1 Introduction↩︎

There is considerable current interest in physical applications of Lie colour algebras and superalgebras, i.e., Lie \((\Gamma, \omega)\)-algebras for general abelian groups \(\Gamma\) with commutative factors \(\omega\) [1]–[6]. Several groups of researchers have worked intensively on the subject in recent years. They have established a Lie colour (super)algebraic framework for parastatistics [7]–[13] (also see [14]), and discovered theoretically measurable effects (see [15] for a brief review and further references) of \({\mathbb{Z}}_2\times{\mathbb{Z}}_2\)-graded paraparticles [11], [12], and more generally of \({\mathbb{Z}}_2^n\)-graded paraparticles [7]. Colour supersymmetries [13], [16]–[18] were further developed in quantum mechanics [7], [19]–[23] and quantum field theory [24], [25]. Some \({\mathbb{Z}}_2\times{\mathbb{Z}}_2\)-graded extensions of the Liouville, Sin-Gordon, and Sinh–Gordon theories [20] were obtained, and KdV and mKdV hierarchies of equations [26] were constructed by Drinfeld-Sokolov reduction for affine Lie colour superalgebras. Another interesting discovery [27] was the \({\mathbb{Z}}_2\times {\mathbb{Z}}_2\)-graded Lie algebra symmetries of the non-relativistic analogues of the Dirac equation known as Lévy-Leblond equations (see [28] and references therein for further developments). The study of physical applications of Lie colour (super)algebras continues to be a very active research area today, see, e.g., [20], [29]–[31] (see [6] for a brief review and references therein).

Various other aspects of Lie colour (super)algebras have also been developed. For example, small Lie colour (super)algebras with grading groups like \(\Gamma={\mathbb{Z}}_2\times{\mathbb{Z}}_2\) and commutative factor \(\omega(\alpha, \beta)= (-1)^{\alpha_1\beta_2\pm \alpha_2 \beta_1}\) [9], [28], [31]–[34], and some associated affine Lie colour (super)algebras [35], [36], have been studied. The paper [37] gave a classification and construction of the finite dimensional simple modules for the Lie colour algebra \({\mathfrak {sl}}_2^c\); and the paper [38] explored general aspects of representations of generic Lie colour (super)algebras from a ring theoretical perspective. Lie colour (super)algebras were also brought into play in various areas of algebra since the late 90s, e.g, the study of ring theoretical aspects of graded associative and Hopf algebras [39]–[41], and group gradings of Lie algebras [42].

We also mention that a cohomology theory for Lie colour (super)algebras was introduced in [5] which has since been widely studied by many researchers. Also, Casimir operators of Lie colour algebras have been constructed very recently [43] (also see [6] for the case of the general linear Lie colour superalgebra in terms of graded tensor calculus [4]).

One should observe the following fundamental fact. In any quantum physics problem with a Lie colour (super)algebra symmetry, the grading is in general the manifestation of the statistics, e.g., parastatistics [44] or generalisations, of the states of the quantum system under study. Thus the grading is in fact physical; one should not do away with it by Klein type transformations – Scheunert’s cocycle twisting [3], [40], [45] and Majid’s bosonisation [46].

The interesting results of recent years call for a systematical development of the theory of Lie colour (super)algebras and related algebraic structures. We embarked on this in the recent publication [6], where a comprehensive treatment of the representation theory and invariant theory of the general linear Lie colour superalgebras and associated colour analogue of algebraic groups were developed. An interesting noncommutative geometry emerged [6], which led to a realisation [6] of simple tensor modules in terms of “holomorphic” sections of some line bundles [47], [48] on noncommutative flag varieties.

The present paper continues this endeavour. Our main aim here is to construct quantised universal enveloping algebras of Lie colour (super)algebras and their affine analogues, generalising the theory of quantum groups [49]–[52] and quantum supergroups [53]–[60] to the colour setting.

There are examples with genesis of “colour quantum groups” in the mathematical physics literature. In [61], a quantum algebra was obtained from \({\rm{U}}_q({\mathfrak {sl}}_3)\) by performing certain Klein transformations in the spirit of [62], [63], whose \(q\to -1\) limit gives rise to \({\mathbb{Z}}_2\times{\mathbb{Z}}_2\)-graded Lie colour algebras. The study [11], [12], [15] of measurable effects of \({\mathbb{Z}}_2\times{\mathbb{Z}}_2\)-graded paraparticles revealed connections with some colour quantum supergroups at roots of unity (see [15] in particular). It appears that one should place these examples in a context of colour quantum groups in order to get an in depth understanding of them.

Let us briefly describe the content and main results of this paper.

(\(i\)). Lie and affine colour algebras realising the Cartan-Weyl paradigm. Recall that the structures of semi-simple Lie (super)algebras and affine Lie (super)algebras are described by root systems, and their representation theories are developed through the study of weight modules. We say that such Lie (affine) algebras and superalgebras fulfil the Cartan-Weyl paradigm.

We re-examine the Cartan-Weyl paradigm in the colour context, the key requirement of which is the existence of Cartan subalgebras which are homogeneous of degree \(0\) (see Section 3.1 for explanation in more precise terms). We demonstrate that the paradigm holds for some Lie colour superalgebras, such as the general and special linear Lie colour superalgebras [6], symplectic Lie colour algebras, and etc., but fails for others. We analyse the structure of the classical Lie colour superalgebras, showing in Theorem 7 that the orthosymplectic Lie colour superalgebra \({\mathfrak {osp}}(V;\kappa)\) fulfils the Cartan-Weyl paradigm if and only if there exists at most one \(2\)-torsion element \(\alpha\in \Gamma\) (i.e., \(\alpha + \alpha=0\)) such that the homogeneous subspace \(V_\alpha \subset V\) of degree \(\alpha\) is odd dimensional.

Our analysis of the classical Lie colour algebras and associated affine Lie colour algebras suggests a way to construct a class of (affine) Lie colour algebras which realise the Cartan-Weyl paradigm. Each such algebra is determined by a reduced Cartan matrix of finite or affine type, and a sequence of elements of \(\Gamma\). We present these algebras in terms of generators and relations in Section 3.6.

(\(ii\)). Quantised universal enveloping algebras of Lie and affine colour algebras. We construct quantised universal enveloping algebras of the Lie colour algebras and their affine analogues described in Section 3.6. These “colour quantum groups” generalise the Drinfeld-Jimbo type quantum (super)groups [50]–[52] [53], [55], [56], [59], [60] and recover the latter when the grading group is trivial. By construction, the colour quantum groups fulfil the Cartan-Weyl paradigm in the same way as the Drinfeld-Jimbo quantum (super)groups do.

We develop in Section 5 the quasi triangular colour Hopf algebraic structure (see Section 6.3) of colour quantum groups through the study of quantum doubles of Hopf \((\Gamma, \omega)\)-algebras (see 6.3). Therefore, there is a universal \(R\)-matrix for each colour quantum group, which satisfies the Yang-Baxter equation and is extremely important for applications.

(\(iii\)). Hopf \((\Gamma, \omega)\)-algebras and quasi triangularity. Since we consider colour quantum groups as Hopf \((\Gamma, \omega)\)-algebras with quasi triangularity, we treat the relevant notions and basic theory carefully in the Appendix (Section 6). In particular, the quantum double construction for Hopf \((\Gamma, \omega)\)-algebras is developed in detail in Section 6.3.

The material in Section 6 generalises the theory of quasi triangular Hopf algebras [51] and superalgebras [64] to the colour setting. This is anticipated conceptually (see further discussions below), but an explicit treatment does not appear to have been written down before.

Part (\(iii\)) above warrants some theoretical reflections, particularly about working with the explicit theories of Lie and Hopf colour (super)algebras instead of cocycle twisted/bosonised versions [3], [46] of them.

We study Lie colour superalgebras and colour quantum groups with a view to physical applications, particularly in quantum physics where the gradings of Lie colour algebras or colour quantum groups are related to statistics of quantum systems, as alluded to in Section 1.2. This requires us treat gradings directly, thus we need to work with the explicit theory (i.e, not cocycle twisted/bosonised).

Another point which we want to make is that the explicit theory reveals deep and intricate aspects of Lie and Hopf colour (super)algebras, which are completely lost in the cocycle twisted/bosonised theory, e.g., the failure of the Cartan-Weyl paradigm (see Theorem 7) in some cases of colour \({\mathfrak {so}}(V; \kappa)\). Also, the example given in [41], showing incompatibility of gradings of a Weyl colour algebra and Lie colour superalgebras realised in it, is too innate to the explicit theory.

Recall that the theory of Nichols algebras over braided vector spaces [65]–[67] (see [68] for a comprehensive treatment) provides an approach to constructing the quantum Borel subalgebras of quantum (super)groups [69]–[72]. [The full quantum groups can then be obtained from the quantum doubles of the de-bosonised[46] Nichols algebras.] We may re-interpret some of our constructions in Sections 5 in terms of Nichols algebras through bosonisation. This is alluded to in Section 5.3.2, where the crucial Lemma 18 is extracted from [71], [72] and [69], [70].

We believe that the development of Lie colour (super)algebras and colour quantum groups will significantly enrich modern Lie theory and its applications.

Since colour quantum groups are quasi triangular Hopf colour algebras with universal \(R\)-matrices, they have all the applications of ordinary quantum groups and quantum supergroups, in particular, to constructing knot invariants [73]–[77], and solving Jimbo’s equations [53], [78] to obtain integrable models in statistical mechanics [79].

Another interesting aspect of colour quantum groups is that they are naturally related to noncommutative geometry [80], [81]. Lie colour (super)algebras themselves are deeply rooted in new geometries [6], which may be related to the non-commutative para-manifolds introduced in [14] (for \(\Gamma={\mathbb{Z}}^n\) and a particular \(\omega\)) and \({\mathbb{Z}}_2^n\)-graded supermanifolds [82] (for \(\Gamma={\mathbb{Z}}_2^n\)). The noncommutative geometries related to colour quantum groups are quantum deformations of these new geometries, which are quite different from those of [80]. It will be very interesting to explore this area.

We conclude this Introduction with a comment on the presentation of this paper. To make the paper accessible to a broad audience interested in colour algebraic structures and/or their physical applications, we prove results by elementary means whenever possible, and present the pertinent steps of all proofs.

2 Background material↩︎

Foundational material on Lie \((\Gamma, \omega)\)-algebras can be found in e.g., [3], [6]. Here we discuss some aspects, which are pertinent for the study of classical Lie \((\Gamma, \omega)\)-algebras.

2.1 The category of \(\Gamma\)-graded vector spaces↩︎

Fix an additive abelian group \(\Gamma\), and let \(\omega: \Gamma\times \Gamma\longrightarrow{\mathbb{C}}^*\) be a commutative factor with the defining properties [83] \[\begin{align} &&\omega(\alpha, \beta) = \omega(\beta, \alpha)^{-1}, \tag{1} \\ &&\omega(\alpha, \beta+\gamma)= \omega(\alpha, \beta) \omega(\alpha, \gamma), \tag{2}\\ &&\omega(\alpha +\beta, \gamma) = \omega(\alpha, \gamma) \omega(\beta, \gamma), \quad \forall \alpha, \beta, \gamma. \tag{3} \end{align}\] Note in particular that \(\omega(\alpha, 0)=\omega(0, \alpha)=1\), and \(\omega(\alpha, \alpha)=\pm 1\) for all \(\alpha\). We let \[\begin{align} \Gamma^\pm=\{\alpha\in \Gamma\mid \omega(\alpha, \alpha)=\pm 1\}, \quad \Gamma_{{\mathbb{Z}}_2}= \{\alpha\in \Gamma\mid \alpha+\alpha=0\}. \end{align}\] It is evident that \(\Gamma^+\) and \(\Gamma_{{\mathbb{Z}}_2}\) are subgroups, where \(\Gamma_{{\mathbb{Z}}_2}\) will be called the \(2\)-torsion subgroup. Note that if \(\alpha\in \Gamma_{{\mathbb{Z}}_2}\), then \(\omega(\alpha, \beta) = \omega(\beta, \alpha)=\pm 1\) for all \(\beta\in\Gamma\).

A \(\Gamma\)-graded vector space \(V\) is the direct sum \(V=\sum_{\alpha\in\Gamma} V_\alpha\) of its homogeneous subspaces \(V_\alpha\). Denote \(m_\alpha=\dim V_\alpha\). For any homogeneous element \(v\in V\), we use \(d(v)\in \Gamma\) to denote the degree of \(v\). Clearly \(V=V_+\oplus V_-\) with \(V_\pm=\sum_{\alpha\in \Gamma^\pm} V_\alpha\). In [6], we introduced the set \[\begin{align} \label{eq:support} \Gamma_R(V)=\{\alpha\in \Gamma\mid V_\alpha\ne 0\}, \end{align}\tag{4}\] which will be referred to as the graded support of \(V\). Let \[\begin{align} \label{eq:torsion} \Gamma_{R, {\mathbb{Z}}_2}(V)=\Gamma_R(V)\cap\Gamma_{{\mathbb{Z}}_2},\quad \Gamma^\pm_{R, {\mathbb{Z}}_2}(V)=\Gamma^\pm_R(V)\cap\Gamma_{{\mathbb{Z}}_2}, \end{align}\tag{5}\] the sets of \(2\)-torsion elements of \(\Gamma_R(V)\) and \(\Gamma^\pm_R(V)\) respectively. If \(\dim V<\infty\), the graded support is necessarily a finite set, and hence so are also \(\Gamma_R^\pm(V)=\Gamma_R(V)\cap \Gamma^\pm\). In this case, we denote \(\aleph(V)=|\Gamma_R(V)|\) and \(\aleph^\pm(V)=|\Gamma_R^\pm(V)|\).

The space of homomorphisms between any two \(\Gamma\)-graded vector spaces \(U\) and \(V\) is naturally \(\Gamma\)-graded. Note that \({\rm{Hom}}_{\mathbb{C}}(U, V)\) can be expressed as \({\rm{Hom}}_{\mathbb{C}}(U, V)=\sum_{\alpha, \beta\in\Gamma}{\rm{Hom}}_{\mathbb{C}}(U_\alpha, V_\beta)\), where \({\rm{Hom}}_{\mathbb{C}}(U_\alpha, V_\beta)\) is of degree \(\beta-\alpha\). Write \({\rm{Hom}}_{\mathbb{C}}(U, V)_\Upsilon=\sum_{\beta-\alpha=\Upsilon}{\rm{Hom}}_{\mathbb{C}}(U_\alpha, V_\beta)\). Then \({\rm{Hom}}_{\mathbb{C}}(U, V)=\sum_{\Upsilon} {\rm{Hom}}_{\mathbb{C}}(U, V)_\Upsilon\).

Denote by \(\text{{\boldsymbol{V}ect}(\Gamma, \omega)}\) the category of \(\Gamma\)-graded vector spaces. It is a strict braided monoidal category, with the monoidal structure given by the tensor product \(\otimes_{\mathbb{C}}\), and the braiding by a functorial map arising from the commutative factor \(\omega\), \[\begin{align} \label{eq:def-tau} \tau_{V, W}: V\otimes_{\mathbb{C}}W\longrightarrow W\otimes_{\mathbb{C}}V, \end{align}\tag{6}\] which is defined by \[\begin{align} \label{eq:tau} \tau_{V, W}(v\otimes w)= \omega(d(v), d(w)) w\otimes v, \end{align}\tag{7}\] for any homogeneous \(v\in V, w\in W\), and linearly extended to inhomogeneous elements.

It is evident that \[\begin{align} \label{eq:invol} \tau_{W, V} \tau_{V, W}={\rm{id}}_{V\otimes W}. \end{align}\tag{8}\] One can also easily verify [6] that the functorial map \(\tau\) indeed defines a braiding for \(\text{{\boldsymbol{V}ect}(\Gamma, \omega)}\), that is, for any \(\Gamma\)-graded vector spaces \(U, V, W\), the following braid relation holds \[\begin{align} \label{eq:braid} &&(\tau_{V,W}\otimes{\rm{id}}_U) \circ (id_V\otimes\tau_{U, W})\circ (\tau_{U, V}\otimes{\rm{id}}_W) \\ &&= (id_W\otimes\tau_{U, W}) \circ (\tau_{U, W}\otimes{\rm{id}}_V)\circ (id_U\otimes\tau_{V, W}). \nonumber \end{align}\tag{9}\]

Observe the following easy facts.

. Given any \(\xi\in \Gamma\), there exists a degree shifting functor \(\digamma_\xi\) on \(\text{{\boldsymbol{V}ect}(\Gamma, \omega)}\) such that \[\begin{align} \label{eq:shift} \digamma_\xi V:=\sum_{\alpha\in \Gamma}(\digamma_\xi V)_\alpha, \quad (\digamma_\xi V)_{\alpha+\xi}=V_\alpha. \end{align}\tag{10}\] Note in particular that if \(\xi\in\Gamma^-\), then \((\digamma_\xi V)_\pm = V_\mp\).

. Given abelian groups \(\Gamma_i\) with commutative factors \(\omega_i: \Gamma_i\times \Gamma_i\longrightarrow{\mathbb{C}}^*\) for \(i=1, 2\), let \(\Gamma=\Gamma_1\times\Gamma_2\), and define the map \[\begin{align} \label{eq:CD} \omega: \Gamma\times \Gamma\longrightarrow{\mathbb{C}}^*, \quad \omega((\alpha, \mu), (\beta, \nu)) =\omega_1(\alpha, \beta)\omega_2(\mu, \nu), \end{align}\tag{11}\] for all \((\alpha, \mu), (\beta, \nu)\in\Gamma\). It gives rise to a commutative factor on \(\Gamma\). There is a functor \[\mathcal{D}: \text{{\boldsymbol{V}ect}(\Gamma_1, \omega_1)\times {\boldsymbol{V}ect}(\Gamma_2, \omega_2)\longrightarrow{\boldsymbol{V}ect}(\Gamma, \omega)}\] such that for any object \(V=\sum_{\alpha\in\Gamma_1}V_\alpha\) of \({\boldsymbol{V}ect}(\Gamma_1, \omega_1)\) and \(W=\sum_{\beta\in\Gamma_2}W_\beta\) of \({\boldsymbol{V}ect}(\Gamma_2, \omega_2)\), we have \(\mathcal{D}(V\times W)\in {\boldsymbol{V}ect}(\Gamma, \omega)\) with \(\mathcal{D}(V\times W)=\sum_{(\alpha, \beta)\in \Gamma} (\mathcal{D}(V\times W))_{(\alpha, \beta)}\), where \((\mathcal{D}(V\times W))_{(\alpha, \beta)}=V_\alpha\otimes W_\beta\). In particular, if we regarded \({\mathbb{C}}\) as homogeneous of degree \(0\), then \(\mathcal{D}(V\times {\mathbb{C}})=V\otimes{\mathbb{C}}\simeq V\) and \(\mathcal{D}({\mathbb{C}}\times W)={\mathbb{C}}\otimes W\simeq W\).

2.2 Bilinear forms↩︎

2.2.1 Bilinear forms↩︎

Fix \(\gamma\in\Gamma\). A bilinear form \(\kappa: V\times V\longrightarrow{\mathbb{C}}\) is said to be homogeneous of degree \(\gamma\) if \(\kappa(V_\alpha, V_\beta)=0\) for all \(\alpha, \beta\) such that \(\alpha+\beta+\gamma\ne 0\). Note in particular that

• if \(\gamma\in \Gamma^-\), then \(\kappa(V_+, V_+)=\kappa(V_-, V_-)=\{0\}\);

• if \(\gamma\in \Gamma^+\), then \(\kappa(V_+, V_-)=\kappa(V_-, V_+)=\{0\}\).

