1 Introduction↩︎

Recently, motivated by three-dimensional \(\mathcal{N}=4\) mirror symmetry and geometric representation theory, Poisson geometry on quiver varieties has attracted growing interest across several areas of mathematics. At the same time, quiver varieties play a fundamental role in noncommutative geometry. Following the Kontsevich–Rosenberg principle—which states that a noncommutative geometric structure is meaningful only when it induces the corresponding classical structure on representation schemes—it is natural to seek an intrinsic, noncommutative description of Poisson geometry for quiver varieties.

Let \(Q\) be a finite quiver. In [1], [2], Crawley-Boevey, Etingof, Ginzburg, and Van den Bergh developed noncommutative symplectic and Poisson geometry and showed that preprojective algebras arise from the path algebra \(\mathbb{K} \overline{Q}\) via noncommutative Hamiltonian reduction. As an immediate consequence, the commutator quotient \(\Pi Q/[ \Pi Q,\Pi Q]\) carries a Lie bracket, and the image of the trace map generates the coordinate ring of the quiver variety as a Poisson algebra. Moreover, this Lie bracket descends to the zeroth reduced cyclic homology \(\overline{\mathsf{HC}}_0(\Pi Q)\). This motivates the following questions:

For non-Dynkin quivers, preprojective algebras are Koszul Calabi–Yau, and hence the works [3], [4] provide affirmative answers to the above questions. However, for Dynkin quivers, preprojective algebras are not Koszul Calabi–Yau; this work presents an approach that uniformly handles both cases.

Our key tool is the Feigin–Tsygan theorem as developed by Berest–Khachatryan–Ramadoss [5]. For an algebra \(A\), the reduced cyclic homology \(\overline{\mathsf{HC}}_\bullet(A)\) can be computed as the cohomology of \(\displaystyle{ \frac{\tilde{A}}{S+[\tilde{A},\tilde{A}]}}\), where \(\tilde{A}\) is a cofibrant resolution of \(A\). For a finite quiver \(Q\) the preprojective algebra \(\Pi Q\) is augmented, and the cobar–bar construction \(\mathbf{\Omega}\mathbf{B}\Pi Q\) provides a natural cofibrant resolution of \(\Pi Q\). One of our main results answers Question 1 positively.

A crucial observation in the proof is the following isomorphism of coaugmented dg coalgebras for any augmented dg algebra \(A\): \[\label{for:32BA32ext} \mathbf{B}A \cong S \oplus \mathfrak{s}\bar A \langle t \rangle.\tag{1}\]

First, we introduce the \(\mathfrak{s}\)-construction. Using (1 ), the cobar–bar construction is identified with the dg algebra \((T_S\,\mathfrak{s}^{-1}(\mathfrak{s}\bar A\langle t\rangle),\partial+b+\delta,\otimes)\). When \(A\) is a dg double Poisson algebra, the \(\mathfrak{s}\)-construction endows \(\mathfrak{s}^dA = A[d]\) with a canonical double Poisson bracket; see Section 5 for details.

Second, we develop a theory of noncommutative Poisson extensions. Given a dg algebra \(A\), form the free product \(A\langle t\rangle=A\ast_S S[t]\). We analyze when double Poisson brackets on \(A\) extend to \(A\langle t\rangle\), when NC Poisson structures (see Definition 2) extend, and how these two extension problems are related. In particular, we show that NC Poisson extensions are compatible with noncommutative Hamiltonian reduction and align with the Kontsevich–Rosenberg principle.

We also prove a more general extension result: the double Poisson bracket extends to the cobar–bar construction.

The latter statement may be regarded as a noncommutative analogue of the fact that the cohomology of a smooth Poisson manifold carries a graded Lie algebra structure.

All constructions above are carried out with respect to an explicit cofibrant resolution. In the derived setting, one expects that noncommutative structures are independent of the choice of cofibrant resolution. We therefore establish:

For completeness, Section 3 explains how shifted double Poisson structures fit naturally into the Kontsevich–Rosenberg principle. In particular, we prove that a shifted double Poisson structure induces a shifted Poisson structure on the derived moduli stack of representations.

1.1 Conventions and notation↩︎

• \(\mathbb{K}\) denotes an algebraically closed field of characteristic zero.

• Algebras are associative algebras over a semisimple ring \(S = \bigoplus_{i\in I}\mathbb{K}e_i\), where the \(e_i\) are mutually orthogonal idempotents. The category of such algebras over \(S\) is denoted by \({\rm Alg}_S\).

• We abbreviate \(-\otimes_S -\) as \(- \otimes -\).

• We use cohomological grading: all complexes are cochain complexes supported in non-positive degrees. For a complex \(V\), the homology is related to cohomology by \(\mathrm{H}_n(V)=\mathrm{H}^{-n}(V^\bullet)\).

• \(\mathrm{Com}_S\) denotes the category of cochain complexes of \(S\)-modules.

• \(\mathrm{DGA}^{\leq 0}_S\) denotes the category of differential graded \(S\)-algebras supported in non-positive degrees. The notation \(\mathrm{ADGA}^{\leq 0}_S\) denotes the category of augmented dg \(S\)-algebras supported in non-positive degrees.

• Let \(A\) be a dg algebra. In this work, by a left (right) \(A\)-module, we mean a left (right) dg module over \(A\). The category of left \(A\)-modules is denoted by \(\mathrm{Mod}(A)\).

• \(\mathfrak{s}\) denotes the shift functor. For a complex \(V^\bullet \in \mathrm{Com}_S\), \((\mathfrak{s}V)^i = V^{i+1}\).

• For any \(n \in \mathbb{N}\), the symmetric group \(\mathcal{S}_n\) acts on \(V^{\otimes n}\): for \(\sigma\in\mathcal{S}_n\) and \(v_1\otimes\cdots\otimes v_n\in V^{\otimes n}\), \[\sigma(v_1\otimes\cdots\otimes v_n) := v_{\sigma^{-1}(1)}\otimes v_{\sigma^{-1}(2)}\otimes\cdots\otimes v_{\sigma^{-1}(n)}.\] In particular, \(\tau\) denotes the cyclic permutation: \[\tau(a_1 \otimes a_2 \otimes \cdots \otimes a_n) = a_n \otimes a_1 \otimes \cdots \otimes a_{n-1}.\]

2 Noncommutative Poisson Geometry↩︎

In classical Poisson geometry, Poisson structures on a smooth manifold \(X\) can be described algebraically via a Lie bracket on the algebra \(C^\infty(X)\) satisfying the Leibniz rule, or geometrically via a bivector field \(\pi\in\Gamma(X,\wedge^2 TX)\) such that \(\{\pi,\pi\}=0\), where \(\{-,-\}\) denotes the Schouten–Nijenhuis bracket. Following this philosophy, this section recalls double Poisson geometry in the sense of Van den Bergh [2] and explains that shifted double Poisson structures behave well in the derived setting.

2.1 Double Poisson brackets↩︎

First, we recall the notion of a double Poisson bracket.

