Group actions on algebras and the graded Lie structure of Hochschild cohomology

TL;DR

This paper investigates the graded Lie algebra structure of Hochschild cohomology for skew group algebras, focusing on deformation theory, bracket formulas, and applications to orbifold and representation theory.

Contribution

It provides new formulas and conditions for the Gerstenhaber bracket in Hochschild cohomology of skew group algebras, especially for abelian groups and polynomial actions.

Findings

Identified conditions when brackets vanish for polynomial skew group algebras.

Derived formulas expressing brackets via group character inner products.

Connected results to graded Hecke algebras and noncommutative Poisson structures.

Abstract

Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms. We examine the Gerstenhaber bracket with a view toward deformations and developing bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We give some general results describing when brackets are zero for polynomial skew group algebras, which allow us in particular to find noncommutative Poisson structures. For abelian groups, we express the bracket using inner products of group characters. Lastly, we interpret results for graded Hecke algebras.

Equations

$$(r\otimes g)(s\otimes h)=r({}^{g}s)\otimes gh$$

$$\cdots \stackrel{{\delta }_3}{⟶}{A}^{\otimes 4}\stackrel{{\delta }_2}{⟶}{A}^{\otimes 3}\stackrel{{\delta }_1}{⟶}{A}^e\stackrel{m}{⟶}A\to 0,$$

$$[f,f^{\prime }]=f\overline{\circ }{f}^{\prime }-(-1{)}^{(p-1)(q-1)}{f}^{\prime }\overline{\circ }f,$$

$$f\overline{\circ }{f}^{\prime }(a_1\otimes \cdots \otimes a_{p+q-1})$$

$$=\sum _{k=1}^p(-1{)}^{(q-1)(k-1)}f(a_1\otimes \cdots \otimes a_{k-1}\otimes f^{\prime }(a_k\otimes \cdots \otimes a_{k+q-1})\otimes a_{k+q}\otimes \cdots \otimes a_{p+q-1})$$

$$[\alpha ⌣\beta ,\gamma ]=[\alpha ,\gamma ]⌣\beta +(-1{)}^{p(r-1)}\alpha ⌣[\beta ,\gamma ].$$

$$a\ast b=ab+{\mu }_1(a\otimes b)t+{\mu }_2(a\otimes b)t^2+\cdots$$

$$\displaystyle{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S\#G)$$

$$\displaystyle\cong\ \ \Biggl{(}\,\bigoplus_{g\in G}{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,S\bar{g})\Biggr{)}^{G}$$

$$\displaystyle\cong\ \ \bigoplus_{g\in{\mathcal{C}}}{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,S\bar{g})^{Z(g)}.$$

$$\mathcal{R}(\gamma ):=\frac{1}{|G|}\sum _{g\in G}{}^g\gamma .$$

$${\Theta }^{\ast }(\kappa )(f_1{\overline{g}}_1\otimes \cdots \otimes f_p{\overline{g}}_p)=\kappa (f_1\otimes {}^{g_1}{f}_2\otimes \cdots \otimes {}^{(g_1\cdots g_{p-1})}{f}_p)\overline{g_1\cdots g_p}$$

$${\Theta }^{\ast }\circ \mathcal{R}:{\mathrm{Hom}}_{\mathbb{C}}(S^{\otimes p},A)$$

$$\bigl{(}(\Theta^{*}\circ\mathcal{R})(\beta)\bigr{)}(f_{1}\overline{g}_{1}\otimes\cdots\otimes f_{p}\overline{g}_{p})=\frac{1}{|G|}\sum_{g\in G}\,^{g}\!\beta(f_{1}\otimes{}^{g_{1}}f_{2}\otimes\cdots\otimes{}^{g_{1}\cdots g_{p-1}}f_{p})\,\overline{g_{1}\cdots g_{p}}$$

$${\mathrm{HH}}^p(S,A{)}^G\stackrel{\sim }{⟶}{\mathrm{HH}}^p(A).$$

$$\Delta :={\oplus }_{g\in G}S\overline{g}\otimes S\overline{g^{-1}},$$

$${\Delta }_p:=\left\{\sum f_0{\overline{g}}_0\otimes \cdots \otimes f_{p+1}{\overline{g}}_{p+1}\mid f_i\in S,g_i\in G\text{mbox}and{g}_0\cdots g_{p+1}=1\right\},$$

$$\cdots \stackrel{{\delta }_3}{⟶}{\Delta }_2\stackrel{{\delta }_2}{⟶}{\Delta }_1\stackrel{{\delta }_1}{⟶}{\Delta }_0\stackrel{m}{⟶}S(V)\to 0$$

$${\mathrm{Hom}}_{S^e}({\Delta }_p,A)\to {\mathrm{Hom}}_{S^e}(S^{\otimes (p+2)},A)$$

$$f_0{\overline{g}}_0\otimes \cdots \otimes f_{p+1}{\overline{g}}_{p+1}\mapsto f_0\otimes {}^{g_0}{f}_1\otimes {}^{g_0{g}_1}{f}_2\otimes \cdots \otimes {}^{(g_0{g}_1\cdots g_p)}{f}_{p+1}$$

$$f_0{\overline{g}}_0\otimes \cdots \otimes f_{p+1}{\overline{g}}_{p+1}\mapsto (1\otimes \overline{g_0\cdots g_{p+1}})\otimes (f_0{\overline{g}}_0\otimes \cdots \otimes f_{p+1}{\overline{g}}_{p+1}(\overline{g_0\cdots g_{p+1}}{)}^{-1}).$$

$$\Theta (f_0{\overline{g}}_0\otimes \cdots \otimes f_{p+1}{\overline{g}}_{p+1})=(1\otimes \overline{g_0\cdots g_{p+1}})\otimes f_0\otimes {}^{g_0}{f}_1\otimes {}^{g_0{g}_1}{f}_2\otimes \cdots \otimes {}^{(g_0{g}_1\cdots g_p)}{f}_{p+1}$$

$${\Psi }_p$$

$${\Phi }_p$$

$$\cdots \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\Theta }^{\ast }\circ \mathcal{R}\circ {\Psi }^{\ast }:{\mathrm{Hom}}_{S^e}(P_p,A)\to {\mathrm{Hom}}_{\mathbb{C}}(A^{\otimes p},A)$$

$${\operatorname{res}}:\operatorname{Hom}_{\mathbb{C}}(A^{\otimes\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}},A)\rightarrow\operatorname{Hom}_{\mathbb{C}}(S^{\otimes\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}},A).$$

$$\Phi^{*}\circ{\operatorname{res}}\ \ \mbox{ and }\ \ \Theta^{*}\circ\mathcal{R}\circ\Psi^{*}\$$

$$\text{on cochains, }{\mathrm{Hom}}_{\mathbb{C}}(A^{\otimes p},A)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A)\stackrel{{\scriptstyle\sim}}{{\longleftrightarrow}}{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A)^{G}\ .$$

$$\mathrm{res}:{\mathrm{HH}}^p(A)\to {\mathrm{HH}}^p(S,A).$$

$${\operatorname{res}}:{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A)^{G}.$$