If \(\kappa\) is non-degenerate, then \(\dim V_\alpha = \dim V_{-\alpha-\gamma}\) for all \(\alpha\).

A bilinear form \(\kappa\) is said to be \[\begin{align} \bullet\quad &\text{\omega-symmetric if } \kappa(v_\alpha, v'_\beta) =\omega(\alpha, \beta) \kappa(v'_\beta, v_\alpha), \\ \bullet\quad &\text{\omega-skew symmetric if } \kappa(v_\alpha, v'_\beta) =-\omega(\alpha, \beta) \kappa(v'_\beta, v_\alpha), \end{align}\] for all \(v_\alpha\in V_\alpha\) and \(v'_\beta\in V_\beta\). We shall often regard a bilinear form on \(V\) as a linear function from \(V\otimes V\) to \({\mathbb{C}}\). Thus we can define \[\kappa^{(s)}:=\frac{1}{2}\kappa\circ({\rm{id}}_V\otimes{\rm{id}}_V+\tau_{V, V}), \quad \kappa^{(a)}:=\frac{1}{2}\kappa\circ({\rm{id}}_V\otimes{\rm{id}}_V-\tau_{V, V}).\] Regard the maps \(\kappa^{(s)}\) and \(\kappa^{(a)}\) as bilinear forms on \(V\), then they are \(\omega\)-symmetric and \(\omega\)-skew symmetric respectively.

2.2.2 The \(\omega\)-trace↩︎

Assume that \(\dim V<\infty\). There is a generalised trace \({\rm tr}_{(\Gamma, \omega)}: {\rm{End}}_{\mathbb{C}}(V)\longrightarrow{\mathbb{C}}\), referred to as the \(\omega\)-trace, which is defined as follows [6]. For any \(\varphi =\sum_{\alpha, \beta\in\Gamma} \varphi(\alpha, \beta)\) with \(\varphi(\alpha, \beta)\in {\rm{Hom}}_{\mathbb{C}}(V_\beta, V_\alpha)\), \[{\rm tr}_{(\Gamma, \omega)}(\varphi) =\sum_{\alpha\in \Gamma} \omega(\alpha, \alpha) {\rm tr}(\varphi(\alpha, \alpha)),\] where \({\rm tr}\) denotes the usual trace. Clearly this is homogeneous of degree \(0\). Write \(\varphi_0= \sum_{\alpha\in \Gamma} \varphi(\alpha, \alpha)\), then \({\rm tr}_{(\Gamma, \omega)}(\varphi) = {\rm tr}_{(\Gamma, \omega)}(\varphi_0).\)

We refer to 12 as the \(\omega\)-symmetry of the generalised trace.

The \(\omega\)-trace leads to a the following bilinear form on \({\rm{End}}_{\mathbb{C}}(V)\). \[\begin{align} \label{eq:biform} &&( \;, \;): {\rm{End}}_{\mathbb{C}}(V)\times {\rm{End}}_{\mathbb{C}}(V)\longrightarrow{\mathbb{C}}, \\ &&( X, Y )={\rm tr}_{(\Gamma, \omega)}(X Y), \quad \forall X, Y\in {\rm{End}}_{\mathbb{C}}(V).\nonumber \end{align}\tag{13}\] This is clearly homogeneous of degree \(0\), and was shown to be non-degenerate in [6]. It follows the \(\omega\)-symmetry of \(tr_{(\Gamma, \omega)}\) that the bilinear form 13 is \(\omega\)-symmetric.

2.2.3 Effects of degree shifting↩︎

It is useful to consider the effect of degree shifting on bilinear forms. Note the following facts.

A homogeneous bilinear form of degree \(\gamma\) on \(V\) gives rise to a homogeneous bilinear form of degree \(\gamma+2\xi\) on \(\digamma_\xi V\).

If \(\xi\in \Gamma^-\cap\Gamma_{{\mathbb{Z}}_2}\), a homogeneous \(\omega\)-symmetric (resp. skew symmetric) bilinear form of degree \(\gamma\) on \(V\) leads to a homogeneous \(\omega\)-skew symmetric (resp. symmetric) bilinear form of the same degree on \(\digamma_\xi V\).

If \(\Gamma^-\cap\Gamma_{{\mathbb{Z}}_2}=0\), we may use the functor \({\mathcal{D}}\) (see 11 ) in the situation of Remark 2 to map \(V\) to an object \({\mathcal{D}}(V\times {\mathbb{C}})=V\) in the category \({\boldsymbol{V}ect}(\Gamma\times{\mathbb{Z}}_2, \omega\times sign)\), and then apply degree shifting to obtain \(\digamma_{(0, 1)} {\mathcal{D}}(V\times {\mathbb{C}})=\digamma_{(0, 1)}V\). Now \(\omega\)-symmetric (resp. skew symmetric) bilinear forms on \(V\) become \((\Gamma\times{\mathbb{Z}}_2, \omega\times sign)\)-skew symmetric (resp. symmetric) bilinear forms on \(\digamma_{(0, 1)}V\).

Hereafter we consider only homogeneous bilinear forms of degree \(0\).

2.3 Classical Lie \((\Gamma, \omega)\)-algebras↩︎

Recall that a \(\Gamma\)-graded Lie \(\omega\)-algebra [1], [3], or Lie \((\Gamma, \omega)\)-algebra for short, is a \(\Gamma\)-graded vector space \({\mathfrak g}=\sum_{\alpha\in\Gamma} {\mathfrak g}_\alpha\) endowed with a bilinear map \([\;, \;]: {\mathfrak g}\times {\mathfrak g}\longrightarrow{\mathfrak g},\) called the Lie \(\omega\)-bracket, which is homogeneous of degree \(0\), and satisfies the following conditions \[\begin{align} &&[X, Y] =- \omega(d(X), d(Y))[Y, X], \tag{14}\\ &&[X, [Y, Z]] = [ [X, Y], Z] + \omega(d(X), d(Y)) [Y, [X, Z]], \tag{15} \end{align}\] for any \(X, Y, Z\in {\mathfrak g}\) (in the convention of Remark 1). We will loosely call it a Lie colour superalgebra if \({\mathfrak g}_-=\sum_{\alpha\in\Gamma^-}{\mathfrak g}_\alpha\ne 0\), and a Lie colour algebra otherwise.

Let \(V\) be a \(\Gamma\)-graded vector space. The general linear Lie \((\Gamma, \omega)\)-algebra \({\mathfrak {gl}}(V)\) is \({\rm{End}}_{\mathbb{C}}(V)\) with the generalised Lie bracket given by the \((\Gamma, \omega)\)-commutator \[\begin{align} [X, Y]=X Y - \omega(d(X), d(Y)) Y X, \quad X, Y\in {\mathfrak {gl}}(V). \end{align}\]

Let \(B(V)=(e_1, e_2, \dots, e_D)\) be an ordered homogeneous bases for \(V\), where \(D= \dim V\). Let \(\gamma_a:= d(e_a)\) the \(\Gamma\)-degree of \(e_a\) for each \(a\). As in [6], we let \({\mathbb{E}}_{a b}\in {\rm{End}}_{\mathbb{C}}(V)\) (for \(a, b=1, 2, \dots, d\)) be the matrix units relative to the basis \(B(V)\). They obey the standard relations \[\begin{align} {\mathbb{E}}_{a b}\cdot e_c = \delta_{b c} e_a, \quad {\mathbb{E}}_{a b} {\mathbb{E}}_{c d} = \delta_{b c} {\mathbb{E}}_{a d}. \end{align}\] The degree of \({\mathbb{E}}_{a b}\) is given by \(\gamma_a-\gamma_b\). Note in particular that \({\mathbb{E}}_{a a}\) is of degree \(0\) for all \(a\). The matrix units form a homogeneous basis of \({\mathfrak {gl}}(V)\), which enable us to express the defining relations of \({\mathfrak {gl}}(V)\) as \[\begin{align} \label{eq:CR} \phantom{XXXX} [{\mathbb{E}}_{a b}, {\mathbb{E}}_{c d} ]= \delta_{b c} {\mathbb{E}}_{a d} -\omega(\gamma_a - \gamma_b, \gamma_c - \gamma_d) \delta_{a d} {\mathbb{E}}_{c b}. \end{align}\tag{16}\]

Fix \(a\ne b\), and let \(V_{a b}={\mathbb{C}}e_a\oplus {\mathbb{C}}e_b\). By [6], the elements \({\mathbb{E}}_{a b}, {\mathbb{E}}_{b a}, {\mathbb{E}}_{a a}\), \({\mathbb{E}}_{b b}\) span a Lie \((\Gamma, \omega)\)-subalgebra \({\mathfrak {gl}}(V_{a b})\), which is isomorphic to

\(\bullet\) \({\mathfrak {gl}}_2({\mathbb{C}})\) if \(\gamma_a - \gamma_b\) belongs to \(\Gamma^+\);

\(\bullet\) \({\mathfrak {gl}}_{1|1}({\mathbb{C}})\) if \(\gamma_a - \gamma_b\) belongs to \(\Gamma^-\).

Since the \(\omega\)-trace \(\tr_{(\Gamma, \omega)}: {\rm{End}}_{\mathbb{C}}(V)\longrightarrow{\mathbb{C}}\) has the property that \(\tr_{(\Gamma, \omega)}([X, Y])=0\) for all \(X, Y\in {\mathfrak {gl}}(V)\), the subspace \[{\mathfrak {sl}}(V)=\{X\in{\mathfrak {gl}}(V)\mid \tr_{(\Gamma, \omega)}(X)=0\}\] forms a Lie \((\Gamma, \omega)\)-subalgebra \({\mathfrak {sl}}(V)\) of \({\mathfrak {gl}}(V)\), the special linear Lie \((\Gamma, \omega)\)-algebra of \(V\). Note that if \(\dim V_+=\dim V_-\), then \({\mathfrak {sl}}(V)\) contains the identity element of \({\mathfrak {gl}}(V)\), which spans an abelian ideal. Thus \({\mathfrak {sl}}(V)\) is not simple in this case.

Now we consider Lie \((\Gamma, \omega)\)-subalgebras of \({\mathfrak {gl}}(V)\) preserving non-degenerate bilinear forms. By discussions in Section 2.2.3, we only need to consider \(\omega\)-symmetric bilinear forms.

We now give an explicit construction of \({\mathfrak {osp}}(V; \kappa)\).

Consider the \(\Gamma\)-graded vector space isomorphism \(\varpi: V\otimes V\longrightarrow{\rm{End}}_{\mathbb{C}}(V)\) defined, for any \(v, v\in V\), by \[\varpi(v\otimes v')(w)= \kappa(v', w) v, \quad \forall w\in V.\] Note in particular that \(\varpi(e_a\otimes\overline{e}_b) ={\mathbb{E}}_{a b}.\) Let \(c_0=\sum_{a} e_a\otimes\overline{e}_a\), then \(\varpi(c_0)={\rm{id}}_V\).

2.3.1 Remarks on invariant theory↩︎

We take a quick digression to the invariant theory of the classical Lie \((\Gamma, \omega)\)-algebras, limiting ourselves to making some brief remarks only.

The invariant theory of the general linear Lie colour superalgera \({\mathfrak {gl}}(V)\) was systematically developed in [6], where a Howe duality of type \(({\mathfrak {gl}}(V), {\mathfrak {gl}}(V'))\) was established, which in particular implies first and second fundamental theorems of invariant theory in a \((\Gamma, \omega)\)-commutative algebra setting, and a Schur-Weyl duality between \({\mathfrak {gl}}(V)\) and the symmetric group \({\rm{Sym}}_r\) for ak \(r\), whose actions on \(V^{\otimes r}\) are constructed from the symmetry \(P:=\tau_{V, V}\) defined by 6 for \(W=V\). One can easily reformulate some of the results in a categorical framework by constructing a full tensor functor from the oriented Brauer category (see, e.g., [85]) to the full subcategory of \({\mathfrak {gl}}(V)\)-modules with objects being repeated tensor products of \(V\) and its dual.

The circle of ideas on applications of the Brauer category [86] to the invariant theory of the orthosymplectic supergroup [87], [88] should also work for the orthosymplectic Lie colour superalgebra \({\mathfrak {osp}}(V; \kappa)\). Now observe that the symmetry \(P\) is clearly an \({\mathfrak {osp}}(V; \kappa)\)-map, and the bilinear form \(\kappa\) also gives rise to the \({\mathfrak {osp}}(V; \kappa)\)-maps \[\begin{align} &&\widehat{C}: V\otimes V\longrightarrow{\mathbb{C}}, \quad v\otimes v'\mapsto \kappa(v, v'), \quad \forall v, v'\in V, \\ &&\check{C}: {\mathbb{C}}\longrightarrow V\otimes V, \quad a\mapsto a c_0, \quad \forall a\in{\mathbb{C}}, \end{align}\] where \(c_0\in V\otimes V\) is defined immediately before Theorem 5, and \(\widehat{C}\) is nothing but \(\kappa\) regarded as a linear map from \(V\otimes V\) to \({\mathbb{C}}\). The maps \(P\), \(\widehat{C}\) and \(\check{C}\) satisfy relations which are formally the same as those in Lemma [85]. This enables one to describe \({\mathfrak {osp}}(V; \kappa)\)-invariants diagrammatically by constructing a tensor functor from the Brauer category to the full subcategory \({\mathcal{T}}(V)\) of \({\mathfrak {osp}}(V; \kappa)\)-modules with objects \(V^{\otimes r}\) for all \(r\in{\mathbb{Z}}_+\).

The tensor functor will not be full in general (but almost full), nevertheless it is expected to product all \({\rm OSp}(V;\kappa)\)-morphisms \(V^{\otimes r}\longrightarrow V^{\otimes s}\) for all \(r, s\), where \({\rm OSp}(V;\kappa)\) is a \((\Gamma, \omega)\)-group scheme, which can be constructed in a similar way as for the general linear colour supergroup \({\rm{GL}}(V)\) given in [6].

It will be very interesting to provide details on the invariant theory of the classical Lie \((\Gamma, \omega)\)-algebras along the lines suggested above.

3 Lie and affine colour algebras fulfilling Cartan-Weyl paradigm↩︎

The Cartan-Weyl paradigm for complex semi-simple Lie algebras describes the structures of the algebras by root systems, and treats the representation theory via weight modules. Here we re-examine it in the \(\Gamma\)-graded context, and define classes of Lie colour algebras and affine Lie colour algebras which realise the paradigm.

3.1 Assessing the Cartan-Weyl paradigm in the colour context↩︎

The structure and representation theories of the general (and hence special) linear Lie colour superalgebras were systematically developed in [6]. The key feature is that

there exist homogeneous Cartan subalgebras of degree \(0\) which are commutative in the usual sense.

They enable us to describe the structure of the Lie colour superalgebras by root systems, and to develop a theory of weight representations. This is the same as for semi-simple Lie algebras and Lie affine algebras. Thus we say that the general and special linear Lie colour superalgebras fulfil the Cartan-Weyl paradigm.

Let us now consider the case of the orthosymplectic Lie colour superalgebra \({\mathfrak {osp}}(V; \kappa)\). It contains the subalgebra \({\mathfrak {osp}}(V; \kappa)_0 ={\mathfrak {so}}(V_+; \kappa|_{V_+})\oplus {\mathfrak {sp}}(V_-; \kappa|_{V_-})\). Since \(\kappa\) is homogeneous of degree \(0\), the orthogonal Lie colour subalgebra \({\mathfrak {so}}(V_+; \kappa|_{V_+})\) and symplectic Lie colour subalgebra \({\mathfrak {sp}}(V_-; \kappa|_{V_-})\) graded commute. It is not difficult to see that \({\mathfrak {osp}}(V; \kappa)\) realises the Cartan-Weyl paradigm if and only if both \({\mathfrak {so}}(V_+; \kappa|_{V_+})\) and \({\mathfrak {sp}}(V_-; \kappa|_{V_-})\) do. We will show in Section 3.3 that the symplectic Lie colour algebra \({\mathfrak {sp}}(V_-; \kappa|_{V_-})\) always realises the Cartan-Weyl paradigm. However, it is not always the case for the orthogonal Lie colour algebra \({\mathfrak {so}}(V_+; \kappa|_{V_+})\). As shown in Sections 3.4.2 and 3.4.1, this depends on the structure of the subspace of \(V\) with graded support \(\Gamma^+_{R, {\mathbb{Z}}_2}(V)\).

The situation for \({\mathfrak {osp}}(V; \kappa)\) can be summarised by the following theorem.

Hereafter we will consider only Lie colour algebras and their quantum analogues, i.e., assuming \(\Gamma_R(V)\subset \Gamma^+\). The super case will be treated in a sequel of this paper.

3.2 General and special linear Lie colour algebras↩︎

Let \(V\) be a finite dimensional \(\Gamma\)-graded vector space such that \(\Gamma_R(V)= \Gamma_R^+(V)\), thus \(V_-=0\). Then \({\mathfrak {gl}}(V)\) and \({\mathfrak {sl}}(V)\) are Lie colour algebras and not superalgebras.

3.2.1 Structure theory↩︎

The material in this section is extracted from [6]. We present it here to set the stage for studying other classical Lie colour algebras.

Consider equation 16 in the special case \(a=b\). It reduces to \[\begin{align} \label{eq:Cartan} [{\mathbb{E}}_{a a}, {\mathbb{E}}_{c d} ]= (\delta_{a c} - \delta_{a d}) {\mathbb{E}}_{c d}. \end{align}\tag{21}\] Let \({\mathfrak h}=\sum_{a=1}^{\dim V}{\mathbb{E}}_{a a}\). It is of crucial importance that \({\mathfrak h}\) is homogeneous of degree \(0\), which forms a usual Abelian Lie subalgebra of \({\mathfrak {gl}}(V)\). Equation 21 shows that \({\mathfrak h}\) is a Cartan subalgebra of \({\mathfrak {gl}}(V)\), which enables one to describe the structure of \({\mathfrak {gl}}(V)\) in the standard way by using root spaces.

We introduce a basis \(\{\varepsilon_a\mid 1\le a \le \dim V\}\) for the dual space \({\mathfrak h}^*\) of \({\mathfrak h}\) such that \[\varepsilon_a({\mathbb{E}}_{b b})=\delta_{a b}, \quad a, b=1, 2, \dots, \dim V.\] Then 21 can be re-written as \[\begin{align} \label{eq:roots} [{\mathbb{E}}_{a a}, {\mathbb{E}}_{c d} ]= (\varepsilon_c - \varepsilon_d)({\mathbb{E}}_{a a}) {\mathbb{E}}_{c d}. \end{align}\tag{22}\] Thus the set of roots of \({\mathfrak {gl}}(V)\) with this choice of the Catan subalgebra is given by \[\begin{align} \Phi =\{\varepsilon_a - \varepsilon_b \mid a\ne b\}. \end{align}\] We take the set \(\Phi^+\) of positive roots to consist of the following elements. \[\Phi^+ =\{\varepsilon_a - \varepsilon_b \mid a< b\}.\] Then \(\Phi=\Phi^+\cup(-\Phi^+)\).