Here, we write \[\mathopen{\{\!\!\{}x , y \mathclose{\}\!\!\}}= \mathopen{\{\!\!\{}x , y \mathclose{\}\!\!\}}' \otimes \mathopen{\{\!\!\{}x , y \mathclose{\}\!\!\}}'' = \mathcal{H}_x (y)' \otimes \mathcal{H}_x (y)'' ;\; \mathopen{\{\!\!\{}x,\mathopen{\{\!\!\{}y,z\mathclose{\}\!\!\}}\mathclose{\}\!\!\}}_L = \mathopen{\{\!\!\{}x, \mathopen{\{\!\!\{}y, z \mathclose{\}\!\!\}}' \mathclose{\}\!\!\}}\otimes \mathopen{\{\!\!\{}y, z \mathclose{\}\!\!\}}''.\] A dg algebra \(A\) is called a dg double Poisson algebra of degree \(n\) if it is equipped with a dg double Poisson bracket of degree \(n\). A complex \(V\in\mathrm{Com}_S\) is called a dg double Lie algebra of degree \(n\) if it has a bracket \(\mathopen{\{\!\!\{}-,-\mathclose{\}\!\!\}}\) that satisfies the axioms in Definition 1, except (2).

The following examples play important roles in this work.

Here \(s(-)\) and \(t(-)\) denote the source and target of an arrow, respectively.

Next, we recall an important property of the double Poisson bracket. In [2], Van den Bergh showed that a double Poisson bracket on \(A\) canonically induces a Lie algebra structure on the quotient space \(A_{cyc}=A/[A,A]\). When \(A\) is a dg double Poisson algebra, one obtains a bracket \(\{-,-\}: A \otimes A \to A\) defined as the composition \[\xymatrix{ A\otimes A \ar[r]^-{\mathopen{\{\!\!\{}-,-\mathclose{\}\!\!\}}} & A\otimes A \ar[r]^-{\mathbf{m}} & A. }\] Here, \(\mathbf{m}\) is the multiplication on \(A\).

Although the Lie bracket in Lemma 1 is induced by the double Poisson bracket, it has since developed into an independent theory, due to Crawley-Boevey [6], Berest, Chen, Eshmatov, and Ramadoss [7]. This notion provides a higher homological extension of the \(\mathrm{H}_0\)-Poisson structure in Crawley-Boevey’s work [6]. Let \(A\) be a dg algebra. The complex \(\underline{\mathrm{Der}}_S(A)\) of \(S\)-derivations on \(A\) is naturally a dg Lie algebra with respect to the commutator bracket. Denote by \(\underline{\mathrm{Der}}_S(A)^\natural\) the subcomplex of derivations with image in \(S+[A,A]\subseteq A\). It is straightforward to verify that \(\underline{\mathrm{Der}}_S(A)^\natural\) is an ideal of \(\underline{\mathrm{Der}}_S(A)\). Therefore, \(\underline{\mathrm{Der}}_S(A)_{\natural}:=\underline{\mathrm{Der}}_S(A)/\underline{\mathrm{Der}}_S(A)^\natural\) is a dg Lie algebra. The canonical action of \(\underline{\mathrm{Der}}_S(A)\) on \(A\) induces a Lie algebra action of \(\underline{\mathrm{Der}}_S(A)_\natural\) on the quotient space \(\displaystyle A_\natural:=\frac{A}{S+[A,A]}\). We write \(\varrho:\underline{\mathrm{Der}}_S(A)_\natural\to\underline{\rm End}_S(A_\natural)\) for the corresponding dg Lie algebra homomorphism.

Following [7], one has the following notion.

In other words, for each \(a\in A\) representing the class \(\bar a \in A_{\natural}\), the map \(\mathrm{ad}_{\bar a} = \{ \bar a,-\}: A_\natural \rightarrow A_\natural\) is induced by a derivation \(H_a: A \rightarrow A\). Throughout this work, a dg algebra \(A\) equipped with an NC Poisson structure of degree \(n\) is called an NC Poisson algebra of degree \(n\).

In the quiver case, one has the necklace Lie bracket.

In Poisson geometry, many important Poisson spaces are obtained from Hamiltonian reduction. In the noncommutative setting, the analogue of a Hamiltonian \(\mathrm{G}\)-space is a double Poisson algebra \(A\) endowed with a noncommutative moment map.

In this case, \(A\) is called a noncommutative Hamiltonian algebra of degree \(n\). Following the Kontsevich–Rosenberg principle, noncommutative reduction theory is developed in [1], [2], [8].

This notion is justified by the following proposition.

2.2 Higher noncommutative Poisson structures↩︎

Double Poisson brackets appear more fundamental than NC Poisson structures. In the derived setting, a natural question is whether double Poisson brackets are stable under quasi-isomorphisms. The key observation is that, under smoothness assumptions, double Poisson brackets can be constructed by solving a Maurer–Cartan equation.

We first recall the construction of noncommutative cotangent bundles in [1], [2] and the correspondence between double Poisson brackets and elements in these bundles. In algebraic geometry, polyvector fields on a smooth variety are identified with functions on the cotangent bundle. Following this intuition, a noncommutative cotangent bundle is defined via noncommutative vector fields. According to Crawley-Boevey-Etingof-Ginzburg and Van den Bergh, noncommutative vector fields on an algebra are double derivations.

Let \(A \in \mathrm{DGA}^{\leq 0}_S\). The noncommutative differential \(\Omega^1(A)\) of \(A\) is defined as the kernel of the multiplication map \(\mathbf{m}: A \otimes A \to A\) in the category \(\mathrm{Mod}(A^e)\) of left \(A^e\)-modules. \(A\) is smooth if \(\Omega^1(A)\) is projective and finitely presented as a left \(A^e\)-module. The complex \(\underline{\mathbb{D}\mathrm{er}}_S(A) := \underline{\mathrm{Der}}_S(A, A \otimes A)\) of double derivations is isomorphic to the bimodule dual of \(\Omega^1(A)\), namely \(\underline{\mathbb{D}\mathrm{er}}_S(A) \cong \Omega^1(A)^{\vee} = \underline{\mathrm{Hom}}_{A^e} (\Omega^1(A), A \otimes A)\) as right \(A^e\)-modules. The right \(A^e\)-module structure on \(\underline{\mathbb{D}\mathrm{er}}_S\) is induced from the inner \(A\)-bimodule structure on \(A \otimes A\).

The noncommutative cotangent bundle is defined as follows.

Note that \(\mathrm{T}^{\ast, [n]}_{ A}\) is a dg algebra with respect to the tensor product, and it carries a natural weight grading: \[\label{for:32weight32grading} \mathrm{T}^{\ast, [n]}_{ A} = \bigoplus_{p\geq 0} \Bigl(\mathrm{T}^{\ast, [n]}_{ A}\Bigr)^{(p)},\; \text{where}\;\Bigl(\mathrm{T}^{\ast, [n]}_{ A}\Bigr)^{(p)} = \Bigl(\underline{\mathbb{D}\mathrm{er}}_S(A)[-n]\Bigr)^{\otimes_A p}.\tag{5}\] It is straightforward to verify that the weight grading (5 ) descends to the quotient space \(\mathrm{T}^{\ast, [n]}_{ A,\,cyc}: = \mathrm{T}^{\ast, [n]}_{ A}/[\mathrm{T}^{\ast, [n]}_{ A}, \mathrm{T}^{\ast, [n]}_{ A}]\). Denote the completion by \(\widehat{\mathrm{T}}^{\ast, [n]} _{ A} : = \prod_{i \geq 0} \Big(\underline{\mathbb{D}\mathrm{er}}_S(A)[-n]\Big)^{\otimes_A i}.\) There is a natural weight filtration on \(\widehat{\mathrm{T}}^{\ast, [n]}_{ A}\): for any \(p \in \mathbb{Z}_{\geq 0}\), \[\label{for:32wei32fil32on32nc32cot} F^p\widehat{\mathrm{T}}^{\ast, [n]}_{ A} : = \prod_{i\geq 0} \Bigl(\underline{\mathbb{D}\mathrm{er}}_S(A)[-n]\Bigr)^{\otimes_A (p+i)}.\tag{6}\] In some literature, for example [9], the (completed) noncommutative cotangent bundle is called shifted noncommutative multiderivations.