Full text

The first author was partially supported by NSF grants #DMS-0402819 and #DMS-0800951 and a research fellowship from the Alexander von Humboldt Foundation. The second author was partially supported by NSA grant #H98230-07-1-0038 and NSF grant #DMS-0800832. Both authors were jointly supported by Advanced Research Program Grant 010366-0046-2007 from the Texas Higher Education Coordinating Board.

Keywords: Hochschild cohomology, Gerstenhaber bracket, symmplectic reflection algebras, deformations, graded Hecke algebras, skew group algebras.

1. Introduction

Let $G$ be a finite group acting on a $\mathbb{C}$-algebra $S$ by automorphisms. Deformations of the natural semi-direct product $S\#G$, the skew group algebra, include many compelling and influential algebras. For example, when $G$ acts linearly on a finite dimensional, complex vector space $V$, it induces an action on the symmetric algebra $S(V)$ (a polynomial ring). Deformations of $S(V)\#G$ play a profound role in representation theory and connect diverse areas of mathematics. Graded Hecke algebras were originally defined independently by Drinfeld [6] and (in the special case of Weyl groups) by Lusztig [15]. These deformations of $S(V)\#G$ include symplectic reflection algebras (investigated by Etingof and Ginzburg [7] in the study of orbifolds) and rational Cherednik algebras (introduced by Cherednik [5] to solve Macdonald’s inner product conjectures). Gordon [11] used these algebras to prove a version of the $n!$ conjecture for Weyl groups due to Haiman.

The deformation theory of an algebra is governed by its Hochschild cohomology as a graded Lie algebra under the Gerstenhaber bracket. Each deformation of the algebra arises from a (noncommutative) Poisson structure, that is, an element of Hochschild cohomology in degree 2 whose Gerstenhaber square bracket is zero. Thus, a first step in understanding an algebra’s deformation theory is a depiction of the Gerstenhaber bracket. In some situations, every noncommutative Poisson structure integrates, i.e., lifts to a deformation (e.g., see Kontsevich [13]). Halbout and Tang [12] investigate some of these structures for algebras $C^{\infty}(M)\#G$ where $M$ is a real manifold, concentrating on the case that $M$ has a $G$-invariant symplectic structure.

In this paper, we explore the rich algebraic structure of the Hochschild cohomology of $S\#G$ with an eye toward deformation theory. We describe the Gerstenhaber bracket on the Hochschild cohomology of $S\#G$, for a general algebra $S$. We then specialize to the case $S=S(V)$ to continue an analysis begun in two previous articles. Hochschild cohomology is a Gerstenhaber algebra under two operations, a cup product and a graded Lie (Gerstenhaber) bracket. In [19], we examined the cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) as a graded algebra under cup product. In this paper, we study its Gerstenhaber bracket and find noncommutative Poisson structures. These structures catalog potential deformations of $S(V)\#G$, most of which have not yet been explored.

For any algebra $A$ over a field $k$, both the cup product and Gerstenhaber bracket on the Hochschild cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A):=\operatorname{Ext}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{A\otimes A^{op}}(A,A) are defined initially on the bar resolution, a natural $A\otimes A^{op}$-free resolution of $A$. The cup product has another description as Yoneda composition of extensions of modules, which can be transported to any other projective resolution. However, the Gerstenhaber bracket has resisted such a general description. In this paper, we use isomorphisms of cohomology which encode traffic between resolutions to analyze {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S\#G) and unearth its Gerstenhaber bracket.

For the case $S=S(V)$, we use Demazure operators on one hand and quantum partial differentiation on the other hand to render the Gerstenhaber bracket and gain theorems which predict its vanishing. For example, the cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S\#G) breaks into a direct sum over $G$. By invoking Demazure operators to implement automorphisms of cohomology, we procure the following main result in Section 9:

Theorem 9.2. The bracket of any two elements $\alpha$, $\beta$ in ${\rm HH}^{2}(S(V)\#G)$ supported on group elements acting nontrivially on $V$ is zero: $[\alpha,\beta]=0\ .$

As a consequence of the theorem, the Gerstenhaber square bracket of every element ${\rm HH}^{2}(S(V)\#G)$ supported off the kernel of the action of $G$ on $V$ defines a noncommutative Poisson structure. In particular, if $G$ acts faithfully, any Hochschild $2$-cocycle with zero contribution from the identity group element defines a noncommutative Poisson structure.

The cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S\#G) is graded not only by cohomological degree, but also by polynomial degree. In [17], we showed that every constant Hochschild 2-cocycle (i.e., of polynomial degree 0) defines a graded Hecke algebra. Since graded Hecke algebras are deformations of $S(V)\#G$ (see Section 11), this immediately implies that every constant Hochschild 2-cocycle has square bracket zero. We articulate our automorphisms of cohomology using quantum partial differentiation to extend this result in Section 8 to cocycles of arbitrary cohomological degree:

Theorem 8.1 The bracket of any two constant cocycles $\alpha$, $\beta$ in {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) is zero: $[\alpha,\beta]=0\ .$

Supporting these two main “zero bracket” theorems, we give formulas for calculating brackets at the cochain level in Theorems 6.12 and 7.2. We apply these formulas to the abelian case in Example 7.6 and express the bracket of 2-cocycles in terms of inner products of characters. We use this example in Theorem 10.2 to show that the hypotheses of Theorem 9.2 can not be weakened and that its converse is false for abelian groups.