For any root \(\Upsilon\in \Phi\), the corresponding root space is \[{\mathfrak g}_\Upsilon =\{X\in {\mathfrak {gl}}(V)\mid [h, X]=\Upsilon(h)X, \;\forall h\in{\mathfrak h}\},\] which is \(1\)-dimensional. Note that \({\mathfrak g}_\Upsilon\) is homogeneous in the \(\Gamma\)-grading. We define the following map for later use. \[\begin{align} \label{eq:xi-map} \xi: \Phi\longrightarrow\Gamma, \quad \xi(\Upsilon)=\text{degree of {\mathfrak g}_\Upsilon}. \end{align}\tag{23}\]

Let \({\mathfrak n}= \sum_{\Upsilon\in\Phi^+} {\mathfrak g}_\Upsilon\) and \(\overline{{\mathfrak n}}=\sum_{\Upsilon\in\Phi^+} {\mathfrak g}_{-\Upsilon}\), which are Lie \((\Gamma, \omega)\)-subalgebras of \({\mathfrak {gl}}(V)\). We have the following triangular decomposition. \[\begin{align} {\mathfrak {gl}}(V)= \overline{{\mathfrak n}}+{\mathfrak h}+{\mathfrak n}. \end{align}\]

By [6], the \(\omega\)-symmetric bilinear \(\omega\)-trace form 13 on \({\rm{End}}_{\mathbb{C}}(V)\) satisfies \[\begin{align} \label{eq:form-basis} ({\mathbb{E}}_{a a}, {\mathbb{E}}_{b b}) = \omega(\gamma_a, \gamma_b) \delta_{a b}, \end{align}\tag{24}\] thus is non-degenerate. It is also \(ad\)-invariant [6], i.e., \[( [X, Y], Z ) = ( X, [Y, Z] ), \quad \forall X, Y, Z\in {\mathfrak {gl}}(V).\] Its restriction \(( \;, \;)|_{\mathfrak h}: {\mathfrak h}\times {\mathfrak h}\longrightarrow{\mathbb{C}}\) to the Cartan subalgebra is non-degenerate and symmetric, thus induces a non-degenerate symmetric bilinear form on \({\mathfrak h}^*\), \[\begin{align} \label{eq:inner-prod} (\;, \;): {\mathfrak h}^*\times {\mathfrak h}^*\longrightarrow{\mathbb{C}}. \end{align}\tag{25}\] It follows [6] that \[\begin{align} \label{eq:basis-prod} (\varepsilon_a, \varepsilon_b)= \delta_{a b}. \end{align}\tag{26}\]

Now \({\mathfrak {sl}}(V)\) inherits the following triangular decomposition from \({\mathfrak {gl}}(V)\), \[{\mathfrak {sl}}(V) =\overline{{\mathfrak n}}+ {\mathfrak h}_s+{\mathfrak n}, \text{ with {\mathfrak h}_s= {\mathfrak h}\cap{\mathfrak {sl}}(V)}.\] Note that \({\mathfrak h}_s\) is spanned by \({\mathbb{E}}_{a a}- {\mathbb{E}}_{b b}\) for all \(a, b\).

We have already seen that each \(\Upsilon\in\Phi^+\) is associated with a Lie subalgebra \({\mathfrak {gl}}_2({\mathbb{C}})\). This enables us to introduce a Weyl group.

The Weyl group \(W\) of \({\mathfrak {sl}}(V)\) is isomorphic to the symmetric group \(Sym_D\) of degree \(D=\dim V\). The set of weights of any weight module [6] of \({\mathfrak {sl}}(V)\) is \(W\)-stable. Definition 2 can be generalised to \({\mathfrak {gl}}(V)\), even though it is not semi-simple. In this case, we replace \(E\) by \(\sum_a {\mathbb{R}}\varepsilon_a\).

3.2.2 Generators and relations↩︎

We take the following set of positive roots \(\Pi=\{\Upsilon_a:= \varepsilon_a - \varepsilon_{a+1}\mid a=1, 2, \dots, \dim V-1\}\) as the simple roots for \({\mathfrak {sl}}(V)\). There is another piece of information associated with the root system, namely, the following sequence of elements of \(\Gamma\). \[\Xi:=(\xi(\Upsilon_1), \xi(\Upsilon_2), \dots, \xi(\Upsilon_{\dim V-1})), \quad \xi(\Upsilon_a)=\gamma_a-\gamma_{a+1}.\]

We define Cartan matrices and Dynkin diagrams as follows.

For \({\mathfrak {sl}}(V)\) with \((\Phi, \Pi, \Xi)\) given above, the Cartan matrix is the usual one of type \(A_{\dim V-1}\), and the Dynkin diagram is as in Figure 1.

The following result is easy to see.

As we have pointed out already, the choice of the basis for \(V\) affects the set \(\Xi\), and hence also the Dynkin diagram. The choices described below lead to the least number of non-zero elements in \(\Xi\).

Fix a total order of \(\Gamma_R(V)\), and write \(\Gamma_R(V)=\{\alpha_1, \alpha_2, \dots, \alpha_{\aleph}\}\) (where \(\aleph=|\Gamma_R(V)|\)) in such a way that \(\alpha_i < \alpha_j\) for all \(i<j\). Choose an ordered basis \(B(\alpha)=(e(\alpha)_1, e(\alpha)_2, \dots, e(\alpha)_{m_\alpha})\) for each \(V_\alpha\) with \(m_\alpha=\dim V_{\alpha_r}\), and construct the following basis for \(V\). \[\begin{align} \label{eq:BV-s} B(V)=(B(\alpha_1), B(\alpha_2), \dots, B(\alpha_{\aleph(V)})). \end{align}\tag{32}\] Then the Dynkin diagram corresponding to this basis has the smallest number, \(\aleph-1\), of double circles. Let \(k_i=m_{\gamma_i}-1\). Then \[\setlength{\unitlength}{0.25mm} \begin{picture}(300, 35)(45, -20) \put(50, 0){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{k_1}} \put(100, 10){\line(1, 0){10}} \put(115, 10){\circle{10}}\put(115, 10){\circle{5}} \put(120, 10){\line(1, 0){10}} \put(130, 0){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{k_2}} \put(180, 10){\line(1, 0){10}} \put(195, 10){\circle{10}}\put(195, 10){\circle{5}} \put(200, 10){\line(1, 0){10}} \put(210, 9){...} \put(220, 10){\line(1, 0){10}} \put(230, 0){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{k_{\aleph}}} \end{picture}\]

3.3 Smplectic Lie colour algebras↩︎

In the case \(V=V_-\), the orthosymplectic Lie colour superalgebra reduces to symplectic Lie colour algebra \({\mathfrak {sp}}(V; \kappa)\). It follows Remark 3 that if \(\alpha \in \Gamma_R(V)\) but \(\alpha \not\in \Gamma_R^{{\mathbb{Z}}_2}(V)\), then \(V_\alpha\) is paired with \(V_{-\alpha}\) under the non-degenerate bilinear form \(\kappa\), thus \(V_{-\alpha}\) is isomorphic to the dual space of \(V_\alpha\). If \(\alpha\in \Gamma_R^{{\mathbb{Z}}_2}(V)\), the restriction of \(\kappa\) to \(V_\alpha\) is non-degenerate. The \(\omega\)-symmetry of \(\kappa\) in effect means skew symmetry on \(V_\alpha\) since \(V=V_-\). Thus \(\dim V_\alpha\) is even dimensional, and can be decomposed into \(V_\alpha = U_\alpha \oplus \overline{U}_\alpha\) such that \(\kappa\) pairs \(U_\alpha\) with \(\overline{U}_\alpha\), and hence \(\overline{U}_\alpha\) is isomorphic to the dual space of \(U_\alpha\). Therefore, there exists \(U=\sum_{\alpha\in \Gamma_R^-(U)} U_\alpha\) with dual space \(U^*= \sum_{\alpha\in \Gamma_R^-(U)} (U^*)_{-\alpha}\) such that \(V= U\oplus U^*\). The bilinear form is now given, for any homogeneous \(v\in U\) and \(\overline{v}'\in U^*\), by \[\kappa(\overline{v}', v) = - \kappa(v, \overline{v}') = \overline{v}'(v).\] Note that \(\Gamma_R(U^*)=-\Gamma_R(U)\subseteq \Gamma^-\).

Denote \(\ell=\dim U\). Choose ordered homogeneous bases \[\begin{align} \label{eq:BU} B(U)=(e_1, e_2, \dots, e_\ell) \text{ for U}, \quad \overline{B}(U^*)=(\overline{e}_1, \overline{e}_2, \dots, \overline{e}_\ell) \text{ for U^*}, \end{align}\tag{33}\] such that \(\overline{e}_i(e_j)=\delta_{i j}\), and introduce the ordered homogeneous basis for \(V\), \[\begin{align} \label{eq:BV-gen} B(V)= (B(U), \overline{B}(U^*)). \end{align}\tag{34}\] Write \(B(V)=(e_1, e_2, \dots, e_{2\ell})\). Then the elements \(e_a\) satisfy \(\kappa(e_{\ell+i}, e_j)=-\kappa(e_i, e_{\ell+j})\) \(= \delta_{i j}\), and hence \((\kappa(e_a, e_b)) = \begin{pmatrix} 0 & - I_\ell\\ I_\ell & 0 \end{pmatrix}\). Denote \(\gamma_a=d(e_a)\) for \(a=1, 2, \dots, 2\ell\), where \(\gamma_i = - \gamma_{\ell+i}\) for all \(i\le \ell\). Note that \(\omega(\gamma_a, \gamma_a)=-1\) for all \(a\).

The matrix units \({\mathbb{E}}_{a b}\in {\rm{End}}_{\mathbb{C}}(V)\) with respect to \(B\) have \(\Gamma\)-degrees \[\begin{align} &d({\mathbb{E}}_{i j}) = d({\mathbb{E}}_{\ell+j, \ell+i}) = \gamma_i - \gamma_j, \\ & d({\mathbb{E}}_{i, \ell+j}) =-d({\mathbb{E}}_{\ell+i, j})= \gamma_i + \gamma_j, \quad \quad \forall i, j\le \ell. \end{align}\] Using the explicit description of \(\kappa\) given above, and bearing in mind that \(\omega(\gamma_a, \gamma_a)=-1\) for all \(a\), we easily deduce the following result from Theorem 5.

The structure of \({\mathfrak {sp}}(V;\kappa)\) is most conveniently described in terms of its root system. Let \({\mathfrak h}=\sum_{i=1}^\ell {\mathbb{C}}\;X_{i i}\), which is homogeneous of degree \(0\). It follows from the lemma that \[\begin{align} {}[X_{i i}, X_{p p}]&=0, \\ {}[X_{i i}, X_{p q}]&= (\delta_{i p} - \delta_{i q})X_{p q}; \quad p\ne q, \\ {}[X_{i i}, X_{p, \ell+ q}]&= (\delta_{i p} + \delta_{i q}) X_{p, \ell+q}, \\ {}[X_{i i}, X_{\ell+ p, q}]&= - (\delta_{i p} +\delta_{i q}) X_{\ell+p, q}, \quad i, p, q\le \ell. \end{align}\] Thus \({\mathfrak h}\) is a Cartan subalgebra of \({\mathfrak {sp}}(V;\kappa)\).

Let \(\delta_i\), for \(i=1, 2, \dots, \ell\), be elements of the dual space \({\mathfrak h}^*\) such that \(\delta_i(X_{j j})=\delta_{i j}\) for all \(i, j=1, 2, \dots, \ell\). They form a basis of \({\mathfrak h}^*\). The last three of the above equations can be re-written as \[\begin{align} {}[X_{i i}, X_{p q}]&=& (\delta_p - \delta_q)(X_{i i})X_{p q}, \quad p\ne q, \tag{35}\\ {}[X_{i i}, X_{p, \ell+ q}]&=& (\delta_p + \delta_q)(X_{i i})X_{p, \ell+q}, \tag{36}\\ {}[X_{i i}, X_{\ell+ p, q}]&=& - (\delta_p + \delta_q)(X_{i i}) X_{\ell+p, q}, \quad i, p, q\le \ell. \tag{37} \end{align}\]

Recall that the \(\omega\)-trace gives rise to a non-degenerate \(\omega\)-symmetric bilinear form on \({\mathfrak {gl}}(V)\), whose restriction to \({\mathfrak {sp}}(V; \kappa)\) is non-degenerate. We define the bilinear form \((\;, \;): {\mathfrak {sp}}(V; \kappa)\times {\mathfrak {sp}}(V; \kappa)\longrightarrow{\mathbb{C}}\) by \((X, Y)=\frac{1}{2}{\rm tr}_{(\Gamma, \omega)}(XY)\) for all \(X, Y\in {\mathfrak {sp}}(V; \kappa)\). It is easy to verify that \((X_{i i}, X_{j j})=-\delta_{i j}\), thus the restriction of \((\;, \;)\) to \({\mathfrak h}\) is non-degenerate. It induces a symmetric biliner form \((\;, \;): {\mathfrak h}^*\times {\mathfrak h}^*\longrightarrow{\mathbb{C}}\) on \({\mathfrak h}^*\) in the usual way, which satisfies \[(\delta_i, \delta_j)=-\delta_{i j}, \quad i, j\in[1, \ell].\]

It follows equations 35 - 37 that the set of roots of \({\mathfrak {sp}}(V;\kappa)\) relative to the Cartan subalgebra \({\mathfrak h}\) is given by \[\Phi=\{\delta_i-\delta_j\mid i, j\in [1, \ell], i\ne j\}\cup \{\delta_p+\delta_q, -(\delta_p+\delta_q)\mid p, q\in [1, \ell], p\le q\}.\]

The map \(\xi: \Phi \longrightarrow\Gamma\) (see 23 ) in this case is defined by \(\xi(\delta_i\pm \delta_j)= \gamma_i\pm \gamma_j\) for all \(i<j\) and \(\xi(2\delta_i) = 2\gamma_i\). Clearly \(\xi(\Upsilon)\in \Gamma^+\) for all \(\Upsilon\in \Phi\) as \(\omega(\xi(\Upsilon), \xi(\Upsilon))=1\).

Note that associated with the root \(\delta_i-\delta_j\), there is an \({\mathfrak {sl}}_2({\mathbb{C}})\) subalgebra in \({\mathfrak {sp}}(V;\kappa)\) spanned by the elements \(X_{i j}, X_{j i}, X_{i i}-X_{j j}\). Also, corresponding to the root \(\delta_i+\delta_j\) for any fixed \(i\le j\), we have \[\begin{align} [X_{i, \ell+j}, X_{\ell+j, i}]&= (1+ \delta_{i j}) (X_{i i} + X_{j j}). \end{align}\] Again the elements \(X_{i, \ell+j}, X_{\ell+j, i}, X_{i i} +X_{j j}\) span an \({\mathfrak {sl}}_2({\mathbb{C}})\). Thus for each root \(\Upsilon\in \Phi\), there is the reflection \(\sigma_\Upsilon\) defined by 27 , which generates the Weyl group of the \({\mathfrak {sl}}_2({\mathbb{C}})\) subalgebra associated with \(\Upsilon\). The Weyl group \(W\) of \({\mathfrak {sp}}(V; \kappa)\) is the subgroup of \({\rm{GL}}(E)\) generated by all \(\sigma_\Upsilon\).

The element \(\hat{\rho}=\sum_{i=1}^\ell (\ell+1-i) X_{i i} \in {\mathfrak h}\) satisfies \(\Upsilon(\hat{\rho})\ne 0\) for all \(\Upsilon\in \Phi\). We use it to define the set \(\Phi^+:=\{\Upsilon\in \Phi\mid \Upsilon(\hat{\rho})>0\}\) of positive roots. We have \[\begin{align} \Phi^+&= \{\delta_i-\delta_j, \delta_i+\delta_j, 2\delta_k\mid 1\le i<j\le \ell, \;1\le k\le \ell\}. \end{align}\] Then \(\Phi=\Phi^+\cup(-\Phi^+)\). The set of simple roots for this choice of the positive roots is given by \[\Pi=\{\Upsilon_i:=\delta_i-\delta_{i+1}, \Upsilon_\ell:=2\delta_\ell\mid 1\le i<\ell\}.\] Let \(\Xi=(\xi(\Upsilon_1), \xi(\Upsilon_2),\dots, \xi(\Upsilon_\ell))\). The Cartan matrix of \({\mathfrak {sp}}(V; \kappa)\) with the root datum \((\Phi, \Pi, \Xi)\) is the same as that of type \(C_\ell\), and the Dynkin diagram is depicted in Figure 2, where the node corresponds to \(\Upsilon_i\) is a double circles if \(\xi(\Upsilon_i)\ne 0\), and a circle otherwise.

We have the following result.

Let us now consider bases for \(V\) which will lead to the least number of double circles in the Dynkin diagram. Denote \(\aleph=|\Gamma_R(U)|\). Fix a total order for \(\Gamma_R(U)=\{\alpha_1, \alpha_2, \dots, \alpha_{\aleph(U)}\}\) with \(\alpha_i< \alpha_{i+1}\) for all \(i\). If \(\Gamma_R^{{\mathbb{Z}}_2}(U)\) is not empty, we further require that \(\alpha_s<\alpha_t\) for all \(\alpha_s\not\in\Gamma_R^{{\mathbb{Z}}_2}(U)\) and \(\alpha_t\in\Gamma_R^{{\mathbb{Z}}_2}(U)\). Choose an ordered basis \(B(\alpha_r)=(e(\alpha_r)_1, e(\alpha_r)_1, \dots e(\alpha_r)_{m_r})\) for each \(U_{\alpha_r}\), where \(m_r=\dim U_{\alpha_r}\). Let \(\overline{B}(-\alpha_r)=(\overline{e}(-\alpha_r)_1, \overline{e}(-\alpha_r)_2, \dots, \overline{e}(-\alpha_r)_{m_r})\) be an ordered basis for \((U^*)_{-{\alpha_r}}\) such that \(\overline{e}(-\alpha_r)_i(e(\alpha_r))_j=\delta_{i j}\). We have the following ordered bases for \(U\), \(U^*\) and \(V\) respectively. \[\begin{align} &B(U)=(B(\alpha_1), B(\alpha_2),\dots, B(\alpha_\aleph)), \tag{38}\\ &\overline{B}(U^*)=(\overline{B}(-\alpha_1), \overline{B}(-\alpha_2), \dots, \overline{B}(-\alpha_\aleph)), \tag{39} \\ &B(V)= (B(U), \overline{B}(U^*)). \tag{40} \end{align}\] Let \(\ell_r=\dim U_{\alpha_r}-1\), for \(i=1, 2, \dots, \aleph\). Then the Dynkin diagram associated with this basis of \(V\) is one of the diagrams shown below, depending on whether \(\alpha_{\aleph(U)}\in \Gamma_R^{{\mathbb{Z}}_2}(U)\) or not. \[\setlength{\unitlength}{0.25mm} \begin{picture}(200, 90)(330, -15) \put(305, 0){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{\ell_1}} \put(355, 10){\line(1, 0){10}} \put(370, 10){\circle{10}}\put(370, 10){\circle{5}} \put(375, 10){\line(1, 0){10}} \put(385, 0){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{\ell_2}} \put(435, 10){\line(1, 0){10}} \put(450, 10){\circle{10}}\put(450, 10){\circle{5}} \put(455, 10){\line(1, 0){10}} \put(465, 9){...} \put(475, 10){\line(1, 0){10}} \put(485, 0){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{\ell_{\aleph(U)}}} \put(534, 12){\line(1, 0){20}} \put(534, 8){\line(1, 0){20}} \put(559, 10){\circle{10}}\put(559, 10){\circle{5}} \put(540, 5){<} \put(305, 60){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{\ell_1}} \put(355, 70){\line(1, 0){10}} \put(370, 70){\circle{10}}\put(370, 70){\circle{5}} \put(375, 70){\line(1, 0){10}} \put(385, 60){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{\ell_2}} \put(435, 70){\line(1, 0){10}} \put(450, 70){\circle{10}}\put(450, 70){\circle{5}} \put(455, 70){\line(1, 0){10}} \put(465, 69){...} \put(475, 70){\line(1, 0){10}} \put(485, 60){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{\ell_{\aleph(U)}}} \put(534, 72){\line(1, 0){20}} \put(534, 68){\line(1, 0){20}} \put(559, 70){\circle{10}}\put(540, 65){<} \put(580, 65) {or} \end{picture}\]

3.4 Orthogonal Lie colour algebras↩︎

We consider \({\mathfrak {so}}(V; \kappa)\), where \(V=V_+\). As we will see, the structure of \({\mathfrak {so}}(V; \kappa)\) depends on the subspace \(\sum_{\alpha\in \Gamma^+_{R, {\mathbb{Z}}_2}(V)} V_\alpha\) in a crucial way. For any \(\alpha\in\Gamma^+_{R, {\mathbb{Z}}_2}(V)\), the restriction of \(\kappa\) to \(V_\alpha\) is non-degenerate by Remark 3. Since it is symmetric in the present case, \(\dim V_\alpha\) can be even or odd. It is the odd dimensional ones that pose problems.