In differential geometry, there is a canonical Poisson structure on a cotangent bundle over a smooth manifold, and this Poisson structure plays a central role in Poisson geometry. Parallel to the noncommutative symplectic approach in [1], Van den Bergh studied the double Poisson geometry on the noncommutative cotangent bundle in [2], [8]. A crucial property is as follows.

In Van den Bergh’s work [2], this bracket is called the double Schouten–Nijenhuis bracket. If \(A\) is assumed to be smooth, then the double Poisson bracket on \(\mathrm{T}^{\ast, [n]}_{ A}\) can be described explicitly. Assume that as a left \(A^e\)-module, \(\Omega^1( A)\) is generated by \(\{\delta a_i = a_i \otimes 1 - 1 \otimes a_i\;\big\vert\;i =1, \dots ,r\}\). Then the right \(A^e\)-module \(\underline{\mathbb{D}\mathrm{er}}_S(A)\) is generated by \(\{\mathcal{D}_{a_i}\}\), where \[\mathcal{D}_{a_i} (\delta a_j) =\delta_{i,j} e_{t(a_i)} \otimes e_{s(a_i)}.\] Then, by the Leibniz rule, the double Schouten–Nijenhuis bracket is determined by evaluation on generators \(\{a_1,\dots,a_r,\mathfrak{s}^{-n}\mathcal{D}_{a_1},\dots,\mathfrak{s}^{-n}\mathcal{D}_{a_r}\}\): \[\label{for:32dpoiss32on32nc32cot} \mathopen{\{\!\!\{}a_i, a_j \mathclose{\}\!\!\}}= 0,\qquad \mathopen{\{\!\!\{}\mathfrak{s}^{-n}\mathcal{D}_{a_i}, \mathfrak{s}^{-n}\mathcal{D}_{a_j} \mathclose{\}\!\!\}}= 0,\qquad \mathopen{\{\!\!\{}\mathfrak{s}^{-n}\mathcal{D}_{a_i}, a_j \mathclose{\}\!\!\}}= \delta_{i,j} e_{t(a_i)} \otimes e_{s(a_i)}.\tag{7}\]

A key observation in Van den Bergh’s work [2] is that elements in noncommutative cotangent bundles correspond to brackets on \(A\). Related higher and homotopy-invariant versions of this viewpoint appear in Pridham [9] and Yeung [10].

The proof of [2] adapts to the dg setting; for completeness, we include a sketch. Before that, we recall the necessary constructions. For any left \(A^e\)-module \(M\) and for any positive integer \(r\), \(M^{\otimes r +1}\) is canonically a left \((A^e)^{\otimes r}\)-module and a left \(A^e\)-module. In particular, the left \((A^e)^{\otimes r}\)-module and the left \(A^e\)-module structure on \(A^{\otimes (r+1)}\) are as follows. Suppose that \(a = a' \otimes a'' \in A^e\), and \(x = x_1 \otimes \cdots \otimes x_{r+1} \in A^{\otimes (r+1)}\), the \({\rm i^{th}}\)-copy of \((A^e)^{\otimes r}\) acts on \(A^{\otimes r+1}\) by \[(a' \otimes a'') \bullet_i (x_1 \otimes \cdots \otimes x_{r+1}) = (-1)^{|a'| |a''| + |a| |x_{< i+1}|} x_1 \otimes\cdots \otimes x_{i} a'' \otimes a' x_{i+1} \otimes \cdots \otimes x_{r+1};\] the canonical left \(A^e\)-module structure on \(A^{\otimes (r+1)}\) is given by \[(a' \otimes a'') . (x_1 \otimes \cdots \otimes x_{r+1}) = (-1)^{|a''| |x|}(a 'x_1 )\otimes \cdots \otimes (x_{r+1} a'').\] Here, \(|x_{< i+1}| = |x_1| + \cdots + |x_i|\). These two module structures clearly commute. Furthermore, there exists a canonical morphism of left \(A^e\)–modules \[\label{for:32Phi} \Phi : M^{\otimes_A r} \longrightarrow \underline{\mathrm{Hom}}_{(A^e)^{\otimes r}}\Bigl(\bigl(M^{\vee}\bigr)^{\otimes r}, A^{\otimes (r+1)}\Bigr).\tag{8}\] For any \(\Theta_1 \otimes \cdots \otimes \Theta_r \in M^{\otimes_A r}\), the image \(\Phi(\Theta_1 \otimes \cdots \otimes \Theta_r)\) is given by \[\alpha_1 \otimes \cdots \otimes \alpha_r \mapsto (\pm 1)\, \Theta''_1(\alpha_1) \otimes \Theta'_1(\alpha_1)\,\Theta''_2(\alpha_2) \otimes \cdots \otimes \Theta'_{r-1}(\alpha_{r-1})\,\Theta''_r(\alpha_r) \otimes \Theta'_r(\alpha_r).\] Here, \((\pm 1)\) follows from the Koszul sign rule.

Next, we recall double Poisson structures in the derived setting. Note that Proposition 9 shows that a dg double Poisson bracket is governed by a Maurer–Cartan equation, a viewpoint that naturally extends to the derived setting ([3], [4], [7], [9], [10]). However, noncommutative \(\text{K\"{a}hler}\) differentials and double derivations in previous constructions are replaced by noncommutative cotangent complexes and noncommutative tangent complexes. Following André–Quillen (co)homology theory, for a dg algebra \(A \in \mathrm{DGA}^{\leq 0}_S\), the (absolute) cotangent complex \(\mathbb{L}_{S \backslash A}\) of \(A\) as an object in \(\mathrm{Ho}(\mathrm{Mod}(A^e))\), is defined to be the homotopy fiber of the multiplication \(\mathbf{m}: A \otimes A \to A\): \[\mathbb{L}_{S \backslash A}\longrightarrow A \otimes A \stackrel{\mathbf{m}}{\longrightarrow} A \stackrel{+1}{\longrightarrow}\] Then, the noncommutative tangent complex \(\mathbb{T}_{S \backslash A}\) of \(A\) is defined to be the bimodule dual: \[\mathbf{R}\underline{\mathrm{Hom}}_{A^e} (\mathbb{L}_{S \backslash A}, A\otimes A).\] Consequently, for an arbitrary cofibrant resolution \(Q \stackrel{\simeq}{\twoheadrightarrow}A\) in \(\mathrm{DGA}^{\leq 0}_S\), the noncommutative cotangent complex is quasi-isomorphic to \(A^e \otimes_{Q^e} \Omega^1(Q)\); the noncommutative tangent complex is quasi-isomorphic to \(\underline{\mathbb{D}\mathrm{er}}_S (Q) \otimes_{Q^e} A^e\). Note that in the derived setting, the \(n\)-shifted noncommutative cotangent bundle \(\mathrm{T}^{\ast, [n]}_{A}\) of \(A\) as an object in \(\mathrm{Ho}(\mathrm{DGA}_S)\) ¹is constructed by choosing an explicit resolution of the noncommutative tangent complex. A standard fact is that \(\mathrm{T}^{\ast, [n]}_{A}\) is quasi-isomorphic to \(T_Q (\underline{\mathbb{D}\mathrm{er}}_S (Q)[-n])\).