We briefly outline our program. In Section 2, we establish notation and definitions. In Sections 3 and 4, we construct an explicit, instrumental isomorphism between {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S\#G) and {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,S\#G)^{G} and lift the Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S\#G) to an arbitrary resolution used to compute {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,S\#G)^{G}. In Sections 5 and 6, we turn to the case $S=S(V)$ and examine isomorphisms of cohomology developed in [18]. We lift the Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) from the bar resolution of $S(V)\#G$ to the Koszul resolution of $S(V)$. In Section 7, we give our closed formulas for the bracket in terms of quantum differentiation and recover the classical Schouten-Nijenhuis bracket in case $G=\{1\}$. These formulas allow us to characterize geometrically some cocycles with square bracket zero in Section 8.

Mindful of applications to deformation theory, we turn our attention to cohomological degree two in Section 9. We prove the above Theorem 9.2 (which applies to cocycles of arbitrary polynomial degree) and focus on abelian groups in Section 10. We then apply our approach to the deformation theory of $S(V)\#G$ and graded Hecke algebras. When a Hochschild 2-cocycle is a graded map of negative degree, corresponding deformations are well understood and include graded Hecke algebras (which encompass symplectic reflection algebras and rational Cherednik algebras). In Section 11, we explicitly present graded Hecke algebras as deformations defined using Hochschild cohomology, thus fitting our previous work on graded Hecke algebras into a more general program to understand all deformations of $S(V)\#G$. We show how to render a constant Hochschild $2$-cocycle into the defining relations of a graded Hecke algebra concretely and vice versa. In fact, we use our results to explain how to convert any Hochschild $2$-cocycle into the explicit multiplication map of an infinitesimal deformation of $S(V)\#G$, allowing for exploration of a wide class of algebras which include graded Hecke algebras as examples.

2. Preliminary material

Let $G$ be a finite group. We work over the complex numbers $\mathbb{C}$; all tensor products will be taken over $\mathbb{C}$ unless otherwise indicated. Let $S$ be any $\mathbb{C}$-algebra on which $G$ acts by automorphisms. Denote by ${}^{g}s$ the result of applying an element $g$ of $G$ to an element $s$ of $S$. The skew group algebra $S\#G$ is the vector space $S\otimes\mathbb{C}G$ with multiplication given by

[TABLE]

for all $r,s$ in $S$ and $g,h$ in $G$. We abbreviate $s\otimes g$ by $s\overline{g}$ ($s\in S$, $g\in G$) and $s\otimes 1,\ 1\otimes g$ simply by $s,\ \overline{g}$, respectively. An element $g$ in $G$ acts on $S$ by an inner automorphism in $S\#G$: $\ \overline{g}s(\overline{g})^{-1}=({}^{g}s)\overline{g}(\overline{g})^{-1}={}^{g}s$ for all $s$ in $S$. We work with the induced group action on all maps throughout this article: For any map $\theta$ and element $h$ in $\operatorname{GL}(V)$, we define ${}^{h}\theta$ by $(^{h}\theta)(v):=\hphantom{\,}^{h}(\theta(^{h^{-1}}v))$ for all $v$.

Hochschild cohomology and deformations

The Hochschild cohomology of a $\mathbb{C}$-algebra $A$ with coefficients in an $A$-bimodule $M$ is the graded vector space {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A,M)=\operatorname{Ext}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{A^{e}}(A,M), where $A^{e}=A\otimes A^{op}$ acts on $A$ by left and right multiplication. We abbreviate {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A)={\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A,A).

One projective resolution of $A$ as an $A^{e}$-module is the bar resolution

[TABLE]

where $\delta_{p}(a_{0}\otimes\cdots\otimes a_{p+1})=\sum_{j=0}^{p}(-1)^{j}a_{0}\otimes\cdots\otimes a_{j}a_{j+1}\otimes\cdots\otimes a_{p+1}$, and $\delta_{0}=m$ is multiplication. For each $p$, $\operatorname{Hom}_{A^{e}}(A^{\otimes(p+2)},*)\cong\operatorname{Hom}_{\mathbb{C}}(A^{\otimes p},*)$, and we identify these two spaces of $p$-cochains throughout this article.

The Gerstenhaber bracket on Hochschild cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A) is defined at the chain level on the bar complex. Let $f\in\operatorname{Hom}_{\mathbb{C}}(A^{\otimes p},A)$ and $f^{\prime}\in\operatorname{Hom}_{\mathbb{C}}(A^{\otimes q},A)$. The Gerstenhaber bracket $[f,f^{\prime}]$ in $\operatorname{Hom}_{\mathbb{C}}(A^{\otimes(p+q-1)},A)$ is defined as

[TABLE]

where

[TABLE]

for all $a_{1},\ldots,a_{p+q}$ in $A$. This induces a bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A) under which it is a graded Lie algebra. The bracket is compatible with the cup product on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A), in the sense that if $\alpha\in{\rm HH}^{p}(A)$, $\beta\in{\rm HH}^{q}(A)$, and $\gamma\in{\rm HH}^{r}(A)$, then

[TABLE]

Thus {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A) becomes a Gerstenhaber algebra.

Let $t$ be an indeterminate. A formal deformation of $A$ is an associative $\mathbb{C}[[t]]$-algebra structure on formal power series $A[[t]]$ with multiplication determined by

[TABLE]

for all $a,b$ in $A$, where $ab=m(a\otimes b)$ and the $\mu_{i}:A\otimes A\rightarrow A$ are linear maps. In case the above sum is finite for each $a,b$ in $A$, we may consider the subalgebra $A[t]$, a deformation of $A$ over the polynomial ring $\mathbb{C}[t]$. Graded Hecke algebras (see Section 11) are examples of deformations over $\mathbb{C}[t]$.

