[C^{\ast}({\mathfrak{g}}_{+1};M) \mathrel{\vcenter{:}}= \bigoplus_{i}C^{i}({\mathfrak{g}}_{+1}, M) \mathrel{\vcenter{:}}= \bigoplus_{i} M \otimes S^{i}({\mathfrak{g}}_{-1}), \qquad d \mathrel{\vcenter{:}}= \sum_{a} \partial_{a} \otimes x_{a} \in \widehat{\operatorname{End}}(C^{\ast}({\mathfrak{g}}_{+1};M)).\] Here, \(d^{2}=0\) and \(d\) commutes with the action of \({\mathfrak{g}_{\bar{0}}}\) by assumption. The associated cohomology group is denoted by \(\operatorname{H}^{\ast}({\mathfrak{g}}_{+1},M)\). It is naturally a \({\mathfrak{g}_{\bar{0}}}\)-module.

References↩︎

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1. Named after Batalin, Fradkin and Vilkovisky [27]–[29].↩︎

2. See also [33], [34].↩︎