Now, we briefly recall the Maurer-Cartan formalism. See [11] for further details. Let \(\Omega(\Delta_n)\) be the commutative dg algebra of polynomial differential forms on the standard \(n\)–simplex \(\Delta_n\). For a nilpotent dg Lie algebra \((L, \partial, \{-,-\})\), set \(\mathrm{MC}(L) = {\{x \in L\;\big\vert\;\partial x + \frac{1}{2} \{x,x\} = 0\}}.\) The Maurer–Cartan space of \(L\) is the simplicial set \(\underline{\mathrm{MC}}_\bullet(L)=\mathrm{MC}\big(L\otimes_\mathbb{K}\Omega(\Delta_\bullet)\big).\) In general, the dg Lie algebra of interest is not nilpotent but pronilpotent. The latter means \(L\) admits a cofiltration \(L_0\leftarrow L_1\leftarrow\cdots,\) such that \(L=\lim_n L_n\) and each \(L_n\) is nilpotent. Then for a pronilpotent \(L\), the Maurer–Cartan space of \(L\) is \(\underline{\mathrm{MC}}_\bullet(L):=\lim_n\underline{\mathrm{MC}}_\bullet (L_n)\), the limit of \(\{\underline{\mathrm{MC}}_\bullet(L_n)\}\). A standard fact is that given a graded dg Lie algebra \(L\), its completion \(L^{\geq 2}\) in weights \(\ge 2\) is pronilpotent. The cofiltration is given by \(L^{\geq 2}/ L^{\geq 3} \leftarrow L^{\geq 2}/L^{\geq 4} \leftarrow \cdots.\)

Now apply the above Maurer–Cartan formalism to the noncommutative shifted cotangent bundles. The following notion serves as the moduli of shifted double Poisson structures.

At first sight, \(\mathsf{DPois}\) seems to depend on the choice of cofibrant resolution. However, we have the following theorem.

A canonical problem in the derived setting is whether a geometric structure is stable under weak equivalences. Note that for two quasi-isomorphic dg algebras \(A, B \in \mathrm{DGA}^{\leq 0}_S\), their cofibrant resolutions \(Q_A,\;Q_B\) are quasi-isomorphic. By the same analysis as in Theorem 10, we have:

Finally, it is natural to seek an analogue of Lemma 1 for shifted double Poisson structures. In [7], Berest-Chen-Eshmatov-Ramadoss introduced the homotopy version of NC Poisson structures.

The following corollary is clear. We remark that the \(L_\infty\)-structure on \(A_{\natural}[n]\) is well known, see [9].

3 Kontsevich–Rosenberg Principle↩︎

The Kontsevich–Rosenberg principle states that a noncommutative geometric structure is meaningful precisely when it induces the classical counterpart on representation schemes. In this section, we recall the construction of derived representation schemes in [5] and prove that shifted double Poisson structures on \(A\) induce shifted Poisson structures on derived representation schemes. In particular, we show that shifted double Poisson structures are related to the shifted Poisson structures on the derived moduli stacks of representations.

3.1 Derived representation schemes↩︎

Let \(V\in \mathrm{Com}_S\) be a complex of finite total dimension, and let \(\underline{\rm End}_S (V)\) denote the complex of endomorphisms of \(V\). For \(A \in \mathrm{DGA}^{\leq 0}_S\), consider the following representation functor: \[\mathrm{DRep}_V^{ A}: \mathrm{CDGA}^{\leq 0}_\mathbb{K}\to \mathrm{Set},\quad C \mapsto \mathrm{DRep}_V ^{ A}(C) : = \mathrm{Hom}_{\mathrm{DGA}^{\leq 0}_S}(A, \underline{\rm End}_S (V)\otimes_\mathbb{K}C).\] As established in [5], this functor is representable; that is, there exists a commutative dg algebra \(A_V\in\mathrm{CDGA}^{\leq 0}_\mathbb{K}\) such that \(\mathrm{DRep}_V^{ A}(C)=\mathrm{Hom}_{\mathrm{CDGA}^{\leq 0}_\mathbb{K}}(A_V, C).\) When \(V\) is fixed, this yields a functor \((-)_V: \mathrm{DGA}^{\leq 0}_S \to \mathrm{CDGA}^{\leq 0}_\mathbb{K}.\) A key result in [5] is the following.

For future reference, we recall the adjunction explicitly.

We recall the explicit constructions of \(\sqrt[V]{-},\;(-)_V\) in Section 3.2. The derived affine scheme \(\mathrm{Spec}\,A_V\) is called the derived representation scheme of \(A\) in \(V\), still denoted \(\mathrm{DRep}_{V}^{ A}\). It appears that the derived representation scheme depends on the choice of \(V\). However, Proposition 2.3 in [5] states that if \(V\) and \(W\) are quasi-isomorphic in \(\mathrm{Com}_S\), then \(\mathrm{DRep}_{V}^{ A}\) and \(\mathrm{DRep}_{W}^{ A}\) are equivalent as derived schemes. Throughout this work, we consider only \(V = \mathbb{K}^\mathbf{d}= \bigoplus_{i \in I} \mathbb{K}^{d_i}\) for \(\mathbf{d}\in \bigoplus_{i\in I} \mathbb{Z}_{\geq 0}\). The corresponding commutative dg algebra is denoted by \(A_\mathbf{d}\).

Since \(\mathrm{GL}_\mathbf{d}(\mathbb{K}) = \prod_i \mathrm{GL}(\mathbb{K}^{d_i})\) acts on \(\mathrm{DRep}^{A} _\mathbf{d}\) by conjugation, we consider the following functor: \[(-)^\mathrm{GL}_\mathbf{d}: \mathrm{DGA}^{\leq 0}_S \to \mathrm{CDGA}^{\leq 0}_\mathbb{K},\quad A \mapsto(A_\mathbf{d})^{\mathrm{GL}_\mathbf{d}(\mathbb{K})}.\] Here \((A_\mathbf{d})^{\mathrm{GL}_\mathbf{d}(\mathbb{K})}\) denotes the subalgebra of \(\mathrm{GL}_\mathbf{d}(\mathbb{K})\)-invariants. \(\mathrm{Spec}\,(A)^\mathrm{GL}_\mathbf{d}\) is called the derived character scheme of \(A\) in \(\mathbb{K}^\mathbf{d}\), denoted by \(\mathrm{Ch}^{ A} _\mathbf{d}\). Although this functor does not appear to admit a right adjoint, Berest et al. proved the following proposition.

Since derived representation schemes and derived character schemes are affine, shifted Poisson structures are constructed by a standard procedure. Throughout this work, we only consider derived affine schemes \(\mathrm{Spec}\,R\) which are derived finitely presentable. We denote by \(\mathbb{L}_{\mathbb{K}\backslash R}\) the cotangent complex and by \(\mathbb{T}_{\mathbb{K}\backslash R}\) the tangent complex. See [11], [13] for details and references.