Associativity of the product $*$ implies in particular that $\mu_{1}$ is a Hochschild 2-cocycle (i.e., $\delta_{3}^{*}(\mu_{1})=0$) and that the Gerstenhaber bracket $[\mu_{1},\mu_{1}]$ is a coboundary (in fact, $[\mu_{1},\mu_{1}]=2\delta_{4}^{*}(\mu_{2})$, see [9, ($4_{2}$)]). If $f$ is a Hochschild 2-cocycle, then $f$ may or may not be the first multiplication map $\mu_{1}$ for some formal deformation. The primary obstruction to lifting $f$ to a deformation is the Gerstenhaber bracket $[f,f]$ considered as an element of ${\rm HH}^{3}(A)$. A (noncommutative) Poisson structure on $A$ is an element $\alpha$ in ${\rm HH}^{2}(A)$ such that $[\alpha,\alpha]=0$ as an element of ${\rm HH}^{3}(A)$. The set of noncommutative Poisson structures includes the cohomology classes of the first multiplication maps $\mu_{1}$ of all deformations of $A$, and thus every deformation of $A$ defines a noncommutative Poisson structure. (For more details on Hochschild cohomology, see Weibel [21], and on deformations, see Gerstenhaber [9]; Poisson structures for noncommutative algebras are introduced in Block and Getzler [2] and Xu [23].)

3. Hochschild cohomology of skew group algebras

We briefly describe various isomorphisms used to compute Hochschild cohomology. Let $S$ be any algebra upon which a finite group $G$ acts by automorphisms, and let $A:=S\#G$ denote the resulting skew group algebra. Let $\mathcal{C}$ be a set of representatives of the conjugacy classes of $G$. For any $g$ in $G$, let $Z(g)$ be the centralizer of $g$. A result of Ştefan [20, Cor. 3.4] implies that there is a $G$-action giving the first in a series of isomorphisms of graded vector spaces:

[TABLE]

The first isomorphism is in fact an isomorphism of graded algebras (under the cup product); it follows from applying a spectral sequence. We take a direct approach in this paper to gain results on the Gerstenhaber bracket. The second isomorphism results from decomposing the $S^{e}$-module $S\#G$ into the direct sum of components $S\,\overline{g}$. The action of $G$ permutes these components via the conjugation action of $G$ on itself. The third isomorphism canonically projects onto a set of representative summands.

In the next theorem, we begin transporting Gerstenhaber brackets on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S\#G) to the other spaces in (3.1) by etching an explicit isomorphism between {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S\#G) and {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,S\#G)^{G}. First, some definitions. We say that a projective resolution P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} of $S^{e}$-modules is G-compatible if $G$ acts on each term in the resolution and the action commutes with the differentials. Let $\mathcal{R}$ be the Reynold’s operator on \operatorname{Hom}_{\mathbb{C}}(S^{\otimes\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}},A),

[TABLE]

We shall use the same notation $\mathcal{R}$ to denote the analogous operator on any vector space carrying an action of $G$. Let $\Theta^{*}$ be the map “move group elements far right, applying them along the way”: Define $\Theta^{*}:\operatorname{Hom}_{\mathbb{C}}(S^{\otimes p},A)^{G}\rightarrow\operatorname{Hom}_{\mathbb{C}}(A^{\otimes p},A)$ by

[TABLE]

for all $f_{1},\ldots,f_{p}$ in $S$ and $g_{1},\ldots,g_{p}$ in $G$. A similar map appears in [12]. In the next theorem, we give an elementary explanation for the appearance of $\Theta^{*}$ in an explicit isomorphism of cohomology; the proof shows that $\Theta^{*}$ is induced from a map $\Theta$ at the chain level.

Remark 3.3**.**

The notion of $G$-invariant cohomology here is well-formulated. If $S$ and $B$ are algebras on which $G$ acts by automorphisms and $B$ is an $S$-bimodule, we may define $G$-invariant cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,B)^{G} via any $G$-compatible resolution. This definition does not depend on choice of resolution. Indeed, since $|G|$ is invertible in our field, any chain map between resolutions can be averaged over the group to produce a $G$-invariant chain map. By the Comparison Theorem [21, Theorem 2.2.6], this yields not only an isomorphism on cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,B) arising from two different resolutions, but a $G$-invariant isomorphism.

We adapt ideas of Theorem 5.1 of [4] to our general situation. It was assumed implicitly in that theorem that the resolution defining cohomology was $G$-compatible and that the cochains fed into the given map were $G$-invariant. We generalize that theorem while simultaneously making it explicit. We apply a Reynold’s operator to turn the cochains fed into the map $\Theta^{*}$ invariant, as they might not be so a priori. This allows us to apply chain maps to arbitrary cocycles, in fact, to cocycles that may not even represent invariant cohomology classes.

Theorem 3.4**.**

Let $S$ be an arbitrary $\mathbb{C}$-algebra upon which a finite group $G$ acts by automorphisms, and let $A=S\#G$. The map

[TABLE]

given by

[TABLE]

for all $f_{1},\ldots,f_{p}\in S$, $g_{1},\ldots,g_{p}\in G$, and $\beta\in\operatorname{Hom}_{\mathbb{C}}(S^{\otimes p},A)$, induces an isomorphism

[TABLE]

We note that the following proof does not require the base field to be $\mathbb{C}$, only that $|G|$ be invertible in the field.

Proof.

We express every $G$-compatible resolution as a resolution over a ring that succinctly absorbs the group and its action: Let

[TABLE]

a natural subalgebra of $A^{e}$ containing $S^{e}$. Any $S^{e}$-module carrying an action of $G$ is naturally also a $\Delta$-module: Each group element $g$ acts as the ring element $\overline{g}\otimes\overline{g^{-1}}$ in $\Delta$. In fact, an $S^{e}$-resolution of a module is $G$-compatible if and only if it is simultaneously a $\Delta$-resolution.