3.4.1 The case with even dimensional \(V_\alpha\) for all \(\alpha\in\Gamma^+_{R, {\mathbb{Z}}_2}(V)\)↩︎

In this case, we again have \(V= U\oplus U^*\) with \(U=\sum_{\alpha\in \Gamma^+(U)} U_\alpha\), and \(U^*= \sum_{\alpha\in \Gamma^+(U)} (U^*)_{-\alpha}\). The bilinear form is given by \(\kappa(\overline{v}', v) = \kappa(v, \overline{v}') = \overline{v}'(v)\) for all \(v\in U, \overline{v}'\in U^*.\) Denote \(\ell=\dim U\).

We construct a homogeneous basis \(B(V)=(e_1, e_2, \dots, e_{2\ell})\) for \(V\) in the same way as described by 33 , 34 . Then \((\kappa(e_a, e_b)) = \begin{pmatrix} 0 & I_\ell\\ I_\ell & 0 \end{pmatrix}\). We have \(\omega(\gamma_a, \gamma_a)=1\) for all \(a=1, 2, \dots, 2\ell\).

The following result is an easy consequence of Theorem 5.

Let \({\mathfrak h}=\sum_{i=1}^\ell {\mathbb{C}}\;X_{i i}\), which is a homogeneous Cartan subalgebra of degree \(0\) for \({\mathfrak {sp}}(V;\kappa)\) (thus is an ordinary abelian Lie algebra). It follows from the lemma that \[\begin{align} {}[X_{i i}, X_{p p}]&=0, \\ {}[X_{i i}, X_{p q}]&= (\delta_{i p} - \delta_{i q})X_{p q}, \\ {}[X_{i i}, X_{p, \ell+ q}]&= (\delta_{i p} + \delta_{i q}) X_{p, \ell+q}, \\ {}[X_{i i}, X_{\ell+ p, q}]&= - (\delta_{i p} +\delta_{i q}) X_{\ell+p, q}, \quad i, p, q\le \ell, \; p\ne q. \end{align}\]

Let \(\varepsilon_i\), for \(i=1, 2, \dots, \ell\), be elements of the dual space \({\mathfrak h}^*\) such that \(\varepsilon_i(X_{j j})=\delta_{i j}\) for all \(i, j=1, 2, \dots, \ell\). They form a basis of \({\mathfrak h}^*\). The last three of the above equations can be re-written as \[\begin{align} {}[X_{i i}, X_{p q}]&=& (\varepsilon_p - \varepsilon_q)(X_{i i})X_{p q}, \tag{41}\\ {}[X_{i i}, X_{p, \ell+ q}]&=& (\varepsilon_p + \varepsilon_q)(X_{i i})X_{p, \ell+q}, \tag{42}\\ {}[X_{i i}, X_{\ell+ p, q}]&=& - (\varepsilon_p + \varepsilon_q)(X_{i i}) X_{\ell+p, q}, \quad i, p, q\le \ell, \; p\ne q. \tag{43} \end{align}\]

The \(\omega\)-trace gives rise to a non-degenerate \(\omega\)-symmetric bilinear form \((\;, \;): {\mathfrak {so}}(V; \kappa)\times {\mathfrak {so}}(V; \kappa)\longrightarrow{\mathbb{C}}\) on \({\mathfrak {so}}(V; \kappa)\), which is defined by \((X, Y)=\frac{1}{2}{\rm tr}_{(\Gamma, \omega)}(XY)\) for all \(X, Y\in {\mathfrak {sp}}(V; \kappa)\). Its restriction of \((\;, \;)\) to \({\mathfrak h}\) is non-degenerate, and satisfies \((X_{i i}, X_{j j})=\delta_{i j}\). It induces a symmetric biliner form \((\;, \;): {\mathfrak h}^*\times {\mathfrak h}^*\longrightarrow{\mathbb{C}}\) on \({\mathfrak h}^*\) in the usual way, which satisfies \((\varepsilon_i, \varepsilon_j)=\delta_{i j}\), for all \(i, j\in[1, \ell].\)

The roots of \({\mathfrak {so}}(V;\kappa)\) relative to the Cartan subalgebra \({\mathfrak h}\) can be read off the equations 35 - 36 , which are the elements of the set \[\Phi=\{\varepsilon_i-\varepsilon_j, \varepsilon_i+\varepsilon_j \mid i, j\in [1, \ell], i\ne j\}.\]

Each root \(\varepsilon_i-\varepsilon_j\) for \(i<j\) is associated with an \({\mathfrak {sl}}(2)\) Lie subalgebra spanned by the elements \(X_{i j}\), \(X_{j i}\), and \(X_{i i}-X_{j j}\). Similarly, each root \(\varepsilon_i+\varepsilon_j\) with \(i\ne j\) is associated with an\({\mathfrak {sl}}(2)\) Lie subalgebra spanned by the elements \(X_{i, \ell+j}, X_{\ell+j, i}\), and \(\widetilde{H}_{i j}:=X_{i i}+X_{j j}\), which satisfy the commutation relations. \[[X_{i, \ell+j}, X_{\ell+j, i}]= \widetilde{H}_{i j}, \quad [\widetilde{H}_{i j}, X_{i, \ell+j}] = 2 X_{i, \ell+j}, \quad [\widetilde{H}_{i j}, X_{\ell+j, i}]=-2 X_{\ell+j, i}.\] Thus the reflection \(\sigma_\Upsilon\) defined by 27 for each root \(\Upsilon\in \Phi^+\) generates the Weyl group of the corresponding \({\mathfrak {sl}}_2({\mathbb{C}})\) subalgebra. The Weyl group \(W\) of \({\mathfrak {sp}}(V; \kappa)\) is the subgroup of \({\rm{GL}}(E)\) generated by all \(\sigma_\Upsilon\), where \(E={\mathbb{R}}\Phi\).

The element \(\hat{\rho}=\sum_{i=1}^\ell (\ell+1-i) X_{i i} \in {\mathfrak h}\) can again be used to define positive roots: \(\Phi^+:=\{\Upsilon\in \Phi\mid \Upsilon(\hat{\rho})>0\}\). We have \[\begin{align} \Phi^+&= \{\varepsilon_i-\varepsilon_j, \varepsilon_i+\varepsilon_j\mid 1\le i<j\le \ell\}. \end{align}\] Then the set of simple roots is given by \[\Pi=\{\Upsilon_i:=\varepsilon_i-\varepsilon_{i+1}, \Upsilon_\ell:=\varepsilon_{\ell-1}+\varepsilon_\ell\mid 1\le i<\ell\},\] and we let \(\Xi=(\xi(\Upsilon_1), \xi(\Upsilon_2),\dots, \xi(\Upsilon_\ell))\). The Cartan matrix is the same as that of type \(D_\ell\), and the Dynkin diagram is as depicted in Figure 3, where the \(i\)-th node is a circle (resp. double circle) if \(\xi(\Upsilon_i)=0\) (resp. \(\xi(\Upsilon_i)\ne 0\)).

Order the elements of \(\Gamma_R(U)\) as in Section 3.3. Let \[\begin{align} \label{eq:k-D} k_r=\dim U_{\alpha_r}-1, \quad i=1, 2, \dots, \aleph(U). \end{align}\tag{44}\] Then the Dynkin diagram is given by one of the two diagrams in Figure 3, depending on whether \(\alpha_{\aleph(U)}\in\Gamma_{{\mathbb{Z}}_2}\). Note that if \(k_{\aleph(U)}=0\), the last two nodes in the Dynkin diagram are connected to a double circle.

\[\setlength{\unitlength}{0.25mm} \begin{picture}(200, 90)(330, -10) \put(305, 0){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{k_1}} \put(355, 10){\line(1, 0){10}} \put(370, 10){\circle{10}}\put(370, 10){\circle{5}} \put(375, 10){\line(1, 0){10}} \put(385, 0){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{k_2}} \put(435, 10){\line(1, 0){10}} \put(450, 10){\circle{10}}\put(450, 10){\circle{5}} \put(455, 10){\line(1, 0){10}} \put(465, 9){...} \put(475, 10){\line(1, 0){10}} \put(485, 0){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{k_{\aleph(U)}-1}} \put(534, 12){\line(2, 1){20}} \put(534, 8){\line(2, -1){20}} \put(559, 23){\circle{10}} \put(559, -3){\circle{10}}\put(559, -3){\circle{5}} \put(305, 60){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{k_1}} \put(355, 70){\line(1, 0){10}} \put(370, 70){\circle{10}}\put(370, 70){\circle{5}} \put(375, 70){\line(1, 0){10}} \put(385, 60){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{k_2}} \put(435, 70){\line(1, 0){10}} \put(450, 70){\circle{10}}\put(450, 70){\circle{5}} \put(455, 70){\line(1, 0){10}} \put(465, 69){...} \put(475, 70){\line(1, 0){10}} \put(485, 60){\underbrace{\begin{picture}(50, 20)(0, 0) \put(5, 10){\circle{10}} \put(10, 10){\line(1, 0){10}} \put(20, 9){...} \put(30, 10){\line(1, 0){10}} \put(45, 10){\circle{10}} \end{picture}}_{k_{\aleph(U)}-1}} \put(534, 72){\line(2, 1){20}} \put(534, 68){\line(2, -1){20}} \put(559, 83){\circle{10}} \put(559, 57){\circle{10}} \put(585, 66){or} \end{picture}\]

3.4.2 The case with one odd dimensional \(V_\eta\) for \(\eta\in\Gamma^+_{R, {\mathbb{Z}}_2}(V)\)↩︎

We assume that there is an \(\eta\in \Gamma^+_{R, {\mathbb{Z}}_2}(V)\) such that \(\dim V_\eta\) is odd, and \(\dim V_\alpha\) is even if \(\eta\ne \alpha\in \Gamma^+_{R, {\mathbb{Z}}_2}(V)\). Then \(V_\eta=U_\eta \oplus U_\eta^*\oplus E_\eta\), where \(E_\eta ={\mathbb{C}}e(\eta)\) is a \(1\)-dimensional homogeneous subspace of degree \(\eta\) with basis \(e(\eta)\), and \(U_\eta\) is a homogeneous subspace of degree \(\eta\) with dual space \(U_\eta^*\). Now \(V= U\oplus U^*\oplus E_\eta\), where \(U\) and its dual space \(U^*\) are the same as in Section 3.4.1, where \(U_\eta\subset U\) and \(U_\eta^*\subset U^*\). The bilinear form \(\kappa\) is given by \(\kappa(v+\overline{v}'+ a e(\eta), u + \overline{u}'+ b e(\eta)) = \overline{v}'(u)+ \overline{u}'(v) + a b\) for all \(u, v\in U, \overline{u}', \overline{v}'\in U^*\) and \(a, b\in {\mathbb{C}}.\)

Denote \(\ell=\dim U\), thus \(\dim V=2\dim U +1=2\ell+1\). We have the ordered bases \(B(U)\) and \(\overline{B}(U^*)\) for \(U\) and \(U^*\) respectively, as defined by 33 . Now we introduce the following ordered basis for \(V\). \[\begin{align} \label{eq:B-odd} B(V)=(e_1, e_2, \dots, e_{2\ell+1}) := (B(U), e(\eta), \overline{B}(U^*)). \end{align}\tag{45}\] Then the bilinear form \(\kappa\) is given by \((\kappa(e_a, e_b)) = \begin{pmatrix} 0 & 0 & I_\ell\\ 0 & 1 & 0 \\ I_\ell & 0 &0 \end{pmatrix}\).

We can easily extract the following relations from the lemma. \[\begin{align} {}[X_{i i}, X_{p q}]&= (\delta_{i p} -\delta_{i q})X_{p q}, \\ [X_{i i}, X_{p, \ell+1+ q}]&= (\delta_{i p} +\delta_{i q})X_{p, \ell+1+q}, \\ [X_{i i}, X_{\ell+1+ p, q}]&=- (\delta_{i p}+\delta_{i q})X_{\ell+1+p, q}, \\ [X_{i i}, X_{p, \ell+1}]&= \delta_{i p} X_{p, \ell+1}, \\ [X_{i i}, X_{\ell+1, p}]&= -\delta_{i p}X_{\ell+1, p}, \quad \forall i, p, q=1, 2, \dots, \ell. \end{align}\] Thus \({\mathfrak h}=\sum_{i=1}^\ell {\mathbb{C}}X_{i i}\) forms a Cartan subalgebra, which is homogeneous of degree \(0\).

Let \(\{\varepsilon_i\mid i=1, 2, \dots, \ell\}\) be a basis of the dual space \({\mathfrak h}^*\) such that \(\varepsilon_i(X_{j j})=\delta_{i j}\) for all \(i, j=1, 2, \dots, \ell\). It follows the above equations that the set of roots of \({\mathfrak {so}}(V; \kappa)\) with respect to the Cartan subalgebra \({\mathfrak h}\) is given by \[\Phi = \{\pm(\varepsilon_i-\varepsilon_j), \pm (\varepsilon_i+\varepsilon_j)\mid i, j\in [1, \ell], i\le j\}\cup \{\pm \varepsilon_i \mid i\in [1, \ell]\}.\]

The \(\omega\)-trace yields a non-degenerate \(\omega\)-symmetric bilinear form \((\;, \;): {\mathfrak {so}}(V; \kappa)\times {\mathfrak {so}}(V; \kappa)\longrightarrow{\mathbb{C}}\) on \({\mathfrak {so}}(V; \kappa)\), whose restriction to \({\mathfrak h}\) is non-degenerate and induces a symmetric bilinear form \((\;, \;): {\mathfrak h}^*\times {\mathfrak h}^*\longrightarrow{\mathbb{C}}\) on \({\mathfrak h}^*\) such that \((\varepsilon_i, \varepsilon_j)=\delta_{i j}\).

There is an \({\mathfrak {sl}}(2)\) subalgebra corresponding to each of the roots \(\varepsilon_i\pm \varepsilon_j\) for \(i<j\). Also, corresponding to each \(\varepsilon_i\), the elements \(X_{i, \ell+1}\), \(X_{\ell+1, i}\) and \(X_{i i}\) span an \({\mathfrak {so}}(3)\simeq {\mathfrak {sl}}()\) subalgebra, with the commutation relations \[[X_{i, \ell+1}, X_{\ell+1, i}]=X_{i i}, \quad [X_{i i}, X_{i, \ell+1}]=X_{i, \ell+1}, \quad [X_{i i}, X_{\ell+1, i}]=-X_{\ell+1, i}.\] Thus the reflection \(\sigma_\Upsilon\) (cf. 27 ) for each \(\Upsilon\in \Phi^+\) generates the Weyl group of the associated \({\mathfrak {sl}}_2({\mathbb{C}})\) subalgebra. The Weyl group \(W\) of \({\mathfrak {so}}(V; \kappa)\) is generated by all \(\sigma_\Upsilon\).

We will take the set of simple roots \[\Pi = \{ \Upsilon_i = \varepsilon_i- \varepsilon_{i+1}, \Upsilon_\ell= \varepsilon_\ell \mid 1\le i<\ell\},\] and let \(\Xi=(\xi(\Upsilon_1), \xi(\Upsilon_2),\dots, \xi(\Upsilon_\ell))\). Then the Cartan matrix is the standard one for \(B_\ell\), and the corresponding Dynkin diagram is as depicted in Figure 4.

Let us consider bases of \(V\) which will lead to the least number of double circles in the Dynkin diagram. Order the elements of \(\Gamma_R(U)=\{\alpha_1, \alpha_2, \dots, \alpha_{\aleph}\}\) so that

i) \(\alpha_i<\alpha_j\) if \(i<j\);

ii) \(\alpha_s< \alpha_t\) if \(\alpha_s\not\in\Gamma^+_{R, {\mathbb{Z}}_2}(U)\) and \(\alpha_t\in\Gamma^+_{R, {\mathbb{Z}}_2}(U)\); and

iii) \(\eta\) is minimal.

Construct bases \(B(U)\) for \(U\) and \(\overline{B}(U^*)\) for \(U^*\) as those given by 38 and 39 . We again take \(B(V)= (B(U), e(\eta), \overline{B}(U^*))\). Then the Dynkin diagram corresponding to this basis has \(\aleph\) double circles if \(\eta\ne 0\), and \(\aleph-1\) double circles if \(\eta= 0\). Note that in the latter case, the last node in Figure 4 should be a circle.

3.4.3 Failure of Cartan-Weyl paradigm↩︎

If there are more than one \(\alpha\in\Gamma_{{\mathbb{Z}}_2}\) such that \(V_\alpha\) are odd dimensional, \({\mathfrak {so}}(V; \kappa)\) fails the Cartan-Weyl paradigm. This is very easy to see in the example below.

Assume that there are \(r\ge 2\) distinct elements \(\gamma_i\in\Gamma^+_{R, {\mathbb{Z}}_2}(V)\) \((i=1, 2, \dots, r)\) such that \(V_{\gamma_i}\) are odd dimensional. We always have \(V=V^{0}\oplus V^{1}\), where \(V^{1}\) is the sum of homogeneous \(1\)-dimensional subspaces of degrees \(\gamma_i\) for \(i=1, 2, \dots, r\) respectively, such that \(V^0\) and \(V^1\) are orthogonal with respect to \(\kappa\). Let \(\ell=(\dim V- r)/2\). We can choose homogeneous bases for \(V^0\) and \(V^1\) such that \(\kappa=\begin{pmatrix} 0 & I_\ell & 0\\ I_\ell & 0 &0\\ 0 & 0 &I_r \end{pmatrix}\). Then \({\mathfrak {so}}(V; \kappa)\) contains the subalgebra \({\mathfrak {so}}(V^0; \kappa|_{V^0}) \oplus {\mathfrak {so}}(V^1; \kappa|_{V^1})\). Now \({\mathfrak {so}}(V^0; \kappa|_{V^0})\) fulfils the Cartan-Weyl paradigm by Section 3.4.1, but \({\mathfrak {so}}(V^1; \kappa|_{V^1})\simeq {\mathfrak {so}}_r(\gamma_1, \dots, \gamma_r)\) fails as discussed in the example above. This implies that \({\mathfrak {so}}(V; \kappa)\) too fails the Cartan-Weyl paradigm.

3.5 Untwisted affine Lie colour algebras↩︎

We note that examples of affine Lie colour (super)algebras were studied in [35], [36].