The space of shifted Poisson structures is defined as follows. Let \(R\in\mathrm{CDGA}^{\leq 0}_\mathbb{K}\) be derived finitely presentable. Let \(n \in \mathbb{Z}\). The complex of \(n\)-shifted polyvector fields on \(\mathrm{Spec}\,R\) is defined to be \[\widehat{\mathrm{Pol}}(R; n) = \prod_{k \geq 0} \mathrm{Sym}^k _R ( \mathbb{T}_{\mathbb{K}\backslash R} [-n-1]).\]

3.2 Explicit presentation↩︎

For simplicity, in this subsection only, we assume \(S = \mathbb{K}\). Let \(A \in \mathrm{DGA}^{\leq 0}_S\). Fix \(N \in \mathbb{N}\) and \(V = \mathbb{K}^N\). Denote \((A \ast_\mathbb{K}\mathfrak{gl}_N(\mathbb{K}))^{\mathfrak{gl}_N (\mathbb{K})}\) by \(\sqrt[N]{A}\). Here \((-)^{\mathfrak{gl}_N (\mathbb{K})}\) denotes the centralizer of \(\mathfrak{gl}_N (\mathbb{K})\). Then the derived representation scheme \(\mathrm{DRep}^{A} _N\) is represented by \[(A)_N : = (\sqrt[N]{A})_\mathrm{ab}: = \sqrt[N]{A}/([\sqrt[N]{A},\sqrt[N]{A}]),\] \((-)_\mathrm{ab}\) denotes the quotient by the ideal generated by commutators.

Let \({\epsilon_{ij}}\) be the basis of elementary matrices in \(\mathfrak{gl}_N (\mathbb{K})\). For any \(a \in A\), define the “matrix coefficients” in \(A \ast \mathfrak{gl}_N (\mathbb{K})\): \[a_{ij} : = \sum_{k = 1} ^N \epsilon_{ki} a \epsilon_{jk}.\] Then \(a_{ij} \in \sqrt[N]{A}\) for all \(i,j=1,\dots,N\). We also write \(a_{ij}\) for the corresponding element in \(A_N\). A standard fact is that \(A_N\) is generated by \(\{a_{ij}\mid a\in A,\;i,j=1,\dots,N\}\).

Equivalently, \(a_{ij}\) is a function on the representation scheme: \[a_{ij}: \rho \in \mathrm{DRep}^{ A} _N \mapsto (\rho_a)_{ij}.\] Here \(\rho_a\) denotes the evaluation of \(\rho\) on \(a\), and \((\rho_a)_{ij}\) denotes its \((i,j)\)-entry. In particular, we set \(\mathrm{Tr} \,a = \sum_i a_{ii}\).

Under this explicit presentation, double Poisson brackets and noncommutative Hamiltonian structures induce classical counterparts on the representation schemes.

Note that Proposition 14 implies \(\mathrm{H}^0 ( \mathbf{L}(A)_N^{\mathrm{GL}}) \cong (A)_N^{\mathrm{GL}}\). The following recovers results from [2], [6].

3.3 Higher traces↩︎

The trace map establishes a correspondence from noncommutative algebras to functions on their representation schemes. This correspondence extends to modules over algebras, connecting noncommutative geometry with geometry on representation spaces.

First, we recall the Van den Bergh functor as introduced in [5], [8]. Van den Bergh proposed this functor as a means to connect noncommutative geometry with sheaf theory on representation schemes. Subsequently, Berest et al. extended this construction into the derived setting.

For a dg algebra \(A \in \mathrm{DGA}^{\leq 0}_S\) and dimension vector \(\mathbf{d}\), the Van den Bergh functor is \[\label{for:32vdb32func} (-)_\mathbf{d}:\;\mathrm{Mod}(A^{e}) \to \mathrm{Mod}(A_\mathbf{d}), \qquad M \mapsto M_\mathbf{d}\cong \big(\mathfrak{gl}(\mathbb{K}^\mathbf{d}) \otimes A_\mathbf{d}\big) \otimes_{A^{e}} M.\tag{10}\]

The following lemma is the module-theoretic analogue of Proposition 13.

Next, we recall the trace map for bimodules: \[\label{for:32trace32formula} \mathrm{Tr} \,_\mathbf{d}: M \to \mathfrak{gl}_\mathbf{d}(\mathbb{K}) \otimes M_\mathbf{d}\to M_\mathbf{d}.\tag{12}\] The first arrow is the canonical map corresponding to the identity under the isomorphism (11 ) for \(L = M_\mathbf{d}\); the second arrow sends a matrix to its trace.

According to [8], for any \(M \in \mathrm{Mod}(A^e)\), \((T_A M)_\mathbf{d}\simeq \mathrm{Sym}_{A_\mathbf{d}} (M)_\mathbf{d}\). Thus, for any \(n\), taking \(M=\underline{\mathbb{D}\mathrm{er}}_S(A)\) yields \((\mathrm{T}^{\ast, [n]}_{ A})_\mathbf{d}\simeq \mathrm{Pol}(A_\mathbf{d}; n-1)\). We now show that shifted double Poisson structures satisfy the Kontsevich–Rosenberg principle.

We end this subsection by presenting an explicit correspondence in the quiver setting.

3.4 Relation with stacks of representations↩︎

In Yeung’s work [10], [14], a pre-Calabi-Yau structure on a dg algebra induces a shifted Poisson structure on derived stacks of representations; on the other hand, pre-Calabi-Yau structures are known to be closely related to double Poisson structures. It is natural to ask whether double Poisson structures directly induce shifted Poisson structures on derived stacks of representations.

First, we recall shifted Poisson structures on derived stacks of representations. Since a derived moduli stack of representations is a derived quotient stack, Toën proposed the following model for its cotangent complex (see [13] for details and references). Yeung studied the shifted symplectic/Poisson structures in [14]. Let \(\mathrm{G}\) be a smooth reductive group scheme acting on a smooth derived affine scheme \(Y=\mathrm{Spec}\,R\), and let \(X:=[Y/\mathrm{G}]\) be the derived quotient stack. The pullback to \(Y\) of the cotangent complex \(\mathbb{L}_{X}\) is the homotopy fiber \(\mathbb{L}\) of the natural morphism \[\rho: \mathbb{L}_{Y} \to \mathscr{O}_Y \otimes \mathfrak{g}^{\vee},\] which is dual to the infinitesimal action of \(\mathrm{G}\) on \(Y\). Since \(\mathrm{G}\) acts on \(Y\), it also acts on \(\mathbb{L}\). By definition, in \(\mathrm{Ho}(\mathrm{Mod}(R))\), \(\mathbb{L}\) is equivalent to the complex \[\mathrm{cone}\Big(\mathbb{L}_{Y} \stackrel{\rho}{\to} \mathscr{O}_Y \otimes \mathfrak{g}^{\ast}\Big)[-1].\]