The bar resolution for $S$ is $G$-compatible, i.e., extends to a $\Delta$-resolution. We define a different resolution of $S$ as a $\Delta$-module that interpolates between the bar resolution of $S$ and the bar resolution of $A$. For each $p\geq 0$, let

[TABLE]

a $\Delta$-submodule of $A^{\otimes(p+2)}$. Note that $\Delta_{0}=\Delta$, each $\Delta_{p}$ is a projective $\Delta$-module, and $\Delta_{p}$ is also a projective $S^{e}$-module by restriction. The complex

[TABLE]

(where $m$ is multiplication and $\delta_{i}$ is the restriction of the differential on the bar resolution of $A$) is a $\Delta$-projective resolution of $S$. By restriction it is also an $S^{e}$-projective resolution of $S$. The inclusion $S^{\otimes(p+2)}\hookrightarrow\Delta_{p}$ induces a restriction map

[TABLE]

that in turn induces an automorphism on the cohomology ${\rm HH}^{p}(S,A)$. The subspace of $G$-invariant elements of either of the above Hom spaces is precisely the subspace of $\Delta$-homomorphisms. Therefore, a $G$-invariant element of $\operatorname{Hom}_{S^{e}}(S^{\otimes(p+2)},A)$ identifies with an element of $\operatorname{Hom}_{\Delta}(\Delta_{p},A)$, and its application to an element of $\Delta_{p}$ is found via the $\Delta$-chain map \Delta_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}\rightarrow S^{\otimes(\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}+2)} given by

[TABLE]

for all $f_{0},\ldots,f_{p+1}\in S$ and $g_{0},\ldots,g_{p+1}\in G$. Finally, one may check that the following map is an isomorphism of $A^{e}$-modules $A^{\otimes(p+2)}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A^{e}\otimes_{\Delta}\Delta_{p}$:

[TABLE]

This gives rise to an isomorphism $\operatorname{Hom}_{\Delta}(\Delta_{p},A)\cong\operatorname{Hom}_{A^{e}}(A^{\otimes(p+2)},A)$. In fact, one may obtain the bar resolution of $A$ from the $\Delta$-resolution \Delta_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} directly by applying the functor $A^{e}\otimes_{\Delta}\otimes-$; the map (3.6) realizes the corresponding Eckmann-Shapiro isomorphism on cohomology, \operatorname{Ext}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{\Delta}(S,A)\cong\operatorname{Ext}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{A^{e}}(A,A), at the chain level. Let $\Theta:A^{\otimes(p+2)}\rightarrow A^{e}\otimes_{\Delta}S^{\otimes(p+2)}$ denote the composition of (3.6) and (3.5), that is

[TABLE]

for all $f_{0},\ldots,f_{p+1}\in S$ and $g_{0},\ldots,g_{p+1}\in G$. The induced map $\Theta^{*}$ on cochains is indeed that defined by equation (3.2). The above arguments show that this induced map $\Theta^{*}$ gives an explicit isomorphism, at the chain level, from ${\rm HH}^{p}(S,A)^{G}$ to ${\rm HH}^{p}(A,A)$. ∎

4. Lifting brackets to other resolutions

The Gerstenhaber bracket on the Hochschild cohomology of an algebra is defined using the bar resolution of that algebra. But cohomology is often computed using some other projective resolution. One seeks to express the induced Gerstenhaber bracket on any other projective resolution giving cohomology. Again, let $S$ be any algebra upon which a finite group $G$ acts by automorphisms, and let $A:=S\#G$ denote the resulting skew group algebra. In this section, we lift the Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A) to other resolutions used to compute cohomology. This task is complicated by the fact that we compute the cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A) as the space {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A)^{G} (using the isomorphisms of (3.1)). Hence, we consider alternate resolutions of $S$, not $A$.

Suppose the Hochschild cohomology of $S$ has been determined using some $S^{e}$-projective resolution P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} of $S$. We assume that P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} is $G$-compatible, so that P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} defines the $G$-invariant cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A)^{G}. Let $\Psi$ and $\Phi$ be chain maps between the bar resolution and P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}, i.e.,

[TABLE]

for all $p\geq 1$, and the following diagram commutes:

[TABLE]

Let $\Psi^{*}$ and $\Phi^{*}$ denote the maps induced by application of the functor $\operatorname{Hom}_{S^{e}}(-,A)$.

Lemma 4.2**.**

The cochain maps $\Psi^{*}$ and $\Phi^{*}$ induce $G$-invariant, inverse isomorphisms on the cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A).

Proof.

Since both P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} and the bar resolution are $G$-compatible, ${}^{g}\Psi^{*}$ and ${}^{g}\Phi^{*}$ are chain maps for any $g$ in $G$. Such chain maps are unique up to chain homotopy equivalence, and so induce the same maps on cohomology. ∎

We represent an element of {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A) at the chain level by a function $f:P_{p}\rightarrow A$. By Theorem 3.4, the corresponding function from $A^{\otimes(p+2)}$ to $A$ is given by $\Theta^{*}\circ\mathcal{R}\circ\Psi_{p}^{*}(f)$. Lemma 4.2 thus implies:

Theorem 4.3**.**

Let $A=S\#G$. Let P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} be any $G$-compatible $S^{e}$-resolution of $S$ and let $\Psi$ be a chain map from the bar resolution of $S$ to P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}. The map

[TABLE]

induces an isomorphism ${\rm HH}^{p}(S,A)^{G}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\rm HH}^{p}(A)$.

We now describe the inverse isomorphism. The inclusion map $S\hookrightarrow A$ induces a restriction map

[TABLE]

Theorem 4.4**.**

Let $A=S\#G$. Let $\Psi,\Phi$ be any chain maps converting between the bar resolution of $S$ and P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}, as above. The maps

[TABLE]

[TABLE]

phisms on cohomology,

[TABLE]

Proof.

By Theorem 4.3, $\Theta^{*}\circ\mathcal{R}\circ\Psi^{*}$ induces an isomorphism from ${\rm HH}^{p}(S,A)^{G}$ to ${\rm HH}^{p}(A)$. We show that $\Phi^{*}\circ{\operatorname{res}}$ defines an inverse map on cohomology.