Consider the Laurent polynomial ring \({\mathbb{C}}[t, t^{-1}]\) in the variable \(t\) as a \(\Gamma\)-graded vector space consisting of the homogeneous subspace of degree \(0\) only. Given a finite dimensional Lie colour algebra \({\mathfrak g}\), we let \({\mathcal{L}}({\mathfrak g})={\mathfrak g}\otimes{\mathbb{C}}[t, t^{-1}]\) as a \(\Gamma\)-graded vector space. The loop Lie colour algebra of \({\mathfrak g}\) is \({\mathcal{L}}({\mathfrak g})\) with the generalised Lie bracket being the bilinear map defined by \[\begin{align} [X\otimes f, Y\otimes g]=[X, Y]\otimes fg, \quad X, Y\in{\mathfrak g},\; f, g\in{\mathbb{C}}[t, t^{-1}]. \end{align}\] The generalised Jacobian identity of \({\mathcal{L}}({\mathfrak g})\) follows that of \({\mathfrak g}\).

Assume that there is a \(\omega\)-symmetric bilinear form \(\kappa\) on \({\mathfrak g}\), which is homogeneous of degree \(0\), and is ad-invariant. The untwisted affine Lie colour algebra \(\widehat{\mathfrak g}= {\mathcal{L}}({\mathfrak g}) +{\mathbb{C}}c\) of \({\mathfrak g}\) is the central extension of the loop algebra \({\mathcal{L}}({\mathfrak g})\) with a generalised Lie bracket such that for any \(X, Y\in{\mathfrak g}\) and \(f, g\in{\mathbb{C}}[t, t^{-1}]\), \[\begin{align} &&[X\otimes f, Y\otimes g]=[X, Y]\otimes fg + c \kappa(X, Y) Res_0\left(\frac{dg}{d t} f\right), \end{align}\] where \(Res_0(F)\) is the coefficient of \(t^{-1}\) in \(F\in{\mathbb{C}}[t, t^{-1}]\). [Bear in mind that \(c\) is central.]

To verify that this indeed defines a Lie colour (super)algebra, we drop the tensor product sign from \(X\otimes f\) etc.. We have \[\begin{align} [Yg, X f] &= [Y, X] f g + c \kappa(Y, X) Res_0(g\frac{d f}{d t})\\ &= -\omega(d Y, d X) \left([X, Y] f g + c \kappa(X, Y) Res_0(f\frac{d g}{d t})\right)\\ &= -\omega(d Y, d X) [Xf, Yg], \end{align}\] proving the \(\omega\)-skew symmetry of the generalised Lie bracket.

The generalised Jacobian identity can be verified in a similar way as for usual affine Lie algebras. Write \[\begin{align} {\mathscr J}&=[X f, [Y g, Z h]]-[[X f, Y g], Z h] \\ &-\omega(d(X), d(Y)) [Y g, [X f, Z h]]. \end{align}\] By using the generalised Jacobian identity of \({\mathcal{L}}({\mathfrak g})\), we obtain \[\begin{align} \label{eq:jac-aff} {\mathscr J}&=& c \kappa(X, [Y, Z]) Res_0\left(\frac{dgh}{d t} f\right) - c \kappa([X, Y], Z]) Res_0\left(\frac{dh}{d t} g f\right) \\ && - c\kappa(Y, [X, Z]) \omega(d(X), d(Y)) Res_0\left(\frac{df h}{d t} g\right). \nonumber \end{align}\tag{46}\] Now \(\kappa([X, Y], Z])=\kappa(X, [Y, Z])\) by ad-invariance of \(\kappa\), and \[\begin{align} \phantom{=}&\omega(d(X), d(Y)) \kappa(Y, [X, Z])\\ &=\omega(d(Y), d(Z)) \kappa([X, Z], Y) &\quad& \text{by \omega-symmetry of \kappa}\\ &= \omega(d(Y), d(Z)) \kappa(X, [Z, Y]) &\quad& \text{by ad-invariance of \kappa}\\ &=-\kappa(X, [Y, Z]), &\quad& \text{by \omega-skew symmetry of [\;, \;].} \end{align}\] Hence the right hand side of 46 is equal to \[\begin{align} \phantom{=}& c \kappa(X, [Y, Z]) \left( Res_0\left(\frac{dgh}{d t} f\right) - Res_0\left(\frac{dh}{d t} g f\right) +Res_0\left(\frac{df h}{d t} g\right)\right)\\ &= c \kappa(X, [Y, Z]) Res_0\left(\frac{dfgh}{d t}\right) =0, \end{align}\] since \(\frac{d F}{d t}\) does not contain a \(t^{-1}\) term for any \(F\in{\mathbb{C}}[t, t^{-1}]\). Hence \({\mathscr J}=0\), proving the generalised Jacobian identity for \(\widehat{\mathfrak g}\).

Assume that the Lie colour algebra \({\mathfrak g}\) fulfils the Cartan-Weyl paradigm, thus \({\mathfrak g}={\mathfrak g}(A; \Xi)\) for some reduced Cartan matrix of finite type. Denote by \(\Upsilon_1, \dots, \Upsilon_\ell\) the simple roots of the Lie colour algebra \({\mathfrak g}(A;\Xi)\). There exist \(k_i\in{\mathbb{Z}}_+\) such that \(\theta=\sum_{i\in I} k_i\Upsilon_i\) is the highest root of \({\mathfrak g}(A;\Xi)\). Let \[\begin{align} \label{eq:Upsil0} \Upsilon_0=-\sum_{i\in I} k_i\Upsilon_i, \quad \xi_0=-\sum_{i=1}^\ell k_i \xi_i, \end{align}\tag{47}\] and set \(\widehat{I}=\{0, 1, 2, \dots, \ell\}\) and \(\hat{\Xi}=(\xi_i)_{i\in\widehat{I}}\). Denote by \(\widehat{A}=(\widehat{A}_{i j})_{i, j\in \widehat{I}}\) the generalised Cartan matrix such that \(\widehat{A}_{0 i}=\frac{2(\Upsilon_0, \Upsilon_i)}{(\Upsilon_0, \Upsilon_0)}\), \(\widehat{A}_{i 0}=\frac{2(\Upsilon_i, \Upsilon_0)}{(\Upsilon_i, \Upsilon_i)}\) for \(i\in \widehat{I}\), and \(\widehat{A}_{i j} = A_{i j}\) for \(i, j\in I\). We can choose root vectors \(X_\theta\) and \(Y_\theta\) of \(\pm \theta\) respectively such that the Cartan element \(h_\theta:=[X_\theta, Y_\theta]\) satisfies \[[h_\theta, X_\theta]= 2 X_\theta, \quad [h_\theta, Y_\theta]=-2Y_\theta.\] Then \(X_0:= Y_\theta t\), \(Y_0:= X_\theta t^{-1}\) and \(Z_0:=\kappa(Y_\theta, X_\theta) c - h_\theta\) satisfy the relations \[\begin{align} &[Z_0, X_i]= -\Upsilon_i(h_\theta) X_i, \quad [Z_0, X_i]= \Upsilon_i(h_\theta) X_i, \\ &[X_i, Y_0]=0, \quad [Y_i, X_0]=0, \quad\quad \forall i=1, 2, \dots, |I|, \\ &[Z_0, X_0] =2X_0, \quad [Z_0, Y_0] = -2 Y_0, \quad [X_0, Y_0] = Z_0. \end{align}\]

3.6 Lie and affine colour algebras fulfilling the Cartan-Weyl paradigm↩︎

Investigations in the last section suggest the existence of the following class of (affine) Lie colour algebras, which realise the Cartan-Weyl paradigm.

We assume that \(\Gamma =\Gamma^+\).

4 Quantised universal enveloping colour algebras↩︎

We construct quantised universal enveloping algebras of the Lie colour algebras and affine Lie colour algebras defined in Section 3.6, which realise the Cartan-Weyl paradigm. The mathematical foundation for this section lies in the theory of Hopf \((\Gamma, \omega)\)-algebras, which are discussed in considerable detail in the appendix, Section 6.

We will work over the field of rational functions \({\mathbb{C}}(q)\) in the indeterminate \(q\). Let \([n]_q=\frac{q^n - q^{-n}}{q-q^{-1}},\) and \(\begin{bmatrix}m\\ r\end{bmatrix}_q= \frac{[m]_q!}{[m-r]_q![r]_q!}.\)

4.1 Colour quantum groups↩︎

We first streamline notation for clarity. We will use \(A=(A_{i j})_{i, j \in I}\) to denote a Cartan matrix either of finite type or affine type, with \(I=\{1, 2, \dots, \ell\}\) in the finite case, and \(I=\{0, 1, 2, \dots, \ell\}\) in the affine case. We also have the sequence \(\Xi=(\xi_i)_{i\in I}\) of elements of \(\Gamma\) in both cases, where we recall that, in the affine case, \(\xi_0\) is defined by 47 . We emphasise that here we only consider the case with \(\xi_i\in \Gamma^+\) for all \(i\in I\). Let \(d_i\) be the smallest positive even integers such that \(B=dia(d_1, \dots, d_\ell) A\) is a symmetric matrix over \({\mathbb{Z}}\). Set \(q_i =q^{d_i/2}\).

The commutative factor appears explicitly in the defining relations 50 , 51 and 52 . Equations 51 and 52 are called colour quantum Serre relations.

Clearly \({\rm{U}}_{q, \Xi}({\mathfrak g})\) has the following \((\Gamma, \omega)\)-subalgebras: \[\begin{align} {\rm{U}}^+_+, &\quad \text{generated by \{e_i\mid i\in I \}}, \\ {\rm{U}}^0, &\quad \text{generated by \{ k^{\pm 1}\mid i\in I\}}, \\ {\rm{U}}^-_-, &\quad \text{generated by \{ f_i\mid i\in I\}}, \\ {\rm{U}}^+_{q, \Xi}({\mathfrak g}), &\quad \text{generated by {\rm{U}}^+_+ and {\rm{U}}^0}, \\ {\rm{U}}^-_{q, \Xi}({\mathfrak g}), &\quad \text{generated by {\rm{U}}^-_- and {\rm{U}}^0}. \end{align}\] By inspecting equations 48 –50 , one can easily see that the multiplication induces the \(\Gamma\)-graded vector space isomorphism \[{\rm{U}}^-_-\otimes{\rm{U}}^0\otimes{\rm{U}}^+_+\stackrel{\sim}\longrightarrow{\rm{U}}_{q, \Xi}({\mathfrak g}).\]

The following important fact will be proven in the next section.

It follows Theorem 17 that \({\rm{U}}^\pm_{q, \Xi}({\mathfrak g})\) are Hopf \((\Gamma, \omega)\)-subalgebras.

4.2 Verification of the Hopf \((\Gamma, \omega)\)-algebra structure↩︎

In this section, we prove Theorem 17 to elucidate the Hopf structure of the colour quantum groups.

Let us introduce the following auxiliary algebra related to \({\rm{U}}_{q, \Xi}(A)\).

It is clear that \(\widetilde{\rm{U}}_{q, \Xi}(A)\) has the following \((\Gamma, \omega)\)-subalgebras. \[\begin{align} \widetilde{\rm{U}}^+_+, &\quad \text{generated by \{e_i\mid i\in I \}}, \\ \widetilde{\rm{U}}^0, &\quad \text{generated by \{ k^{\pm 1}\mid i\in I\}}, \\ \widetilde{\rm{U}}^-_-, &\quad \text{generated by \{ f_i\mid i\in I\}}, \\ \widetilde{\rm{U}}^+_{q, \Xi}({\mathfrak g}), &\quad \text{generated by \widetilde{\rm{U}}^+_+ and \widetilde{\rm{U}}^0}, \\ \widetilde{\rm{U}}^-_{q, \Xi}({\mathfrak g}), &\quad \text{generated by \widetilde{\rm{U}}^-_- and \widetilde{\rm{U}}^0}. \end{align}\]

The following fact is easy to see.

Define the following elements of \(\widetilde{\rm{U}}_{q, \Xi}({\mathfrak g})\): \[\begin{align} S^+_{i j} = Ad_{e_i}^{1-A_{i j}} (e_j), \quad S^-_{i j}=Ad_{f_i}^{1-A_{i j}} (f_j), \quad i\ne j. \end{align}\] The proof of Lemma 5 can be adapted to show that \[\begin{align} S_{i j}^+&=\sum_{r=0}^{1-A_{i j}} (- \omega(\xi_i, \xi_j))^r\begin{bmatrix}1-A_{i j}\\ r\end{bmatrix}_{q_i} e_i^{1-A_{i j}-r} e_j e_i^r, \tag{61}\\ S_{i j}^-&= \sum_{r=0}^{1-A_{i j}} (- \omega(\xi_i, \xi_j))^r\begin{bmatrix}1-A_{i j}\\ r\end{bmatrix}_{q_i} f_i^{1-A_{i j}-r} f_j f_i^r. \tag{62} \end{align}\] We have the following technical results.

Recall that the \(\omega\)-commutator is defined by \([X, Y]_\omega= X Y - \omega(d(X), d(Y)) Y X\).

The proofs of the two lemmas above will be given in Section 4.3.

The following results immediately follow from Lemmas 7 and 8.

Equipped with Corollary 2, we can now prove Theorem 17

4.3 Proofs of technical lemmas↩︎

We prove Lemmas 7 and 8 in this section.

4.3.1 Some useful formulae↩︎

We collect here some useful formulae, some of which will be used later in this section.

The following relations hold, which are familiar from ordinary quantum groups. \[\begin{align} [e_i, f_i^r] &=&[r]_{q_i} f_i^{r-1} \frac{k_i q_i^{-r +1} - k_i^{-1}q_i^{ r-1}}{q_i-q_i^{-1}}, \tag{66} \\ {}[f_i, e_i^r] &=&-[r]_{q_i} e_i^{r-1} \frac{k_i q_i^{ r-1}- k_i^{-1} q_i^{-r +1} }{q_i-q_i^{-1}}; \tag{67}\\ \Delta(e_i^n) &=& \sum_{s=0}^n q_i^{s(n-s)}\begin{bmatrix}n \\ s\end{bmatrix}_{q_i} e_i^{n-s} \otimes e_i^s k_i^{n-s},\tag{68}\\ \Delta(f_i^n) &=& \sum_{s=0}^n q_i^{s(n-s)}\begin{bmatrix}n \\ s\end{bmatrix}_{q_i} f_i^s k_i^{-n+s}\otimes f_i^{n-s}. \tag{69} \end{align}\]

The the first and third formulae can be proven by inductions. One can deduce the second relation from the first, and the fourth relation from the third, by using the following algebra automorphism.

Finally, we have the following formulae.

4.3.2 Proof of Lemmas 7 and 8↩︎

5 Quasi triangularity of colour quantum groups↩︎

The theory of quasi triangular Hopf algebras [50], [51] and superalgebras [64], including Drinfeld’s quantum double construction, is generalised to Hopf \((\Gamma, \omega)\)-algebras in Section 6. We will prove the quasi triangular Hopf algebra structure of the quantised universal enveloping algebra \({\rm{U}}_{q, \Xi}(A)\) of any Lie colour algebra or an affine analogue. For this purpose, it is best to work over the formal Laurent series ring \({\mathbb{C}}((\hbar))\) equipped with the \(\hbar\)-adic topology. The general setup of topological Hopf \((\Gamma, \omega)\)-algebras is discussed in Section 6.4, which we refer to for details.

In this section, we will freely make use of material from Section 6.

We denote \({\mathbb{K}}={\mathbb{C}}((\hbar))\) throughout this section.

5.1 Colour quantum groups over Laurent series ring↩︎

We adopt the convention and notation for finite and affine root systems in Section 4.1. In particular, we use \(A=(A_{i j})_{i, j \in I}\) to denote either a finite type Cartan matrix with \(I=\{1, 2, \dots, \ell\}\) or affine type Cartan matrix with \(I=\{0, 1, 2, \dots, \ell\}\). We also have the sequence \(\Xi=(\xi_i)_{i\in I}\) in each case, where we recall that \(\xi_0\) is defined by 47 in the affine case. Let \((\Phi, \Pi, \Xi)\) be the root datum associated with \(A\), where \(\Phi\) is the set of roots, and \(\Pi=\{\Upsilon_i\mid i\in I\}\) is the set of simple roots.

Fix the ring homomorphism \[\begin{align} \psi: {\mathbb{C}}(q)\longrightarrow{\mathbb{K}}, \quad q\mapsto \exp(\hbar):=\sum_{i\ge 0} \frac{\hbar^i}{i!}. \end{align}\] We consider the specialisation \({\rm{U}}_{q, \Xi}(A)\otimes_\psi{\mathbb{K}}\) to \({\mathbb{K}}\) of the Hopf \((\Gamma, \omega)\)-algebra \({\rm{U}}_{q, \Xi}(A)\) with respect \(\psi\). It can be thought as a Hopf \((\Gamma, \omega)\)-algebra over \({\mathbb{K}}\) generated by \(e_i, f_i, k_i^{\pm 1}\) with \(i\in I\) and defining relations given in Definition 6, and the same Hopf algebra structure as described in Theorem 17. Consider the topological Hopf \((\Gamma, \omega)\)-algebra over \({\mathbb{K}}\) generated by it and the additional generators \(h_i\) (\(i\in I\)), which are related to the specialisations of \(k_i^{\pm 1}\) as follows. \[\begin{align} \label{eq:ks} k_i^{\pm 1}= \exp(\pm \frac{\hbar d_i}{2} h_i):=\sum_{n=0}^\infty \frac{\hbar^n}{ n!} \left(\frac{\pm h_i d_i}{2}\right)^n. \end{align}\tag{72}\] It then follows that the elements \(h_i\) commute among themselves and satisfy the following relations with the elements \(e_i, f_i\): \[h_i e_j - e_j h_i= A_{i j} e_j, , \quad h_i f_j - f_j h_i=- A_{i j} f_j.\] Their co-products are given by \(\Delta(h_i)=h_i\otimes 1+ 1\otimes h_i\).

We will show that \({\rm{U}}_\hbar(A; \Xi)\) has the structure of a quasi triangular topological Hopf \((\Gamma, \omega)\)-algebra, by reconstructing it from the quantum double (see Definition 12) of a quotient of the topological Hopf \((\Gamma, \omega)\)-algebra \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)\) to be defined presently.

5.2 Auxiliary topological \((\Gamma, \omega)\)-algebras↩︎

For convenience, we write \(q=\exp(\hbar)\) \(:=\sum_{i\ge 0} \frac{\hbar^i}{i!}\) and \(q_i= \exp(\hbar d_i/2)\), for all \(i\in I\), throughout this section.

Let us introduce the following topological \((\Gamma, \omega)\)-algebra.