Assume \(R\) is almost cofibrant, meaning that the \(\text{K\"{a}hler}\) differential \(\Omega^1_{\rm com}(R)\in\mathrm{Mod}(R)\) is cofibrant and \(\Omega^1_{\rm com}(R)\to\mathbb{L}_{\mathbb{K}\backslash R}\) is a quasi-isomorphism. In [14], the model for shifted polyvector fields on \(X=[Y/\mathrm{G}]\) is: \[\begin{align} \label{for:32pol32on32drep} \mathrm{Pol}^r_{\rm Car}(X; n) := & \bigoplus_{p+q = r} \Bigl(\underline{\mathrm{Hom}}_{R} \big(\mathrm{Sym}_R (\Omega^1_{\rm com}(R)[n+1]), R\big)\otimes {\rm CoSym}^q\bigl(\mathfrak{g}[-n]\bigr)\Bigr)^{\mathrm{G}}\\ = & \bigoplus_{p+q = r} \Big( \mathrm{Pol}(R ; n) \otimes {\rm CoSym}^q\bigl(\mathfrak{g}[-n]\bigr) \Big)^{\mathrm{G}}. \end{align}\tag{14}\] The Lie bracket on \(\mathrm{Pol}_{\rm Car}(X; n)\) is defined by \[\label{for:32Lie32on32polcar} \{-, -\} := \{-,-\}_{SN} \otimes \xi,\tag{15}\] where \(\{-,-\}_{SN}\) is the Schouten–Nijenhuis bracket on \(\mathrm{Pol}(R ; n)\) and \(\xi\) is the shuffle product on \({\rm CoSym}(\mathfrak{g}[-n])\). Under this model, shifted Poisson structures on derived quotient stacks are given as follows. Let \(\widehat{\mathrm{Pol}}_{\rm Car}(X; n)[n+1] := \prod_{r \geq 0} \mathrm{Pol}^r_{\rm Car}(X; n)[n+1]\) be the completion with the weight filtration: for any \(k \in \mathbb{Z}_{\geq 0}\), \[F^k\widehat{\mathrm{Pol}}_{\rm Car}(X; n)[n+1] := \prod_{i \geq 0} \mathrm{Pol}^{k+i}_{\rm Car}(X; n) [n+1].\]

By the Maurer–Cartan formalism, one has

Now, fix a cofibrant resolution \(\pi : Q \stackrel{\simeq}{\twoheadrightarrow}A\) in \(\mathrm{DGA}^{\leq 0}_S\), and apply the above analysis to the case of the derived moduli stack of representations \[\mathpzc{DRep}^{ A}_{\mathbf{d}} := [\mathrm{DRep}^{ Q}_{\mathbf{d}} / \mathrm{GL}_\mathbf{d}].\] Then \(\mathbb{L}= \mathrm{cone}\Bigl( \mathbb{L}_{\mathrm{DRep}^{ Q}_{\mathbf{d}}} \stackrel{\rho}{\to} Q_\mathbf{d}\otimes \mathfrak{gl}^\ast _\mathbf{d}\Bigr)[-1].\) A key observation in Yeung’s work [10] is that the noncommutative cotangent complex of \(A\) should be \(\mathscr{S}_{ A}[-1]\), where \(\mathscr{S}_{ A} := \mathrm{cone}(\Omega^1( A) \to A \otimes A).\) Applying the derived Van den Bergh functor to \(\mathscr{S}_{ A}[-1]\), one obtains an isomorphism in \(\mathrm{Ho}(\mathrm{Mod}(Q_\mathbf{d}))\): \(\mathbf{L}(\, \mathbf{R}\pi^\ast\mathscr{S}_{ A}[-1])_\mathbf{d}\simeq \mathbb{L}.\)

As mentioned at the beginning of this section, Yeung [10] proved that pre-Calabi-Yau structures induce shifted Poisson structures on derived stacks of representations; in particular, he showed that a shifted double Poisson structure induces a pre-Calabi-Yau structure in [10]. The trace compatibility used below is a derived analogue of Van den Bergh’s compatibility between the double Schouten–Nijenhuis bracket and the Schouten–Nijenhuis bracket on representation schemes.

4 Noncommutative Poisson Extensions↩︎

In this section we systematically examine extensions of noncommutative Poisson structures. The goal of this section is to develop the mechanism that allows double Poisson and NC Poisson brackets to interact with the cobar–bar construction used in Section 5. We study criteria for extending double Poisson brackets on \(A\) to \(A\langle t\rangle\), for extending NC Poisson structures, and for relating these two extension problems. In particular, we show that noncommutative Poisson extensions are compatible with noncommutative Hamiltonian reduction and fit the Kontsevich–Rosenberg principle.

4.1 Double Poisson algebra extension↩︎

Let \((A, \mathopen{\{\!\!\{}-,-\mathclose{\}\!\!\}})\) be a dg double Poisson algebra. We aim to extend the double bracket to the free product \(A\langle t\rangle := A\ast_S S[t]\), where the degree of \(t\) is not assumed to be zero. By the Leibniz rule, such an extension is determined by the double derivation \(\mathopen{\{\!\!\{}t, - \mathclose{\}\!\!\}}\). We therefore introduce the following definition.

Here, \(\kappa_1,\;\kappa_2\) are determined by the Koszul sign rule: \[\begin{align} \kappa_1 & = |\Theta| |\mathfrak{s}^n a| + |\Theta(b)'| |\Theta(b)''| + |\Theta(b)'| |\mathcal{H}_a (\Theta(b)'') |;\\ \kappa_2 &= |\mathfrak{s}^n b| (|\Theta| -n -|a|) + |\mathcal{H}_b (\Theta(a)' )'| |\mathcal{H}_b (\Theta(a)')'' \Theta(a)''| +1. \end{align}\]

Let \((A,\partial,\mathopen{\{\!\!\{}-,-\mathclose{\}\!\!\}})\) be a dg double Poisson algebra of degree \(n\). Assume \(t\) has degree \(n\) and let \(\Theta\) be a dg double derivation of degree zero. We extend the dg double bracket to the free product \(A\langle t\rangle := A\ast_S S[t]\) by setting

1. \(\mathopen{\{\!\!\{}t,t \mathclose{\}\!\!\}}:= t\otimes 1-1\otimes t\);

2. \(\mathopen{\{\!\!\{}t,a \mathclose{\}\!\!\}}:= \Theta(a)\), for \(a\in A\).

The following theorem provides a sufficient condition for this extension to be a dg double Poisson bracket.

Throughout this work, the dg algebra \(A\langle t\rangle\) endowed with the dg double bracket defined above is called the double Poisson algebra extension of \(A\) by \(\Theta\), denoted \(A\langle t;\Theta\rangle\).

4.2 NC Poisson algebra extension↩︎

It is natural to consider the extension problem in the setting of NC Poisson structures. Moreover, since a double Poisson bracket induces an NC Poisson structure, one must clarify how the two extension procedures interact. Since an NC Poisson structure is not merely a dg Lie bracket on \(A_\natural\); rather, the adjoint action in each argument is induced by a derivation, we introduce the following definition.

Let \((A,\partial,\{-,-\})\) be an NC Poisson algebra of degree \(n\). Assume \(t\) has degree \(n\), and let \(\nu\) be an NC Poisson derivation of degree zero. We extend the NC Poisson bracket \(\{-,-\}\) to \(A\langle t\rangle\) by setting, for any \(\bar a\in A_\natural\), \[\label{for:32H95032ext} \{\bar t, \bar a \} :=\overline{\, \nu (a) \,},\qquad \{\bar t, \bar t\} = 0.\tag{16}\] The following theorem provides a sufficient condition for this extension.

Throughout this work, the dg algebra \(A\langle t\rangle\) endowed with the bracket defined above is called the NC Poisson algebra extension of \(A\) by \(\nu\), denoted \(A\langle t;\nu\rangle\).

Next, we show that a dg double Poisson algebra extension canonically induces an NC Poisson algebra extension.