By functoriality of Hochschild cohomology (see [14, §1.5.1]), ${\operatorname{res}}$ induces a linear map:

[TABLE]

We saw in the proof of Theorem 3.4 that the Eckmann-Shapiro isomorphism realized at the chain level in (3.6) induces a map from $\operatorname{Hom}_{A^{e}}(A^{\otimes(p+2)},A)$ to $\operatorname{Hom}_{S^{e}}(\Delta_{p},A)^{G}$. The restriction of (3.6) to $S^{\otimes(p+2)}$ is essentially the identity map, and it induces a map to $\operatorname{Hom}_{S^{e}}(S^{(p+2)},A)^{G}$ agreeing with ${\operatorname{res}}$. Hence the image of ${\operatorname{res}}$ is $G$-invariant. Note that ${\operatorname{res}}\circ\Theta^{*}=1$ on cochains, and thus on cohomology. By the proof of Theorem 3.4, $\Theta^{*}$ is invertible on cohomology, and so we conclude that ${\operatorname{res}}=(\Theta^{*})^{-1}$ and ${\operatorname{res}}$ also yields an isomorphism on cohomology:

[TABLE]

Lemma 4.2 then implies that $\Phi^{*}\circ{\operatorname{res}}$ induces a well-defined map on cohomology with $G$-invariant image. The fact that it is inverse to $\Theta^{*}\circ\mathcal{R}\circ\Psi^{*}$ follows from the observations that $\mathcal{R}$ commutes with $\Phi^{*}$ (as $\Phi^{*}$ is invariant), $\Phi^{*}$ and $\Psi^{*}$ are inverses on cohomology, and $\mathcal{R}$ is the identity on $G$-invariants. ∎

Remark 4.5**.**

We may replace $\Psi^{*}$ in Theorems 4.3 and 4.4 above by any other cochain map (with the same domain and range), provided that map induces an automorphism inverse to $\Phi^{*}$ on the cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A), even if it is not induced by a map on the original projective resolution. (As $\Phi^{*}$ is $G$-invariant on cohomology, so too is its inverse.)

Remark 4.6**.**

Some comments are in order before we give a formula for the Gerstenhaber bracket. By Theorem 4.4, a cocycle in $\operatorname{Hom}_{\mathbb{C}}(A^{\otimes p},A)$ is determined by its values on $S^{\otimes p}$, and so we may compute the bracket of two cocycles by determining how that bracket acts on elements in $S^{\otimes p}$. The ring $S$ embeds canonically in $A$, and we identify this ring with its image when convenient. We also note that, for the purpose of using Theorem 4.4, it suffices to start with a (not necessarily $G$-invariant) cocycle $\alpha$ in \operatorname{Hom}_{S^{e}}(P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}},A) and apply $\Theta^{*}\circ\mathcal{R}\circ\Psi^{*}$, since $\mathcal{R}\circ\Psi^{*}(\alpha)$ is cohomologous to $\mathcal{R}\circ\Psi^{*}\circ\mathcal{R}(\alpha)$: Indeed, observe that $\mathcal{R}(\Psi^{*}(\mathcal{R}(\alpha)))=\mathcal{R}(\mathcal{R}(\Psi^{*})(\alpha))$ and that $\mathcal{R}({\mathcal{R}}(\Psi^{*})(\alpha))\sim\mathcal{R}(\Psi^{*}(\alpha))$ since $\Psi^{*}$ and ${\mathcal{R}}(\Psi^{*})$ are both chain maps.

We now lift the Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A) to the cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A)^{G} expressed in terms of any resolution of $S$. Again, we assume the resolution is $G$-compatible, or else it may not define {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A)^{G}. Theorem 4.4 implies the following formula for the graded Lie bracket at the cochain level, on cocycles in \operatorname{Hom}_{S^{e}}(P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}},A)^{G}. We use Definition 2.2 of the bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A) at the cochain level.

Theorem 4.7**.**

Let P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} be any $G$-compatible $S^{e}$-resolution of $S$. The isomorphism {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A)\cong{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A)^{G} induces the following bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S,A)^{G} (expressed via P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}): For $\alpha,\beta$ in \operatorname{Hom}_{S^{e}}(P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}},A) representing $G$-invariant cohomology classes, the cohomology class of $[\alpha,\beta]$ is represented at the cochain level by

[TABLE]

where the bracket $[\ \ ,\ \,]$ on the right side is the Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A).

Proof.

We apply the inverse isomorphisms of Theorem 4.4 to lift the bracket from the bar complex of $A$ to P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}: The above formula gives the resulting bracket $\Phi^{*}{\operatorname{res}}\,[\Theta^{*}\mathcal{R}\Psi^{*}(\alpha),\Theta^{*}\mathcal{R}\Psi^{*}(\beta)]$. Note that the restriction map ${\operatorname{res}}$ is not needed in the formula since the output of $\Phi$ is automatically in S^{\otimes\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}. Also note that it is not necessary to apply the Reynolds operator to $\alpha$ and $\beta$ before applying $\Psi^{*}$ since we are interested only in the bracket at the level of cohomology (see Remark 4.6). ∎

5. Koszul resolution

Let $G$ be a finite group and $V$ a (not necessarily faithful) $\mathbb{C}G$-module of finite dimension $n$. The Hochschild cohomology of the skew group algebra {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) is computed using the Koszul resolution of the polynomial ring $S(V)$ while the cup product and Gerstenhaber bracket are defined on the bar resolution of $S(V)\#G$. In this section, we use machinery developed in Section 4 to translate between spaces and between resolutions.

First, some preliminaries. We denote the image of $v$ in $V$ under the action of $g$ in $G$ by ${}^{g}v$. Let $V^{*}$ denote the contragredient (or dual) representation. For any basis $v_{1},\ldots,v_{n}$ of $V$, let $v_{1}^{*},\ldots,v_{n}^{*}$ be the dual basis of $V^{*}$. Denote the set of $G$-invariants in $V$ by $V^{G}:=\{v\in V:{}^{g}v=v\text{ for all }g\in G\}$ and the $g$-invariant subspace of $V$ by $V^{g}:=\{v\in V:{}^{g}v=v\}$ for any $g$ in $G$. Since $G$ is finite, we may assume $G$ acts by isometries on $V$ (i.e., $G$ preserves a Hermitian form $\langle\ ,\ \rangle$). If $h$ lies in the centralizer $Z(g)$, then $h$ preserves both $V^{g}$ and its orthogonal complement $(V^{g})^{\perp}$ (defined with respect to the Hermitian form). We shall frequently use the observation that $(V^{g})^{\perp}=\operatorname{im}(1-g)$.