Consider the topological subalgebras \(\widetilde{\rm{U}}_0=\langle h_i\mid i\in I\rangle\) and \(\widetilde{\rm{U}}_+=\langle e_i\mid i\in I\rangle\) of \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)\). The algebra \(\widetilde{\rm{U}}_0\) is the completion of the algebra of polynomials in \(h_i\)’s with respect to the \(\hbar\)-adic topology. For any \({\boldsymbol{m}}=(m_1, m_2, \dots, m_{|I|})\in{\mathbb{Z}}_+^{|I|}\), let \(h^{\boldsymbol{m}}=\prod_{i\in I} h_i^{m_i}\). Then \(\widetilde{\rm{U}}_0\) has the monomial basis \(\{h^{\boldsymbol{m}}\mid {\boldsymbol{m}}\in{\mathbb{Z}}_+^{|I|}\}\). The \((\Gamma, \omega)\)-subalgebra \(\widetilde{\rm{U}}_+\) is also \({\mathbb{Z}}_+\Pi\)-graded, with \(\widetilde{\rm{U}}_+=\sum_{\mu\in {\mathbb{Z}}_+\Pi}(\widetilde{\rm{U}}_+)_\mu\), where \((\widetilde{\rm{U}}_+)_\mu\) will be called the weight \(\mu\)-subspace. For any \(\nu=\sum_i \ell_i \Upsilon_i\) in \({\mathbb{Z}}_+\Phi^+\), we let \(|\nu|=|-\nu|=\sum_i \ell_i\). Let \(E({\boldsymbol{j}})= e_{j_1}e_{j_2}\dots e_{j_n}\), where \({\boldsymbol{j}}=(j_1, \dots, j_n)\) \(\in\) \(I^n\) and \(n\in{\mathbb{Z}}_+\). If \(n=0\), we set \(E({\boldsymbol{j}})=1\). Call \(\mu({\boldsymbol{j}}) =\sum_{t=1}^n \Upsilon_{j_t}\) the weight, and \(n\) the length, of \(E({\boldsymbol{j}})\). The elements \(E({\boldsymbol{j}})\) form a basis of \(\widetilde{\rm{U}}_+\). Note that any \(E_\mu\in (\widetilde{\rm{U}}_+)_\mu\) with \(\mu=\sum_{i\in I} \mu_i\Upsilon_i\) is spanned by \(E({\boldsymbol{j}})\)’s of length \(|\mu|=\sum_i \mu_i\) and weight \(\mu=\mu({\boldsymbol{j}})\). Furthermore, we have the following basis for \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)\). \[\begin{align} h^{\boldsymbol{m}} E({\boldsymbol{j}}), \quad \text{for {\boldsymbol{m}}\in{\mathbb{Z}}_+^{|I|}, {\boldsymbol{j}}\in I^n, n\in{\mathbb{Z}}_+}. \end{align}\]

We have the following formula for the co-product of \(h^{\boldsymbol{m}}\). \[\Delta(h^{\boldsymbol{m}}) =\Delta'(h^{\boldsymbol{m}})=\sum_{\boldsymbol{r}} \begin{pmatrix} {\boldsymbol{m}}\\ {\boldsymbol{r}} \end{pmatrix} h^{{\boldsymbol{m}}-{\boldsymbol{r}}}\otimes h^{\boldsymbol{r}}, \quad \begin{pmatrix} {\boldsymbol{m}}\\ {\boldsymbol{r}} \end{pmatrix}=\prod_{i\in I} \begin{pmatrix} m_i\\ r_i \end{pmatrix}.\]

We introduce some further notation for later use. For any \(E({\boldsymbol{j}})\), we define \[\begin{align} \hat{E}({\boldsymbol{j})}_{;i}^{R, \pm}&= \sum_{t=1}^n \delta_{i, j_t}\omega\big(\xi_i, \sum_{s>t} \xi_{j_s}\big) q^{\pm(\Upsilon_i, \sum_{s<t} \Upsilon_{j_s})} e_{j_1} e_{j_2}\dots e_{j_{t-1}}e_{j_{t+1}} \dots e_{j_n}, \\ \hat{E}({\boldsymbol{j})}_{;i}^{L, \pm}&= \sum_{t=1}^n \delta_{i, j_t} \omega\big(\sum_{s=1}^{t-1} \xi_{j_s}, \xi_i\big) q^{\pm (\Upsilon_i, \sum_{s<t} \Upsilon_{j_s})} e_{j_1} e_{j_2}\dots e_{j_{t-1}} e_{j_{t+1}} \dots e_{j_n }\\ \end{align}\] and generalise this linearly to any \(E_\mu= \sum_{\boldsymbol{j}} c_{\boldsymbol{j}}E({\boldsymbol{j}})\in (U_+)_\mu\) with \(c_{\boldsymbol{j}}\in{\mathbb{K}}\). \[\begin{align} \label{eq:deriv-i} \hat{E}_{\mu, i}^{R, \pm} = \sum c_{\boldsymbol{j}} \hat{E}({\boldsymbol{j}})_i^{R, \pm}, \quad \hat{E}_{\mu, i}^{L, \pm} = \sum c_{\boldsymbol{j}} \hat{E}({\boldsymbol{j}})_i^{L, \pm}. \end{align}\tag{73}\] Note that \[\hat{E}_{\mu, i}^{R, \pm}= \omega\big(\xi_i, d E_\mu -\xi_i\big) \hat{E}_{\mu, i}^{L, \pm}.\]

The topological dual space \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)^*\) of \({\rm{U}}^+_\hbar(A;\Xi)\) (i.e., space of continuous linear functions) is a \((\Gamma, \omega)\)-algebra with the multiplication \(\Delta^*: \widetilde{\rm{U}}^+_\hbar(A;\Xi)^*\widehat\otimes\widetilde{\rm{U}}^+_\hbar(A;\Xi)^*\longrightarrow\widetilde{\rm{U}}^+_\hbar(A;\Xi)^*.\) Denote by \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)'\) the topological \((\Gamma, \omega)\)-subalgebra of \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)^*\) generated by the homogeneous elements \(f_i, \widetilde{H}_i\) (\(i\in I\)) of \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)^*\), with \(\Gamma\)-degrees \(d(f_i)=-\xi_i\) and \(d(\widetilde{H}_i)=0\) respectively, such that \[\varphi_i =(q_i-q_i^{-1}) f_i, \quad \rho_i=\hbar \frac{ d_i}{2} \widetilde{H}_i, \quad i\in I,\] satisfy the following relations for any \(b_\mu\in (\widetilde{\rm{U}}_+)_\mu\): \[\begin{align} \label{eq:pair-def} \phantom{XXX} \left\langle \rho_i, h^{\boldsymbol{n}} b_\mu\right\rangle =\delta_{\mu 0} \delta_{{\boldsymbol{n}} {\boldsymbol{1}}_i}, \quad \langle \varphi_i, h^{\boldsymbol{n}} e_j\rangle=\delta_{i j} \delta_{\boldsymbol{n} 0}, \quad \langle \varphi_i, h^{\boldsymbol{n}} b_\mu\rangle=0 \text{ if } \mu\not\in\Pi, \end{align}\tag{74}\] where \(\delta_{\boldsymbol{n} m}=\prod_{r\in I} \delta_{n_r, m_r}\), and \({\boldsymbol{1}}_i=(0, \dots, 0, \underbrace{1}_i, 0, \dots, 0)\in {\mathbb{Z}}_+^{|I|}\).

Let \(\widetilde{\rm{U}}'_0\) be the topological subalgebra of \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)'\) generated by the \(\rho_i\)’s, and \(\widetilde{\rm{U}}'_-\) that generated by the \(\varphi_i\)’s. A basis for \(\widetilde{\rm{U}}_0'\) is given by \(\{\rho^{\boldsymbol{m}}=\prod_{i\in I}\rho_i^{m_i} \mid {\boldsymbol{m}}=(m_i)_{i\in I}\in{\mathbb{Z}}_+^{|I|}\}\). Now \(\widetilde{\rm{U}}'_+\) is \(-{\mathbb{Z}}_+\Pi\)-graded. For any \({\boldsymbol{j}}=(j_1, \dots, j_n)\) \(\in\) \(I^n\), where \(n\in{\mathbb{Z}}_+\), let \(\phi({\boldsymbol{j}})= \varphi_{j_1}\varphi_{j_2}\dots \varphi_{j_n}\). Call \(\mu({\boldsymbol{j}}) =-\sum_{t=1}^n \Upsilon_{j_t}\) the weight, and \(n\) the length, of \(\phi({\boldsymbol{j}})\). The elements \(\phi({\boldsymbol{j}})\) form a basis of \(\widetilde{\rm{U}}'_+\), and we have the following basis for \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)'\). \[\begin{align} \rho^{\boldsymbol{m}} \phi({\boldsymbol{j}}), \quad \text{for {\boldsymbol{m}}\in{\mathbb{Z}}_+^{|I|}, {\boldsymbol{j}}\in I^n, n\in{\mathbb{Z}}_+}. \end{align}\]

Note that \(\widetilde{h}_i=\sum_j A_{i j}\widetilde{H}_j\) satisfies \[\widetilde{h}_i \varphi_j - \varphi_j \widetilde{h}_i = - A_{i j} \varphi_j, \quad \forall j.\]

5.3 Quantised enveloping algebras of Borel subalgebras↩︎

5.3.1 Colour quantum Serre relations↩︎

Let \(\mathcal{J}^+=\{x\in \widetilde{\rm{U}}^+_\hbar(A;\Xi)\mid \langle \widetilde{\rm{U}}^+_\hbar(A;\Xi)', x\rangle =\{0\} \}\). Clearly \(\mathcal{J}^+\) is a two-sided ideal of \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)\), which is homogeneous since dual space pairing preserves the grading. If \(x\in \mathcal{J}^+\), we have \(\langle \beta\otimes\beta', \Delta(x)\rangle = \langle \beta\beta', x\rangle = 0\) for all \(\beta, \beta'\in \widetilde{\rm{U}}^+_\hbar(A;\Xi)'\), thus \(\Delta(x)\in \mathcal{J}^+\otimes\widetilde{\rm{U}}^+_\hbar(A;\Xi) + \widetilde{\rm{U}}^+_\hbar(A;\Xi)\otimes\mathcal{J}^+\). This shows that \(\mathcal{J}^+\) is a homogeneous Hopf ideal of \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)\).

Now we have the Hopf \((\Gamma, \omega)\)-algebra \({\rm{U}}^+_\hbar(A;\Xi):=\widetilde{\rm{U}}^+_\hbar(A;\Xi)/\mathcal{J}^+\). We will still denote the images of \(e_i\) and \(k_i\) in the quotient by the same symbols.

An element \(E_\mu\in (\widetilde{\rm{U}}_+)_\mu\), where \(\mu>0\), is called skew primitive if its co-product is of the form \(\Delta(E_\mu)=E_\mu\otimes k_\mu + 1 \otimes E_\mu\), where \(k_\mu\) is a product of \(k_i\)’s such that \(k_\mu e_i k_\mu^{-1}= q^{(\Upsilon_i, \mu)} e_i\) for all \(i\in I\). Clearly \(e_i\)’s are skew primitive. Let \(S^+_{i j}\), for \(i\ne j\), be the elements in \(\widetilde{\rm{U}}_+\) defined by the formula 61 but with \(e_\ell\)’s being elements of \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)\). The proof of Lemma 10 is still valid in the current context, showing that \(S^+_{i j}\) are skew primitive.

The following fact is easy to prove.

The following result is an immediate consequence of the lemma.

Define \(\mathcal{J}^-=\{f\in \widetilde{\rm{U}}^+_\hbar(A;\Xi)'\mid \langle f, \widetilde{\rm{U}}^+_\hbar(A;\Xi)\rangle =\{0\} \}\). We can similarly show, as in the case of \(\mathcal{J}^+\), that \(\mathcal{J}^-\) is a homogeneous Hopf ideal of \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)'\). Let \(S^-_{i j}\), for \(i\ne j\), be the elements in \(\widetilde{\rm{U}}^+_\hbar(A;\Xi)'\) defined by the formula 62 with \(f_\ell\)’s replaced by \(\varphi_\ell\). Then \(S^-_{i j}\) are skew primitive.

We have the following result, whose proof is similar to that of Lemma 16.

The following important fact immediately follows from the definitions of \({\rm{U}}^+_\hbar(A;\Xi)\) and \({\rm{U}}^+_\hbar(A;\Xi)'\).

5.3.2 Relationship to Nichols algebras↩︎

Since the quantised universal enveloping algebras defined in Section 4.1 are associated with (affine) Lie colour algebras fulfilling the Cartan-Weyl paradigm, results obtained in previous subsections are closely related to Nichols algebras [69]–[72] (see [68] for a review) through a de-bosonisation process, even though our treatment is within the framework of Hopf colour algebras.

Let \(\mathfrak{B}_Q(I)\) be the Nichols algebra of a \(|I|\)-dimensional braided vector space with diagonal braiding given by the matrix \(Q=(q_{i j})_{i, j\in I}\) with \(q_{i j} =\omega(\gamma_i, \gamma_j) q^{A_{i j}}\), and denote by \({\mathcal{B}}\mathfrak{B}_Q(I)\) the bosonisation of \(\mathfrak{B}_Q(I)\) (using a multiplicative abelian group, see e.g., [69], [70]). By inspecting the definitions of \(\mathfrak{B}_Q(I)\) and \({\rm{U}}^+_\hbar(A;\Xi)\), one can easily see that \({\mathcal{B}}\mathfrak{B}_Q(I)\) is isomorphic to the subalgebra \({\rm{U}}_{q; \Xi}^+(A)\) of \({\rm{U}}^+_\hbar(A;\Xi)\) generated by \(e_i+\mathcal{J}^+\) and \(k_i^{\pm 1}+\mathcal{J}^+\) (\(i\in I\)), where \(\mathfrak{B}_Q(I)\) is identified with the subalgebra \({\rm{U}}_{q; \Xi}^{++}\subset{\rm{U}}_{q; \Xi}^+(A)\) generated by the elements \(e_i+\mathcal{J}^+\). The following fact is extracted from a well-known result in the theory of Nichols algebras (see e.g., [68]–[70]).

A detailed proof of the lemma requires considerable background material on Nichols algebras, thus we will refrain from spelling it out here. We merely point out that in the usual non-graded case, the analogous result was first proved by Ross [71] for finite type Cartan matrices, and by Lusztig [72] for affine type Cartan matrices. Work of Andruskiewitsch and co-workers (see e.g., [69], [70]) enables one to treat the grading, thus to deduce the lemma from [71], [72]. [The usual proof for Serre presentations of finite dimensional simple Lie algebras (see, e.g., [89]) was generalised to the Nichols algebra setting using the notion of Weyl groupoid (see e.g., [69], [70]).]

An analogous statement is true for the dual Hopf algebra \({\rm{U}}_{q; \Xi}^+(A)'\).

5.4 Colour quantum groups as quantum doubles↩︎

In this section, the Hopf \((\Gamma, \omega)\)-algebra \({\rm{U}}^+_\hbar(A;\Xi)'\) is taken to be \(({\rm{U}}^+_\hbar(A;\Xi)', \Delta_0=(\Delta^0)', \varepsilon_0=\varepsilon^0, S_0=(S^0)^{-1})\). We can read off the structure maps from Lemma 13. In particular, \[\begin{align} \Delta_0(\widetilde{h}_i) = \widetilde{h}_i\otimes 1 + 1\otimes\widetilde{h}_i, \quad \Delta_0(f_i) =f_i\otimes 1 + \widetilde{\nu}_i\otimes f_i. \end{align}\]

5.4.1 Actions of dual Hopf algebras↩︎

Let us make some preparations for studying the quantum double of \({\rm{U}}^+_\hbar(A;\Xi)\). It follows Section 6.2.2 that there are the following left and right \({\rm{U}}^+_\hbar(A;\Xi)\)-actions on \({\rm{U}}^+_\hbar(A;\Xi)'\), \[\begin{align} \rhd: &\; {\rm{U}}^+_\hbar(A;\Xi)\otimes{\rm{U}}^+_\hbar(A;\Xi)'\longrightarrow{\rm{U}}^+_\hbar(A;\Xi)', \\ \lhd: &\; {\rm{U}}^+_\hbar(A;\Xi)'\otimes{\rm{U}}^+_\hbar(A;\Xi) \longrightarrow{\rm{U}}^+_\hbar(A;\Xi)', \\ \end{align}\] and left and right \({\rm{U}}^+_\hbar(A;\Xi)'\)-actions on \({\rm{U}}^+_\hbar(A;\Xi)\), \[\begin{align} \blacktriangleright: &\; {\rm{U}}^+_\hbar(A;\Xi)'\otimes{\rm{U}}^+_\hbar(A;\Xi)\longrightarrow{\rm{U}}^+_\hbar(A;\Xi), \\ \blacktriangleleft: &\; {\rm{U}}^+_\hbar(A;\Xi)\otimes{\rm{U}}^+_\hbar(A;\Xi)' \longrightarrow{\rm{U}}^+_\hbar(A;\Xi), \\ \end{align}\] which are respectively defined, for any \(x\in {\rm{U}}^+_\hbar(A;\Xi)\) and \(f\in {\rm{U}}^+_\hbar(A;\Xi)'\), by \[\begin{align} &x\rhd f=\sum_{(f)} \omega(d x, d f_{(1)}) \langle f_{(1)}, x\rangle f_{(2)}, \\ &f\lhd x=\sum_{(f)} f_{(1)} \langle f_{(2)}, x\rangle,\\ &f\blacktriangleright x=\sum_{(x)} \langle S_0(f), x_{(1)}\rangle x_{(2)}, \\ & x\blacktriangleleft f=\sum_{(x)} \omega(d x_{(2)}, d f) x_{(1)}\langle S_0(f), x_{(2)}\rangle. \end{align}\] Clearly \[\begin{align} \label{eq:U-act-dU-0} 1\blacktriangleright x=x, \quad 1\lhd x=\epsilon(x), \quad f\blacktriangleright 1=\epsilon_0(f), \quad f\lhd 1=f. \end{align}\tag{83}\] One can show that for any \(E\in ({\rm{U}}_+)_{\mu}\) with \(\mu>0\), \[\begin{align} \varphi_i\blacktriangleright E = -k_i \hat{E}_{; i}^{L, -}, \quad E\blacktriangleleft\varphi_i =- \omega(\gamma_i, \gamma_i) \hat{E}_{; i}^{R, +}. \end{align}\] We have \[\begin{align} \label{eq:Delta40E41} \quad \Delta(E)= E\otimes k_\mu + \sum _i e_i\otimes k_i \hat{E}_{;i}^{L, -} +\dots+ \sum _i \hat{E}_{;i}^{R, +} \otimes e_i k_{\mu-\Upsilon_i} + 1\otimes E. \end{align}\tag{84}\]

The lemma below can be easily verified by direct calculations.

There is also a left action \(R: {\rm{U}}^+_\hbar(A;\Xi)'\otimes{\rm{U}}^+_\hbar(A;\Xi)\longrightarrow{\rm{U}}^+_\hbar(A;\Xi)\) defined by \[\begin{align} f\otimes x\mapsto R_f(x):=\sum_{(x)} \omega(d f, d x_{(1)}) x_{(1)} \langle f, x_{(2)}\rangle, \end{align}\] which is the analogue of right translations in the context of Lie groups.

One can show that for any \(E\in ({\rm{U}}_+)_\mu\) with \(\mu>0\), \[\begin{align} R_{\varphi_j} (E)&=\sum_{(E_\mu)} \omega( d E_{(1)}, \gamma_j) E_{(1)} \langle \varphi_j, E_{(2)}\rangle\\ &= \omega( d {E}-\gamma_j, \gamma_j) \widehat{E}_{; j}^{R, +} q^{-(\Upsilon_j, \mu-\Upsilon_j)}\\ &= \omega( d E-\gamma_j, \gamma_j) Ad_{k_j^{-1}}(\widehat{E}_{; j}^{R, +}), \end{align}\] that is, \(\widehat{E}_{; j}^{R, +} =\omega(\gamma_j, d E) Ad_{k_j}( R_{\varphi_j} (E)).\)

5.4.2 Quantum double construction↩︎

Theorem 19 gives a non-degenerate Hopf pairing between \({\rm{U}}^+_\hbar(A;\Xi)\) and \(U^+_\hbar(A;\Xi)'\). It enables us to construct the quantum double (see Defintion 12) of \({\rm{U}}^+_\hbar(A;\Xi)\), \[\begin{align} {\mathscr D}_\hbar(A; \Xi)={\rm{U}}^+_\hbar(A;\Xi)\otimes_{{\mathbb{K}}} {\rm{U}}^+_\hbar(A;\Xi)', \end{align}\] where \({\rm{U}}^+_\hbar(A;\Xi)'\) has the opposite Hopf algebra structure. We denote by \({\mathscr R}\) the universal \(R\)-matrix of the topological quasi triangular Hopf \((\Gamma, \omega)\)-algebra \({\mathscr D}_\hbar(A; \Xi)\).