4.3 Reduction commutes with extension↩︎

Let \(A\) be a noncommutative Hamiltonian algebra and let \(A_\mathbf{w}\) denote its noncommutative Hamiltonian reduction. It is natural to ask whether extensions are compatible with reduction. In general, \(A_\mathbf{w}\) need not carry a dg double Poisson bracket. We show that certain dg double Poisson algebra extensions of \(A\) descend to NC Poisson algebra extensions of \(A_\mathbf{w}\), and that this descent is compatible with the Kontsevich–Rosenberg principle.

Since an NC Poisson extension is determined by the derivation associated with \(t\), the following lemma gives a criterion for when the NC Poisson derivation descends to the noncommutative Hamiltonian reduction.

Hence, if \(|\Theta|=0\), then \(A_\mathbf{w}\langle t;\theta\rangle\) is an NC Poisson algebra extension. Moreover, their NC Poisson structures are compatible as follows.

In other words, noncommutative reduction commutes with extension. Next, we prove that this commutativity fits into the Kontsevich–Rosenberg principle for ordinary algebras.

5 \(\mathfrak{s}\)-constructions↩︎

For a dg double Poisson algebra \(A\), a natural question is whether the shifted complex \(\mathfrak{s}^d A = A[d]\) inherits a dg double Poisson structure. We give an affirmative answer by introducing the \(\mathfrak{s}\)-construction. As a main application, we explain how double Poisson brackets induce Lie algebra structures on reduced cyclic homology. This phenomenon may be viewed as a noncommutative analogue of the classical fact that the cohomology of a smooth Poisson manifold carries a graded Lie algebra structure.

5.1 Main constructions↩︎

We begin by recalling the bar and cobar constructions; see [15] for further details. Let \(A = S \oplus \bar{A} \in \mathrm{ADGA}^{\leq 0}_S\) be an augmented dg \(S\)-algebra. On the cocomplete coaugmented graded coalgebra \(T^c_S(\mathfrak{s}\bar{A})=\bigoplus_{n=0}(\mathfrak{s}\bar{A})^{\otimes n} = S\oplus\mathfrak{s}\bar{A}\oplus(\mathfrak{s}\bar{A})^{\otimes 2}\oplus\cdots,\) there are two differentials, \(\partial\) and \(b\). The differential \(\partial\) is induced by the internal differential \(\partial_A\) on \(A\). More precisely, \[\partial(\mathfrak{s}a_1\otimes\cdots\otimes\mathfrak{s}a_n)=\sum_{i=1}^{n} (-1)^{|\mathfrak{s}a_{<i}|+1} \mathfrak{s}a_1\otimes\cdots\otimes\mathfrak{s}(\partial_A a_i)\otimes\cdots\otimes\mathfrak{s}a_n.\] The differential \(b\) is given by: \[b(\mathfrak{s}a_1\otimes\cdots\otimes\mathfrak{s}a_n)=\sum_{i=1}^{n-1} (-1)^{|\mathfrak{s}a_{<i+1}| +1} \mathfrak{s}a_1\otimes\cdots\otimes\mathfrak{s}(a_i a_{i+1})\otimes\cdots\otimes\mathfrak{s}a_n.\] One checks that \(\partial\circ b+b\circ\partial=0\). Hence, the total differential \(\partial+b\) is well-defined on \(T_S^c(\mathfrak{s}\bar{A})\). The cocomplete coaugmented dg \(S\)-coalgebra \(T_S^c(\mathfrak{s}\bar{A})\) is called the bar construction of \(A\), denoted \(\mathbf{B}A\).

Let \(C=S\oplus\bar{C}\) be a coaugmented dg \(S\)-coalgebra. On the augmented graded \(S\)-algebra \[T_S(\mathfrak{s}^{-1}\bar{C})=\bigoplus_{n=0}(\mathfrak{s}^{-1}\bar{C})^{\otimes n} = S\oplus\mathfrak{s}^{-1}\bar{C}\oplus(\mathfrak{s}^{-1}\bar{C})^{\otimes 2}\oplus\cdots,\] there are two differentials, \(\partial\) and \(\delta\). \(\partial\) is induced from the internal differential \(\partial_C\) on \(C\) by \[\partial(\mathfrak{s}^{-1}c_1\otimes\cdots\otimes\mathfrak{s}^{-1}c_n) :=\sum_{i=1} (-1)^{|\mathfrak{s}^{-1}c_{< i}| +1}\mathfrak{s}^{-1}c_1\otimes\cdots\otimes\mathfrak{s}^{-1}(\partial_C c_i)\otimes\cdots\otimes\mathfrak{s}^{-1}c_n.\] The differential \(\delta\) is given by \[\delta(\mathfrak{s}^{-1}c_1\otimes\cdots\otimes\mathfrak{s}^{-1}c_n) =\sum_{i=1}^{n} (-1)^{|\mathfrak{s}^{-1}c_{<i}|+|\mathfrak{s}^{-1}c_{i1}|} \mathfrak{s}^{-1}c_1\otimes\cdots\otimes\mathfrak{s}^{-1}c_{i1}\otimes\mathfrak{s}^{-1}c_{i2}\otimes\cdots\otimes\mathfrak{s}^{-1}c_n,\] where \(\Delta(c_i)=c_{i1}\otimes c_{i2}\). One checks that \(\partial\circ\delta+\delta\circ\partial=0\). Thus, the total differential \(\partial+\delta\) is well-defined on \(T_S(\mathfrak{s}^{-1}\bar{C})\). The augmented dg \(S\)-algebra \(T_S(\mathfrak{s}^{-1}\bar{C})\) is called the cobar construction of \(C\), denoted \(\mathbf{\Omega}C\).

A key property of the cobar–bar construction is that it provides canonical cofibrant resolutions in the categories of dg algebras and dg coalgebras. A proof of the following proposition can be found in [16].

A key observation is the following:

As a corollary, the canonical cofibrant resolution \(\mathbf{\Omega}\mathbf{B}A\) provided by the cobar–bar construction is isomorphic to \[(T_S\,\mathfrak{s}^{-1}(\mathfrak{s}\bar A\langle t\rangle),\partial+b+\delta,\otimes)\] as dg algebras. Furthermore, the reason to replace the bar construction by \(S \oplus \mathfrak{s}\bar A \langle t \rangle\) is as follows. Using the shifted multiplication \(\widetilde{\mathbf{m}}\) (see (17 )), one can verify that there exists a free product \(\ast\) on \(\mathfrak{s}\bar A \langle t \rangle\); this free product is not the same as the underlying tensor product in the bar construction. For \(\mathfrak{s}a,\;\mathfrak{s}b \in \mathfrak{s}\bar A \langle t \rangle\), one has \(\mathfrak{s}a \ast \mathfrak{s}b = (-1)^{d|a| + d}\mathfrak{s}(ab)\).

When \(A\) is a dg double Poisson algebra, it is natural to ask whether \(T\mathfrak{s}^{-1}(\mathfrak{s}\bar A\langle t\rangle)\) inherits a dg double Poisson structure. The crucial point is to verify the existence of a double Lie bracket on \(\mathfrak{s}\bar A\langle t\rangle\). It is known that if \(L\) is a dg Lie algebra of degree \(n\), then for any \(d\in\mathbb{Z}\), the shifted complex \(\mathfrak{s}^d L\) is a dg Lie algebra of degree \(n-d\). However, for a dg double Poisson algebra \(A\) of degree \(n\), it does not follow in general that \(\mathfrak{s}^d A\) is a dg double Poisson algebra of degree \(n-d\). Intuitively, one should define a double bracket on \(\mathfrak{s}^d A\) by shifting the bracket on \(A\).