The Koszul resolution K_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)) is given by $K_{0}(S(V))=S(V)^{e}$ and

[TABLE]

for $p\geq 1$, with differentials defined by

[TABLE]

for any choice of basis $v_{1},\ldots,v_{n}$ of $V$ (e.g., see Weibel [21, §4.5]). We identify $\operatorname{Hom}_{\mathbb{C}}(\textstyle\bigwedge^{p}V,S(V)\overline{g})$ with $S(V)\overline{g}\otimes\textstyle\bigwedge^{p}V^{*}$ for each $g$ in $G$.

We fix the set of cochains arising from the Koszul resolution (from which the cohomology classes emerge) as vector forms on $V$ tagged by group elements: Let

[TABLE]

We refer to C_{g}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} as the set of cochains supported on $g$. Similarly, if $X$ is a subset of $G$, we set C^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{X}:=\oplus_{g\in X}C^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{g}, the set of cochains supported on $X$. Note that group elements permute the summands of C^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} via the conjugation action of $G$ on itself.

We apply the notation of Section 4 to the case $S=S(V)$. Let P_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} be the Koszul resolution (5.1) of $S(V)$. Since the bar and Koszul complexes of $S(V)$ are both $S(V)^{e}$-resolutions, there exist chain maps $\Phi$ and $\Psi$ between the two complexes,

[TABLE]

for all $p\geq 1$, such that Diagram 4.1 commutes. Let $\Phi$ be the canonical inclusion of the Koszul resolution (5.1) into the bar resolution (2.1):

[TABLE]

for all $v_{j_{1}},\ldots,v_{j_{p}}$ in $V$. (See [18] for an explicit chain map $\Psi$ in this case.) We obtain the following commutative diagram of induced cochain maps:

[TABLE]

Note that both the Koszul and bar resolutions are $G$-compatible. Using Lemma 4.2, we identify $\Phi^{*}$ and $\Psi^{*}$ with their restrictions to

[TABLE]

Given any basis $v_{1},\ldots,v_{n}$ of $V$, let $\partial/\partial v_{i}$ denote the usual partial differential operator with respect to $v_{i}$. In addition, given a complex number $\epsilon\neq 1$, we define the $\epsilon$-quantum partial differential operator with respect to $v:=v_{i}$ as the scaled Demazure (BGG) operator $\partial_{v,\epsilon}:S(V)\rightarrow S(V)$ given by

[TABLE]

where $s\in\operatorname{GL}(V)$ is the reflection whose matrix with respect to the basis $v_{1},\ldots,v_{n}$ is $\operatorname{diag}(1,\ldots,1,\epsilon,1,\ldots,1)$ with $\epsilon$ in the $i$th slot. Set $\partial_{v,\epsilon}=\partial/\partial v$ when $\epsilon=1$. The operator $\partial_{v,\epsilon}$ coincides with the usual definition of quantum partial differentiation:

[TABLE]

where $[k]_{\epsilon}$ is the quantum integer $[k]_{\epsilon}:=1+\epsilon+\epsilon^{2}+\cdots+\epsilon^{k-1}.$

Next, we recall an explicit map $\Upsilon$, involving quantum differential operators, that replaces $\Psi^{*}$:

[TABLE]

For each $g$ in $G$, fix a basis $B_{g}=\{v_{1},\ldots,v_{n}\}$ of $V$ consisting of eigenvectors of $g$ with corresponding eigenvalues $\epsilon_{1},\ldots,\epsilon_{n}$. Decompose $g$ into reflections according to this basis: Let $g=s_{1}\cdots s_{n}$ where each $s_{i}$ in $\operatorname{GL}(V)$ is the reflection (or the identity) defined by $s_{i}(v_{j})=v_{j}$ for $j\neq i$ and $s_{i}(v_{i})=\epsilon_{i}v_{i}$. Let $\partial_{i}:=\partial_{v_{i},\epsilon_{i}}$, the quantum partial derivative with respect to $B_{g}$.

Definition 5.8**.**

We define a linear map $\Upsilon$ from the dual Koszul complex to the dual bar complex for $S(V)$ with coefficients in $A:=S(V)\#G$,

[TABLE]

Fix $g$ in $G$ with basis $B_{g}$ of $V$ as above. Let $\alpha=f_{g}\overline{g}\otimes v_{j_{1}}^{*}\wedge\cdots\wedge v_{j_{p}}^{*}$ lie in $C^{p}$ with $f_{g}$ in $S(V)$ and $1\leq j_{1}<\ldots<j_{p}\leq n$. Define $\Upsilon(\alpha):S(V)^{\otimes p}\rightarrow S(V)\bar{g}$ by

[TABLE]

Then $\Upsilon$ is a cochain map (see [18]) and thus induces an endomorphism $\Upsilon$ of cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V),A). Denote the restriction of $\Upsilon$ to the $g$-component of C^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} and of {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V),A)\cong\oplus_{g\in G}\,{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V),S(V)\bar{g}) by $\Upsilon_{g}=\Upsilon_{g,B_{g}}$ so that

[TABLE]

In formulas, we wish to highlight group elements explicitly instead of leaving them hidden in the definition of $\Upsilon$, and thus we find it convenient to define a version of $\Upsilon$ untagged by group elements:

Definition 5.9**.**

Let $\operatorname{proj}_{G\rightarrow 1}:S(V)\#G\rightarrow S(V)$ be the projection map that drops group element tags: $f\overline{g}\mapsto f$ for all $g$ in $G$ and $f$ in $S(V)$. Define

[TABLE]

We shall use the following observation often.

Remark 5.10**.**

For the fixed basis $B_{g}=\{v_{1},\ldots,v_{n}\}$ and $\alpha=f_{g}\overline{g}\otimes v_{j_{1}}^{*}\wedge\cdots\wedge v_{j_{p}}^{*}$ in $C^{p}_{g}$ (with $j_{1}<\ldots<j_{p}$), note that

[TABLE]

Generally, $\Upsilon(\alpha)(f_{1}\otimes\cdots\otimes f_{p})=0$ whenever ${\frac{\partial}{\partial v_{j_{k}}}(f_{k})}=0$ for some $k$.

The following proposition from [18] provides a cornerstone for our computations.

Proposition 5.11**.**

The map $\Upsilon$ induces an isomorphism on the Hochschild cohomology of the skew group algebra

[TABLE]

Specifically, $\Upsilon$ and $\Phi^{*}$ are $G$-invariant, inverse isomorphisms on cohomology converting between expressions in terms of the Koszul resolution and the bar resolution.