The Hopf algebras \({\rm{U}}^+_\hbar(A;\Xi)\) and \({\rm{U}}^+_\hbar(A;\Xi)'\) are naturally embedded in \({\mathscr D}_\hbar(A; \Xi)\) with images \({\rm{U}}^+_\hbar(A;\Xi)\otimes 1\) and and \(1\otimes{\rm{U}}^+_\hbar(A;\Xi)'\) respectively. For any \(y\in {\rm{U}}^+_\hbar(A;\Xi)\) and \(f\in {\rm{U}}^+_\hbar(A;\Xi)'\), we shall simply write their images \(y\otimes 1\) and \(1\otimes f\) as \(y\) and \(f\) respectively. Then \(y f \in {\mathscr D}_\hbar(A; \Xi)\) is understood as \(y\otimes f\), and \(f y\in {\mathscr D}_\hbar(A; \Xi)\) as \((1\otimes f)(y \otimes 1)\). Recall from Section 6.3 that in \({\mathscr D}_\hbar(A; \Xi)\), \[f y =\sum_{(f), (y)} \omega(d f_{(2)}, d y_{(1)}) (f_{(1)} \blacktriangleright y_{(1)}) (f_{(2)} \lhd y_{(2)}).\]

We now work out the relations between the generators of \({\rm{U}}^+_\hbar(A;\Xi)\) and \({\rm{U}}^+_\hbar(A;\Xi)'\) in \({\mathscr D}_\hbar(A; \Xi)\). By using equation 83 , one can easily show that \[\begin{align} \rho_i h_j &= \sum \left( ({\rho_i}\blacktriangleright{h_j}_{(1)}) (1\lhd {h_j}_{(2)}) + (1\blacktriangleright{h_j}_{(1)}) (\rho_i \lhd {h_j}_{(2)})\right)\\ &= {\rho_i}\blacktriangleright{h_j} + \sum {h_j}_{(1)} (\rho_i \lhd {h_j}_{(2)})\\ &= {\rho_i}\blacktriangleright{h_j} + {h_j} \rho_i + \rho_i \lhd {h_j}; \end{align}\] \[\begin{align} \rho_i e_j &= \sum \left( ({\rho_i}\blacktriangleright{e_j}_{(1)}) (1\lhd {e_j}_{(2)}) + (1\blacktriangleright{e_j}_{(1)}) (\rho_i \lhd {e_j}_{(2)})\right)\\ &= {\rho_i}\blacktriangleright{e_j}+ \sum{e_j}_{(1)}(\rho_i \lhd {e_j}_{(2)})\\ &= {\rho_i}\blacktriangleright{e_j}+ \rho_i \lhd {e_j} + {e_j} (\rho_i \lhd k_j); \end{align}\] \[\begin{align} \varphi_i h_j &= \sum \left( ({\varphi_i}_{(1)}\blacktriangleright{h_j}) ({\varphi_i}_{(2)}\lhd 1) +({\varphi_i}_{(1)}\blacktriangleright 1) ({\varphi_i}_{(2)}\lhd {h_j}) \right)\\ &= \sum ({\varphi_i}_{(1)}\blacktriangleright{h_j}) {\varphi_i}_{(2)} +{\varphi_i}\lhd {h_j} \\ &= {\varphi_i}\blacktriangleright{h_j} + ({\widetilde{\nu}_i}\blacktriangleright{h_j}) \varphi_i +{\varphi_i}\lhd {h_j} = ({\widetilde{\nu}_i}\blacktriangleright{h_j}) \varphi_i, \end{align}\] \[\begin{align} \varphi_i e_j &=\sum \left( ({\varphi_i}_{(1)}\blacktriangleright{e_j}) ({\varphi_i}_{(2)}\lhd k_i) +\omega(\gamma_j, \gamma_i) ({\varphi_i}_{(1)}\blacktriangleright 1) ({\varphi_i}_{(2)}\lhd {e_j}) \right)\\ &= ({\varphi_i}\blacktriangleright{e_j}) (1\lhd k_i) + \omega(\gamma_j, \gamma_i) (\widetilde{\nu_i}\blacktriangleright{e_j}) ({\varphi_i}\lhd k_i) +\omega(\gamma_j, \gamma_i) {\varphi_i}\lhd {e_j} \\ &= {\varphi_i}\blacktriangleright{e_j} + \omega(\gamma_j, \gamma_i) (\widetilde{\nu_i}\blacktriangleright{e_j}) ({\varphi_i}\lhd k_i) +\omega(\gamma_j, \gamma_i) {\varphi_i}\lhd {e_j}. \end{align}\] Applying Lemma 19, we obtain \[\begin{align} &\rho_i h_j = h_j \rho_i; \\ &\rho_i e_j = e_j\left(\rho_i + \delta_{i j}\frac{d_i}{2} \hbar \right), \quad \varphi_i h_j = h_j \varphi_i + A_{j i}\varphi_i, \\ &\varphi_i e_j =\omega(\gamma_j, \gamma_i) e_j \varphi_i +\delta_{i j} (\widetilde{\nu_i} - k_i ), \end{align}\] which immediately leads to the following result.

It is clear from the definitions of \({\rm{U}}^+_\hbar(A;\Xi)\) and \({\rm{U}}^+_\hbar(A;\Xi)'\), and Lemmas 11, 13, 16, 17 and 21 that the elements \(h_i-\widetilde{h_i}\), for all \(i\in I\), generate a Hopf ideal in \({\mathscr D}_\hbar(A; \Xi)\), which will be denoted by \(\langle h_i-\widetilde{h_i}\mid i\in I\rangle\). Let \[\mathcal{p}: {\mathscr D}_\hbar(A; \Xi)\longrightarrow\frac{{\mathscr D}_\hbar(A; \Xi)}{\langle h_i-\widetilde{h_i}\mid i\in I\rangle}\] be the canonical surjection, which is a quasi triangular topological Hopf algebra homomorphism, where the the universal \(R\)-matrix of the image of \(\mathcal{p}\) is given by \(R:=\mathcal{p}\otimes\mathcal{p}({\mathscr R})\).

In view of Lemma 18 for \({\rm{U}}_{q; \Xi}^+(A)\) and the analogous fact for the dual Hopf algebra \({\rm{U}}_{q; \Xi}^+(A)'\), we have the following result.

Finally, we make connection of the discussion here with Lemma 7. We note that for any \(E\in ({\rm{U}}_+)_\mu\) with \(\mu>0\), \[\begin{align} [f_i, E]_\omega&= - \omega(\xi_i, \xi_i) \frac{k_i \hat{E}_{; i}^{L, -} - k_i^{-1} \hat{E}_{; i}^{L, +}}{q_i-q_i^{-1}}, \end{align}\] which is equivalent to \([E, f_i]_\omega= \frac{k_i \hat{E}_{; i}^{R, -} - k_i^{-1} \hat{E}_{; i}^{R, +}}{q_i-q_i^{-1}}\). In view of 84 , any skew primitive element \(E\in ({\rm{U}}_+)_\mu\), with \(0<\mu\not\in\Pi\), graded \(\omega\)-commutes with all \(f_i\).

6 Quasi triangular Hopf \((\Gamma, \omega)\)-algebras↩︎

Drinfeld’s theory of quasi triangular Hopf algebras [50], [51] provides a theoretical framework for Baxter’s \(R\) matrices [49]. It has been generalised to Hopf superalgebras in [64]. Quantum groups [49], [50], [52], quantum supergroups [53], [55], [56], [59], [60] and their affine analogues are the most important examples of quasi triangular Hopf (super)algebras.

Here we describe quasi triangular Hopf \((\Gamma, \omega)\)-algebras for an arbitrary grading group \(\Gamma\) equipped with a commutative factor \(\omega\), giving detailed proofs of vital results including the quantum double construction.

We work in the algebraic setting first for simplicity, and then extend the results to the topological setting.

Fix a commutative ring \({\mathbb{K}}\) with identity. Let \(\Gamma\) be an additive abelian group with a commutative factor \(\omega\). Recall that \(\omega(\gamma, \gamma)=\pm 1\).

. We will write the \(\Gamma\)-degree of a homogeneous element \(v\) in a vector space as \(d v\) in this section, as the original notation \(d(v)\) is too cumbersome in long handed computations.

6.1 Quasi-triangular Hopf \((\Gamma, \omega)\)-algebras↩︎

This section and the next section give a more detailed treatment of the relevant material in [6], and also define the notion of quasi-triangular Hopf \((\Gamma, \omega)\)-algebras.

6.1.1 Hopf \((\Gamma, \omega)\)-algebras↩︎

Let \(H\) be an associative \((\Gamma, \omega)\)-algebra over \({\mathbb{K}}\), with multiplication \(\mu: H\otimes H\longrightarrow H\) and unit map \(u: {\mathbb{K}}\longrightarrow H\). Let \(1_H\) be the unit element of \(H\), then \(u: a\mapsto a 1_H\). The \((\Gamma, \omega)\)-algebra \(H\) is a \((\Gamma, \omega)\)-bi-algebra if there exist \((\Gamma, \omega)\)-algebra homomorphisms \(\varepsilon: H\longrightarrow{\mathbb{C}}\) and \(\Delta: H\longrightarrow H\otimes H\) such that \[\begin{align} &&(\Delta\otimes{\rm{id}})\Delta= ({\rm{id}}\otimes\Delta)\Delta: H\longrightarrow H\otimes H\otimes H, \tag{85}\\ &&(\varepsilon\otimes{\rm{id}})\Delta ={\rm{id}}= ({\rm{id}}\otimes\varepsilon)\Delta: H\longrightarrow H, \tag{86} \end{align}\] where the second condition involves the canonical identifications \({\mathbb{C}}\otimes H\simeq H\simeq H\otimes{\mathbb{C}}\). Call \(\varepsilon\) the co-unit and \(\Delta\) the co-multiplication. Equation 85 is the co-associativity of the co-multiplication.

We will adopt Sweedler’s notation to write \(\Delta(x)=\sum_{(x)}x_{(1)}\otimes x_{(2)}\), \((\Delta\otimes{\rm{id}})\Delta(x)= \sum_{(x)}x_{(1)}\otimes x_{(2)}\otimes x_{(3)}\), and etc. for any \(x\in H\). We denote by \(\Delta'\) the opposite co-multiplication. For any \(x\in H\), \[\Delta'(x)=\sum_{(x)}\omega(d x_{(1)}, d x_{(2)}) x_{(2)}\otimes x_{(1)}.\] The following formula will be useful. \[\begin{align} \label{eq:D2-H} \begin{aligned} &({\rm{id}}\otimes\Delta)\Delta(y z) =(\Delta\otimes{\rm{id}})\Delta(y z)\\ &=\sum \omega(d y_{(2}+d y_{(3)}, d z_{(1)}) \omega(d y_{(3)}, d z_{(2)}) \\ & \times y_{(1)} z_{(1)} \otimes y_{(2)} z_{(2)}\otimes y_{(3)} z_{(3)}, \end{aligned} \end{align}\tag{87}\] where we note that \(\omega(d y_{(2}+d y_{(3)}, d z_{(1)})=\omega(d y -d y_{(1)}, d z_{(1)})\).

A Hopf \((\Gamma, \omega)\)-algebra [6] is a \((\Gamma, \omega)\)-bi-algebra \((H, \mu, u, \Delta, \varepsilon)\) with a \((\Gamma, \omega)\)-algebra anti-homomorphism \(S: H\longrightarrow H,\) called the antipode, such that \[\begin{align} \label{eq:Del-S} \mu(S\otimes\Delta) = \mu(\Delta\otimes S)= \varepsilon. \end{align}\tag{88}\] The fact that \(S\) is an \((\Gamma, \omega)\)-algebra anti-homomorphism means that \(S(x y)=\omega(d x, d y)S(y) S(x)\) for \(x, y\in H\). Equation 88 can be expressed in Sweedler’s notation as \(\sum_{(x)}S(x_{(1)}) x_{(2)} = \sum_{(x)}x_{(1)}S(x_{(2)})= \varepsilon(x)\) for all \(x\in A\).

If the antipode is bijective, \((H, \mu, u, \Delta', \epsilon, S^{-1})\) is a Hopf algebra, the opposite Hopf algebra of \((H, \mu, u, \Delta, \epsilon, S)\).

A \(\Gamma\)-graded module for a Hopf \((\Gamma, \omega)\)-algebra \(H\) is a \(\Gamma\)-graded module for \(H\) as an associative \((\Gamma, \omega)\)-algebra. If \(V\) and \(W\) are \(\Gamma\)-graded \(H\)-modules, \(V\otimes W\) is a \(\Gamma\)-graded \(H\otimes H\)-module with the action \((H\otimes H) \otimes(V\otimes W)\longrightarrow V\otimes W\) defined by \[(x\otimes y)\cdot(v\otimes w)= \omega(d y, d v) x\cdot v\otimes y\cdot w, \quad x, y\in H, v\in V, w\in W.\] As the co-multiplication \(\Delta: H\longrightarrow H\otimes H\) is a \((\Gamma, \omega)\)-algebra homomorphism, it induces an \(H\)-action \(H \otimes(V\otimes W)\longrightarrow V\otimes W\) on \(V\otimes W\), which is given by \[\begin{align} x\otimes(v\otimes w)\mapsto \Delta(x)\cdot(v\otimes w). \label{eq:act-tensor} \end{align}\tag{89}\]

6.1.2 Quasi-triangular Hopf \((\Gamma, \omega)\)-algebras↩︎

It is straightforward to devise a notion of quasi-triangular Hopf \((\Gamma, \omega)\)-algebras by generalising from the \({\mathbb{Z}}_2\)-grading [64] to arbitrary grading.

The following important theorem is a consequence of 90 and 91 .

6.2 Finite duals of Hopf \((\Gamma, \omega)\)-algebras↩︎

We adapt results on finite duals of Hopf algebras [90] to the \(\Gamma\)-graded setting.

6.2.1 Finite duals of Hopf \((\Gamma, \omega)\)-algebras↩︎

Let \((H, \mu, u, \Delta, \varepsilon, S)\) be a Hopf \((\Gamma, \omega)\)-algebra, whose opposite Hopf algebra is \((H, \mu, u, \Delta', \epsilon, S^{-1})\).

Denote by \(H^*={\rm{Hom}}_{\mathbb{K}}(H, {\mathbb{K}})\) and \((H\otimes H)^*={\rm{Hom}}_{\mathbb{K}}(H\otimes H, {\mathbb{K}})\) the \(\Gamma\)-graded dual \({\mathbb{K}}\)-modules of \(H\) and \(H\otimes H\) respectively. Let \[\begin{align} &\Delta^*: H^* \otimes H^*\longrightarrow H^*, \quad \mu^*: H^*\longrightarrow(H\otimes H)^*, \label{eq:maps42}\\ & u^*: H^*\longrightarrow{\mathbb{K}}, \quad \epsilon^*: {\mathbb{K}}\longrightarrow H^*, \quad S^*: H^*\longrightarrow H^* \end{align}\tag{94}\] be the adjoint maps of \(\Delta\), \(M\), \(u\), \(\epsilon\) and \(S\) respectively, defined, for any \(f, g\in H^*\), by \[\begin{align} &&\langle\Delta^*(f\otimes g), x \rangle = \langle f\otimes g, \Delta(x) \rangle, \quad \forall x\in H, \tag{95}\\ &&\langle \mu^*(f), x\otimes y\rangle =\langle f, x y\rangle, \quad \forall x, y \in H \\ &&u^*(f) = \langle f, 1_H \rangle, \\ &&\epsilon^*(a) = a\epsilon, \quad \forall a\in {\mathbb{K}}, \\ &&\langle S^*(f), x\rangle = \langle f, S(x)\rangle, \quad \forall x\in H. \tag{96} \end{align}\] The maps are all homogeneous of degree \(0\). We denote \(\epsilon^*\) by \(u_{H^*}\).

Now \((H^*, \Delta^*, u_{H^*})\) is a \((\Gamma, \omega)\)-algebra with multiplication \(\Delta^*\) and unit map \(u_{H^*}\), where the associativity of \(\Delta^*\) follows from the co-associativity of \(\Delta\). Now \(H^*\otimes H^*\) is a subalgebra of \((H\otimes H)^*\), but in general \({\rm{Im}}(\mu^*)\not\subset H^*\otimes H^*\), thus \(H^*\) is not a Hopf algebra.

A co-finite homogeneous ideal \(\mathcal{J}\) of \(H\) (as a \((\Gamma, \omega)\)-algebra) is a homogeneous \(2\)-sided ideal such that \(H/\mathcal{J}\) is a finitely generated \({\mathbb{K}}\)-module. Let \[H^0 = \{f\in H^*\mid \ker(f) \text{ contains a co-finite homogeneous ideal}\},\] and call it the finite dual of \(H\). Some comments are in order.

(1). If for any non-vanishing \(x\in H\), there exists \(f\in H^0\) such that \(\langle f, x\rangle\ne 0\), we say that \(H^0\) is dense in \(H^*\). It is clear that this occurs if and only if for any non-zero \(x\in H\), there is a homogeneous co-finite ideal which does not contain \(x\). In this case, \(H\) is called a proper Hopf \((\Gamma, \omega)\)-algebra.

(2). We call an \(H\)-module \({\mathbb{K}}\)-finite if it is finitely generated over \({\mathbb{K}}\). Given a \({\mathbb{K}}\)-finite \(H\)-module \(V\) generated by the generators \(v_1, \dots, v_r\) over \({\mathbb{K}}\), there are elements \(\rho_{i j} \in H^*\), such that \(x\cdot v_i=\sum_j \rho_{j i}(x) v_j\). Call \(\rho_{j i}\) the matrix coefficients of \(V\) relative to this generating set. Clearly \(\rho_{i j}\in H^0\).

(3). If \(f\in H^0\), let \(\mathcal{J}_f\) be a co-finite homogeneous ideal contained in \(\ker(f)\). Then \(H/\mathcal{J}_f\) is a \({\mathbb{K}}\)-finite \(H\)-module with the action \(y\cdot(x+\mathcal{J}_f)= xy + \mathcal{J}_f\) for all \(x, y\in H\). If \(H/\mathcal{J}_f=0\), we have \(f=0\). If \(H/\mathcal{J}_f\ne 0\), let \(t_i\in H\), for \(i=1, 2, \dots, r\), be elements such that their images \(t_i+\mathcal{J}_f\) generate \(H/\mathcal{J}_f\) over \({\mathbb{K}}\). Then \(1+\mathcal{J}_f = \sum_{i} a_i (t_i + \mathcal{J}_f)\) for some \(a_i\in{\mathbb{K}}\), and hence \(x+\mathcal{J}_f = \sum_{i, j} a_i \rho_{j i} (x) (t_j + \mathcal{J}_f)\) for all \(x\in H\), where \(\rho_{i j}\) are the matrix coefficients of \(H/\mathcal{J}_f\) with respect to the generating set \(\{ t_i +\mathcal{J}_f\}\). Then \(f(x) = \sum_{i, j} a_i f(t_j) \rho_{j i}(x)\). This shows that \(f = \sum_{i, j} a_i f(t_j) \rho_{j i}.\) Therefore \(f\in H^0\) if and only if it is a \({\mathbb{K}}\)-linear combination of matrix coefficients of some \({\mathbb{K}}\)-finite \(H\)-modules.