Although shifting is not an endomorphism in the category of dg algebras, it nevertheless equips \(\mathfrak{s}^d A\) with a shifted multiplication \[\label{for:32tbm} \mathfrak{s}^d\circ\mathbf{m}\circ(\mathfrak{s}^{-d}\otimes\mathfrak{s}^{-d}): \mathfrak{s}^d A \otimes \mathfrak{s}^d A \to \mathfrak{s}^d A.\tag{17}\] When no ambiguity arises we denote \(\mathfrak{s}^d\circ\mathbf{m}\circ(\mathfrak{s}^{-d}\otimes\mathfrak{s}^{-d})\) by \(\widetilde{\mathbf{m}}\). Note that \(\mathfrak{s}^d A\) is not an algebra in the usual sense, since the shifted multiplication \(\widetilde{\mathbf{m}}\) has degree \(d\) rather than zero. When \(A\) admits a dg double Poisson bracket, the dg double Lie bracket from Proposition 24 is compatible with the shifted multiplication:

Thus, when \(A\) admits a dg double Poisson bracket, the above lemma shows that \(\mathfrak{s}^d A\) still carries a dg double Poisson bracket. Moreover, Lemma 1 implies that double Poisson brackets induce Lie brackets on the cyclic quotient. We verify that this phenomenon persists in the \(\mathfrak{s}\)-construction as well. The shifted multiplication induces a bracket on \(\mathfrak{s}^d A\): \[\{-,-\}_{\mathfrak{s}^d A}:=(-1)^{d}\mathfrak{s}^d\circ\{-,-\}_A\circ(\mathfrak{s}^{-d}\otimes\mathfrak{s}^{-d}) : \mathfrak{s}^d A \otimes \mathfrak{s}^d A \to \mathfrak{s}^d A.\] We define the commutator of \(\mathfrak{s}^d A\) to be \([\mathfrak{s}^d A,\mathfrak{s}^d A]:=(\widetilde{\mathbf{m}}-\widetilde{\mathbf{m}}\circ\tau)(\mathfrak{s}^d A\otimes \mathfrak{s}^d A)\). Following the previous convention, \((\mathfrak{s}^d A)_{cyc}:=\mathfrak{s}^d A/[\mathfrak{s}^d A,\mathfrak{s}^d A]\).

We now return to the cobar–bar construction \(T_S\,\mathfrak{s}^{-1}(\mathfrak{s}\bar A\langle t\rangle)\). If \(A\) is a dg double Poisson algebra of degree \(n\), by the noncommutative Poisson extension discussed in Section 4, we may choose the zero double derivation to extend \(\mathopen{\{\!\!\{}-,-\mathclose{\}\!\!\}}\) onto \(\mathfrak{s}\bar A\langle t\rangle\).

Furthermore, the cobar–bar construction carries a dg double Poisson structure.

At the end, since multiplications \(\mathbf{m}\) and \(\widetilde{\mathbf{m}}\) commute with above differentials, by the same argument as in Lemma 1, one has the following corollary.

5.2 Application to preprojective algebra↩︎

First, we briefly recall cyclic homology theory. See [15] for further details. Let \(A \in \mathrm{ADGA}^{\leq 0}_S\) be a dg algebra. The Hochschild complex \(\mathrm{CH}(A)_\bullet\) is a model to compute the Hochschild homology of \(A\), which is the cohomology of \(A \stackrel{\mathbf{L}}{\otimes}_{A^e} A\) (under the cohomological grading). It is obtained by replacing \(A\) with its bar resolution. It is a mixed complex with respect to the differential \(\partial+ b\) on \(\mathrm{CH}(A)_\bullet\) and Connes’ \(\mathsf{B}\) operator. The cyclic complex is defined to be \(\mathrm{CC}(A)_\bullet : = (\mathrm{CH}(A)_\bullet [u^{-1}], \partial+ b +u\mathsf{B})\). Here, \(u\) is of degree \(2\).

On the other hand, the cyclic permutation operator \(\tau\) acts on the Hochschild complex \(\mathrm{CH}(A)\). The Connes’ complex is defined to be \(C^\lambda (A) _\bullet : = (\mathrm{coker}(id - \tau), \partial+ b)\), see [15] for details. Since we work over an algebraically closed field of characteristic zero, cohomologies \(\mathrm{H}^\bullet (\mathrm{CC}(A))\) and \(\mathrm{H}^\bullet (C^\lambda(A))\) coincide and are called the cyclic homology \(\mathsf{HC}_\bullet(A)\). See [15] for details.

Berest–Khachatryan–Ramadoss introduced the functor \[\mathcal{C}:\mathrm{DGA}_S\to\mathrm{Com}_S,\qquad A\mapsto\mathrm{cone}(S_{cyc}\to A_{cyc}).\] Here the map \(S_{cyc}\to A_{cyc}\) is induced by the unit map of \(A\). One of the main results in [5] is that the functor \(\mathcal{C}\) admits a total left derived functor \(\mathbf{L}\mathcal{C}:\mathrm{Ho}(\mathrm{DGA}_S)\to\mathrm{Ho}(\mathrm{Com}_S)\). Berest et al. proved that \(\mathbf{L}\mathcal{C}( S \backslash A)\cong\mathrm{cone}(C^\lambda(S) _\bullet\to C^\lambda (A)_\bullet)\). Here, by \(\;S \backslash A\), we emphasize that \(A\) is a dg algebra over \(S\). See [5] for details. Combining Berest–Khachatryan–Ramadoss’ result with Corollary [cor: cobar bar has loday], we obtain

For a finite quiver \(Q\), Crawley-Boevey, Etingof, Ginzburg and Van den Bergh showed the quotient space \(\Pi Q / [\Pi Q, \Pi Q]\) carries a Lie bracket. Note that this Lie bracket indeed descends to the zeroth reduced cyclic homology \(\overline{\mathsf{HC}}_0(\Pi Q)\). This raises the following natural questions: (1) Does there exist a Lie algebra structure on the full reduced cyclic homology \(\overline{\mathsf{HC}}_\bullet(\Pi Q)\)? (2) How can such a Lie algebra structure be explained at the cochain level? Since there is no canonical double Poisson bracket on \(\Pi Q\), one cannot deduce a graded Lie algebra structure on \(\overline{\mathsf{HC}}_\bullet(\Pi Q)\) directly from Section 5. However, we still have the following result.

It is straightforward to verify that the above results hold for deformed preprojective algebras, since for any \(\mathbf{r}\in S\), the deformed preprojective algebra \(\Pi^\mathbf{r}Q = \mathbb{K} \overline{Q}/(\mathbf{w}- \mathbf{r})\) is also a noncommutative Hamiltonian reduction.

Finally, it is natural to ask whether, when \(A \in \mathrm{ADGA}^{\leq 0}_S\) is endowed with a shifted double Poisson structure (see Definition 6), there is a homotopy version of Corollary 5? In forthcoming work, we present an affirmative answer by proving the following theorem.

References↩︎

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1. \(\mathrm{DGA}_S\) is the category of dg algebras over \(S\).↩︎