(4). If \(\rho^{(1)}_{i j}\) and \(\rho^{(2)}_{rs}\) are the matrix coefficients of two \({\mathbb{K}}\)-finite \(H\)-modules \(V_1\) and \(V_2\) respectively, \(\sum_{i, j; r, s} {\mathbb{K}}\rho^{(1)}_{i j}\rho^{(2)}_{rs}\) tis he \({\mathbb{K}}\)-span of the matrix coefficients of the tensor product \(H\)-module \(V_1\otimes V_2\), which is again \({\mathbb{K}}\)-finite. Hence \(H^0\) is a subalgebra of \(H^*\).

(5). As \(H/\mathcal{J}_f\) is finitely generated over \({\mathbb{K}}\), and so is also \((H/\mathcal{J}_f)^*\). There is a canonical embedding \(\iota_f: (H/\mathcal{J}_f)^*\hookrightarrow H^0\), \(\bar{g}\mapsto \iota(\bar{g})\), such that \(\langle \iota_f(\bar{g}), x\rangle = \langle \bar{g}, x+\mathcal{J}_f\rangle\) for all \(x\in H\). Now it is easy to see that \(\mu^*(f)\in \iota_f((H/\mathcal{J}_f)^*)\otimes\iota_f((H/\mathcal{J}_f)^*)\subset H^0\otimes H^0\).

The following result generalises a basic fact of usual Hopf algebras [90] to Hopf \((\Gamma, \omega)\)-algebra.

Assume that the antipode of the Hopf \((\Gamma, \omega)\)-algebra \((H, \mu, u, \Delta, \epsilon, S)\) is bijective. Then \(S^0\) is bijective. The opposite of the finite dual Hopf \((\Gamma, \omega)\)-algebra \(H^0\) of \(H\) has the structure maps \[\begin{align} &\text{multiplication } \mu_0=\mu^0, \quad \text{unit map } u_0=u^0, \tag{97}\\ &\text{co-multiplication } \Delta_0 = \tau_{H^0, H^0}\circ \Delta^0, \quad \text{co-unit } \epsilon_0=\epsilon^0, \tag{98}\\ &\text{antipode } S_0=(S^0)^{-1}. \tag{99} \end{align}\] Note in particular that \[\begin{align} \langle \Delta_0(f), x\otimes y\rangle &= \langle \Delta^0(f),\tau_{H, H}(x\otimes y)\rangle =\omega(d x, d y)\langle f, y x\rangle, \\ \langle S_0(f), x\rangle &=\langle f, S^{-1}(x)\rangle, \quad f\in H^0, x, y\in H. \end{align}\] We denote this Hopf \((\Gamma, \omega)\)-algebra by \((H^0, \mu_0, u_0, \Delta_0, \epsilon_0, S_0)\).

Hereafter we will always refer to \((H^0, \mu_0, u_0, \Delta_0, \epsilon_0, S_0)\) whenever \(H^0\) is considered as a Hopf \((\Gamma, \omega)\)-algebra. We write \(\Delta_0(f)=\sum_{(f)} f_{(1)} \otimes f_{(2)}\) for any \(f\in H^0\).

6.2.2 Technical results on dual Hopf \((\Gamma, \omega)\)-algebras↩︎

The dual Hopf \((\Gamma, \omega)\)-algebras \(H\) and \(H^0\) act on each other. We have the following

\[\begin{align} &\text{left action \;} H\otimes H^0\longrightarrow H^0, \quad x\otimes f\mapsto x\rhd f, \\ &\text{right action \;} H^0\otimes H\longrightarrow H^0, \quad f\otimes x \mapsto f\lhd x, \end{align}\] and \[\begin{align} &\text{left action \;} H^0\otimes H\longrightarrow H, \quad f\otimes x \mapsto f\blacktriangleright x, \\ &\text{right action \;} H\otimes H^0\longrightarrow H, \quad x\otimes f\mapsto x\blacktriangleleft f, \end{align}\] which are respectively defined, for any \(x\in H\) and \(f\in H^0\), by \[\begin{align} &x\rhd f=\sum_{(f)} \omega(d x, d f_{(1)}) \langle f_{(1)}, x\rangle f_{(2)}, \\ &f\lhd x=\sum_{(f)} f_{(1)} \langle f_{(2)}, x\rangle,\\ &f\blacktriangleright x=\sum_{(x)} \langle S_0(f), x_{(1)}\rangle x_{(2)}, \\ & x\blacktriangleleft f=\sum_{(x)} \omega(d x_{(2)}, d f) x_{(1)}\langle S_0(f), x_{(2)}\rangle. \end{align}\] It is easy to show that these indeed define left or right actions.

The following fact can also be easily proven.

Now observe that both \(H^0\otimes H\) and \(H\otimes H^0\) are Hopf \((\Gamma, \omega)\)-algebras with respective co-multiplications \[\begin{align} \widehat\Delta= ({\rm{id}}_{H^0}\otimes\tau_{H^0, H}\otimes{\rm{id}}_H)(\Delta_0\otimes\Delta), \\ \widehat\Delta'= ({\rm{id}}_H\otimes\tau_{H, H^0}\otimes{\rm{id}}_{H^0})(\Delta\otimes\Delta_0). \end{align}\] [Warning: \(\widehat\Delta'\) is not the opposite co-multiplication of \(\widehat\Delta\).] They are co-associative, i.e., \(({\rm{id}}_{H^0\otimes H}\otimes\widehat\Delta)\widehat\Delta = (\widehat\Delta\otimes{\rm{id}}_{H^0\otimes H})\widehat\Delta\) and \(({\rm{id}}_{H\otimes H^0}\otimes\widehat\Delta')\widehat\Delta\) \(= (\widehat\Delta'\otimes{\rm{id}}_{H\otimes H^0})\widehat\Delta'\). We write \(\widehat\Delta^{(k)}\) and \({\widehat\Delta'}{^{(k)}}\) for the \(k\)-th iterated co-multiplications. In particular, \(\widehat\Delta^{(2)}=({\rm{id}}_{H^0\otimes H} \otimes\widehat\Delta)\widehat\Delta\) and \({\widehat\Delta'}{^{(2)}}=({\rm{id}}_{H\otimes H^0} \otimes\widehat\Delta')\widehat\Delta'\). Note that \[\begin{align} \widehat\Delta^{(2)}= ({\rm{id}}_{H^0}\otimes\widehat\Delta' \otimes{\rm{id}}_{H})\widehat\Delta, \tag{102}\\ {\widehat\Delta'}{^{(2)}}= ({\rm{id}}_{H}\otimes\widehat\Delta \otimes{\rm{id}}_{H^0})\widehat\Delta'. \tag{103} \end{align}\]

The following result is an easy consequence of Lemma 25.

We also have the following result.

Define \({\mathbb{K}}\)-linear maps \(\psi: H^0\otimes H\longrightarrow H\otimes H^0\) and \(\eta: H\otimes H^0\longrightarrow H^0\otimes H\) by the following compositions \[\begin{align} &\psi: H^0\otimes H \stackrel{\widehat\Delta}\longrightarrow H^0\otimes H\otimes H^0\otimes H \stackrel{\blacktriangleright\otimes\lhd}\longrightarrow H\otimes H^0, \tag{108}\\ &\eta: H\otimes H^0\stackrel{\widehat\Delta'}\longrightarrow H\otimes H^0\otimes H\otimes H^0\stackrel{\rhd\otimes\blacktriangleleft}\longrightarrow H^0\otimes H. \tag{109} \end{align}\] It is easy to see that for any \(x\in H\) and \(f\in H^0\), \[\begin{align} \psi(f\otimes x)&= \sum_{(f), (x)} \omega(d f_{(2)}, d x_{(1)}) f_{(1)}\blacktriangleright x_{(1)}\otimes f_{(2)}\lhd x_{(2)}\\ &=\sum_{(f), (x)} \omega(d f-d f_{(1)}, d x-d x_{(3)}) \\ &\times \langle S_0(f_{(1)}), x_{(1)}\rangle \langle f_{(3)}, x_{(3)}\rangle x_{(2)}\otimes f_{(2)}, \\ \eta(x\otimes f) &= \sum_{(f), (x)} \omega(d x_{(2)}, d f_{(1)}) x_{(1)}\rhd f_{(1)}\otimes x_{(2)}\blacktriangleleft f_{(2)}\\ &= \sum_{(f), (x)} \omega(d x-d x_{(1)}, d f-d f_{(3)}) \omega(d x_{(3)}, d f_{(3)})\\ & \times \langle f_{(1)}, x_{(1)}\rangle \langle S_0(f_{(3)}), x_{(3)}\rangle f_{(2)}\otimes x_{(2)}. \end{align}\] The following result will play a crucial role in the quantum double construction.

6.3 Quantum double construction↩︎

Assume that \((H, \mu, u, \Delta, \epsilon, S)\) is a proper Hopf \((\Gamma, \omega)\)-algebra with bijective antipode, and denote by \((H^0, \mu_0, u_0, \Delta_0, \epsilon_0, S_0)\) the opposite of the finite dual Hopf \((\Gamma, \omega)\)-algebra. Recall that the structure maps are defined by equations 97 , 98 and 99 . Let \[\begin{align} \label{eq:D} D(H)=H\otimes_{\mathbb{K}}H^0. \end{align}\tag{110}\] We have the following results.

For any \(x, y\in H\) and \(f, g\in H^0\), \[\begin{align} \label{eq:mult-d} \phantom{XXX} \overline{\mu}(x\otimes f\otimes y\otimes g) =\sum_{(f), (y)} \omega(d f_{(2)}, d y_{(1)}) x (f_{(1)} \blacktriangleright y_{(1)}) \otimes(f_{(2)} \lhd y_{(2)}) g, \end{align}\tag{112}\] and in particular, \[\begin{align} &\overline{\mu}((x\otimes\epsilon)\otimes(y\otimes g)) = x y\otimes g, \tag{113}\\ &\overline{\mu}((x\otimes f)\otimes(1\otimes g)) = x\otimes fg, \tag{114}, \\ &\overline{\mu}((1\otimes f)\otimes(y\otimes\epsilon)) = \psi(f\otimes y). \end{align}\]

Let us now prove the theorems.

The proof of Theorem 26 makes essential use of Lemma 27.

By inspecting the definitions of the structure maps of the quantum double, particularly equations 113 , 114 , 117 , 118 and 119 , we can easily see that \(D(H)\) contains \(H\) and \(H^0\) as Hopf \((\Gamma, \omega)\)-sub-algebras. More explicitly,

6.3.1 Quasi-triangular structure of the quantum double↩︎

Let \((H, \mu, u, \Delta, \epsilon, S)\) be a proper Hopf \((\Gamma, \omega)\)-algebra over \({\mathbb{K}}\), with \(H\) being a free \({\mathbb{K}}\)-module. Let \(B=\{x_i\mid i\in I\}\) (for an index set \(I\)) be a homogeneous basis for \(H\), and let \(\overline{B}=\{f_i\mid i\in I\}\) be a homogeneous basis for \(H^0\) such that \(\langle f_i, x_j\rangle =\delta_{i j}\) for all \(i, j\in I\). Then for any \(y\in H\) and \(f\in H^0\), \[\begin{align} \label{eq:ident} y= \sum_{i\in I} \langle f_i, y\rangle x_i, \quad f= \sum_{i\in I} \langle f, x_i\rangle f_i. \end{align}\tag{125}\]

6.4 Topological Hopf \((\Gamma, \omega)\)-algebras↩︎

There are various definitions of topological Hopf algebras depending on the contexts. We consider a special kind of topological Hopf \((\Gamma, \omega)\)-algebras over the formal power series ring. A detailed treatment can be found in [91].

6.4.1 Topological Hopf \((\Gamma, \omega)\)-algebras over formal power series ring↩︎

The formal power series ring in \(\hbar\) is \({\mathbb{C}}[[\hbar]]=\left\{\sum_{i=0}^\infty c_i \hbar^i \mid c_i \in {\mathbb{C}}\right\}\) endowed with the \(\hbar\)-adic topology, which is defined by a neighbourhood base \(\{a+ {\mathfrak m}^k \mid k=0, 1, \dots\}\) of each element \(a\in {\mathbb{C}}[[\hbar]]\), where \({\mathfrak m}=\hbar {\mathbb{C}}[[\hbar]]\) (the maximal ideal), and hence \({\mathfrak m}^k= \hbar^k {\mathbb{C}}[[\hbar]]\). As translation by \(a\) is a homeomorphism, the topology is in fact defined by the neighbourhood base \(\{ {\mathfrak m}^k \mid k=0, 1, \dots\}\) of \(0\). This is also the metric topology defined by the following non-archimedean norm. Fix a positive number \(\rho>1\). For any \(a =\sum_{i=0}^\infty c_i \hbar^i\in {\mathbb{C}}[[\hbar]]\) with \(c_i\in {\mathbb{C}}\), the norm is defined by \[\|a\|= \left\{ \begin{align} &\rho^{-r}, && c_r\ne 0, \;c_i=0, \; \forall i<r, \\ &0, && a=0. \end{align} \right.\] It is easy to see that \({\mathbb{C}}[[\hbar]]\) is complete, e.g., by considering Cauchy sequences.

Topological \({\mathbb{C}}[[\hbar]]\)-modules are equipped with the \(\hbar\)-adic topology defined by the neighbourhood base \(\{\hbar^k V\mid k=0, 1, \dots\}\) at \(0\). For any topological \({\mathbb{C}}[[\hbar]]\)-modules \(V\) and \(W\), we denote by \(V\hat{\otimes}_{{\mathbb{C}}[[\hbar]]} W\) the topological tensor product given by \(V\hat{\otimes}_{{\mathbb{C}}[[\hbar]]} W=\{\sum_{i\ge 0} \hbar^i v_i\otimes w_i\mid v_i\in V, w_i\in W\}\).

Observe that every \({\mathbb{C}}[[\hbar]]\)-module homomorphism is continuous with respect to the \(\hbar\)-adic topology. Note in particular that \(V^*={\rm{Hom}}_{{\mathbb{C}}[[\hbar]]}(V, {\mathbb{C}}[[\hbar]])\) is the continuous dual module of \(V\), i.e., the set of continuous linear maps \(V\longrightarrow{\mathbb{C}}[[\hbar]]\).

A \({\mathbb{C}}[[\hbar]]\)-module [91] is topologically free if it is isomorphic to \(V[[\hbar]]=\{\sum_{i=0}^\infty h^i v_i \mid v_i\in V\}\) for some vector space \(V\), where the module structure of \(V[[\hbar]]\) is defined, for any \(a=\sum_{j\ge 0} a_j h^j\) with \(a_j\in{\mathbb{C}}\), and \(v=\sum_{i=0}^\infty h^i v_i\in V[[\hbar]]\), by \[a\cdot v = \sum_{i=0}^\infty h^i \sum_{j+k=i} a_j v_k.\] It is easy to see that topologically free modules are complete with respect to the \(\hbar\)-adic topology. A topologically free \({\mathbb{C}}[[\hbar]]\)-module \(M\) is isomorphic to \(M_0[[\hbar]]\) with \(M_0=\frac{M}{\hbar M}\), and its dual module \(M^*\) is isomorphic to \(M_0^*[[\hbar]]\). If \(M\) and \(N\) are topologically free \({\mathbb{C}}[[\hbar]]\)-modules, \(M\hat{\otimes} N\simeq (M_0\otimes N_0)[[\hbar]]\).

Consider the dual \({\mathbb{C}}[[\hbar]]\)-module \(H^*\) of \(H\) and the adjoint maps of \(\Delta\), \(\mu\), \(u\), \(\epsilon\) and \(S\) respectively defined by analogues of 95 –96 , where \(H^*\hat{\otimes} H^*\) and \((H\hat{\otimes} H)^*\) are used for the definitions of \(\Delta^*\) and \(\mu^*\) in 94 . Inspired by [92], we take \[H^0=\{ f\in H^*\mid \mu^*(f)\in H^*\hat{\otimes} H^*\}.\] This is a topological Hopf \((\Gamma, \omega)\)-algebra. Given any sequence \((g_0, g_1, g_2, \dots)\) of elements in \(H^0\), let \(g=\sum_i h^i g_i\). Then \(\mu^*(g) = \sum_i h^i \mu^*(g_i)\in H^*\hat{\otimes} H^*\), and hence \(H^0\) is complete with respect to the \(\hbar\)-adic topology. We call \(H\) proper if there is a positive integer \(D\) such that for any \(x\in \hbar^k H\) with non-zero image in \(H^0/ \hbar^{k+1}H\), we have \(H^0(x)/\hbar^{k+D}{\mathbb{C}}[[\hbar]]\ne 0\), where \(H^0(x)=\{f(x)\mid f\in H^0\}\).

6.4.2 Topological Hopf \((\Gamma, \omega)\)-algebras over formal Laurent series ring↩︎

The formal Laurent series ring \({\mathbb{C}}((\hbar)) =\left\{\hbar^{-k} \sum_{i=0}^\infty c_i \hbar^i \mid c_i \in {\mathbb{C}}, k\in{\mathbb{Z}}_+\right\}\) is the quotient field of \({\mathbb{C}}[[\hbar]]\). It is also equipped with the \(\hbar\)-adic topology. We can also regard \({\mathbb{C}}((\hbar))\) as the localisation of \({\mathbb{C}}[[\hbar]]\) with respect to the multiplicative set \({\mathcal{S}}=\{1, \hbar, \hbar^2, \hbar^3, \dots\}\). Given a topological \({\mathbb{C}}[[\hbar]]\)-module \(V\), we denote the localisation \({\mathcal{S}}^{-1}V\) by \(V[\hbar^{-1}]\), and let \(\mathfrak{j}: V\longrightarrow V[\hbar^{-1}]\) be the canonical map \(v\mapsto \frac{v}{1}\).

Recall that \(V[\hbar^{-1}]=\frac{V}{N}[\hbar^{-1}]\), where \(N=\{v\in V\mid \hbar^{k} v=0 \;\text{for some k> 0} \}\). Call \(V\) torsion free if \(N=0\). Topologically free \({\mathbb{C}}[[\hbar]]\)-modules are clearly torsion free.

Consider \(M[\hbar^{-1}]\) and \(N[\hbar^{-1}]\) which are the localisations of topological \({\mathbb{C}}[[\hbar]]\)-modules \(M\) and \(N\). Their topological tensor product \(M[\hbar^{-1}]\hat{\otimes} N[\hbar^{-1}]\) is taken to be \({\mathcal{S}}^{-1}(\mathfrak{j}(M)\hat{\otimes}\mathfrak{j}(N))\).

Now given any topological algebra \(H\) over \({\mathbb{C}}[[\hbar]]\) of any type (e.g., associative, Hopf, and etc.), we have a corresponding topological algebra \(H[\hbar^{-1}]\) over \({\mathbb{C}}((\hbar))\) of the same type, the structure maps of which are obtained by \({\mathbb{C}}((\hbar))\)-linearly extending those of \(H\).

. We thank N. Aizawa, Z. Kuznetsova, F. Toppan and J. Van der Jeugt for sharing their insights on Lie colour algebras. This work was reported at the 2025 Autumn Conference on Lie Theory, Nov. 14–16, 2025, East China Normal University, Shanghai. We thank the audience for comments.

. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References↩︎