In fact, the map $\Upsilon$ is easily seen to be a right inverse to $\Phi^{*}$ on cochains, not just on cohomology; see [19, Prop. 5.4].

Remark 5.12**.**

The cochain map $\Upsilon=\bigoplus_{g\in G}\Upsilon_{g,B_{g}}$ depends on the choices of bases $B_{g}$ of eigenvectors of $g$ in $G$, but $\Upsilon$ induces an automorphism on cohomology which does not depend on the choice of basis. (This follows from Proposition 5.11 above, as $\Phi$ is independent of basis.)

6. Brackets for polynomial skew group algebras

Let $G$ be a finite group and let $V$ be a finite dimensional $\mathbb{C}G$-module. We apply the previous results to determine the Gerstenhaber bracket of {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) explicitly. We lift the Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G), which is defined via the bar complex of $S(V)\#G$, to the cohomology {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V),S(V)\#G)^{G}, which is computed via the Koszul complex of $S(V)$.

The next theorem allows one to replace the Hochschild cohomology of $S(V)\#G$ with a convenient space of forms. Set

[TABLE]

where a negative exterior power is defined to be 0. We regard these spaces as subsets of C^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}, the space of cochains arising from the Koszul complex (see (5.3)), after making canonical identifications. Note that for each $g$ in $G$, the centralizer $Z(g)$ acts diagonally on the tensor factors in H_{g}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}. The third factor $\textstyle\bigwedge^{\operatorname{codim}V^{g}}((V^{g})^{\perp})^{*}$ of H_{g}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} is a vector space of dimension one with a possibly nontrivial $Z(g)$-action. The theorem below is due to Ginzburg and Kaledin [10, (6.4)] for faithful group actions in a more general geometric setting. See also Farinati [8, Section 3.2] in the algebraic setting. Our formulation is from [17, (3.3) and (3.4)].

Theorem 6.2**.**

Set $A:=S(V)\#G$. Cohomology classes arise as tagged vector forms:

• •

H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}* is a set of cohomology class representatives for {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V),A) arising from the Koszul resolution: The inclusion map H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}\hookrightarrow C^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} induces an isomorphism H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}\cong{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V),A).*

• •

{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(A)\ \cong\ {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V),A)^{G}\ \cong\ (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G}\ .**

Remark 6.3**.**

The only contribution to degree two cohomology ${\rm HH}^{2}(S(V)\#G)$ comes from group elements acting either trivially or as bireflections:

[TABLE]

Indeed, by Definition 6.1, (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G}\cong\oplus_{g\in{\mathcal{C}}}(H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{g})^{Z(g)} where $\mathcal{C}$ is a set of representatives of the conjugacy classes of $G$. Since each group element $g$ in $G$ centralizes itself, the determinant of $g$ on $V$ is necessarily $1$ whenever (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{g})^{Z(g)} is nonzero (since $g$ acts on $\textstyle\bigwedge^{\operatorname{codim}V^{g}}((V^{g})^{\perp})^{*}$ by the inverse of its determinant). In fact, if $g$ acts nontrivially on $V$, it does not contribute to cohomology in degrees 0 and 1.

Our results from previous sections allow us to realize the isomorphism of Theorem 6.2 explicitly at the chain level in the next theorem, which will be used extensively in our bracket calculations:

Theorem 6.4**.**

Let $A=S(V)\#G$. The map

[TABLE]

induces an isomorphism

[TABLE]

Proof.

The statement follows from Theorem 4.3, Remark 4.5, Proposition 5.11, and Theorem 6.2. ∎

Remark 6.5**.**

We explained explicitly in [17] how a constant Hochschild 2-cocycle defines a graded Hecke algebra. More generally, Theorem 6.4 offers a direct conversion from vector forms in (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G} to functions on tensor powers of $S(V)\#G$ representing elements in {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G): For all $\alpha$ in $H^{p}$,

[TABLE]

for all $f_{1},\ldots,f_{p}$ in $S(V)$ and $g_{1},\ldots,g_{p}$ in $G$. In particular, if $p=2$, then $\Gamma(\alpha)(f_{1}\overline{g}_{1}\otimes f_{2}\overline{g}_{2})=\frac{1}{|G|}\sum_{g\in G}{}^{g}(\Upsilon\alpha)(f_{1}\otimes{}^{g_{1}}f_{2})\,\overline{g_{1}g_{2}}$.

We now record some inverse isomorphisms that will facilitate finding bracket formulas. For each $g$ in $G$, let $\{v_{1},\ldots,v_{n}\}$ be a basis of $V$ consisting of eigenvectors of $g$ and let

[TABLE]

be the map which takes any $\mathbb{C}$-basis element $(v_{1}^{m_{1}}\cdots v_{n}^{m_{n}})\,\overline{g}\otimes v_{i_{1}}\wedge\cdots\wedge v_{i_{p}}$ of C_{g}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} to itself if it lies in H_{g}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} and to zero otherwise. Set ${\operatorname{Proj}}_{H}:=\bigoplus_{g\in G}\operatorname{Proj}_{H_{g}}$. Then

[TABLE]

projects each cocycle in C^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} to its cohomology class representative in H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}. (See, for example, the computation of [8, Section 3.2].) In fact, the map ${\operatorname{Proj}}_{H}$ splits with respect to the canonical embedding H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}\hookrightarrow C^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}. Now consider projection just on the polynomial part of a form: Let

[TABLE]

be the canonical projection arising from the isomorphism $S(V^{g})\cong S(V)/I((V^{g})^{\perp})$ for each $g$ in $G$. Then $\operatorname{Proj}_{H_{g}}=\operatorname{Proj}_{H_{g}}\circ\operatorname{Proj}_{V^{g}}$ and thus any cocycle in $C_{g}$ which vanishes under $\operatorname{Proj}_{V^{g}}$ is zero in cohomology. (In fact, we can write $\operatorname{Proj}_{H}$ as a composition of $\bigoplus_{g}\operatorname{Proj}_{V^{g}}$ with a similar projection map on the exterior algebra.)

The inclusion map $S(V)\hookrightarrow A$ again induces a restriction map (as in Section 4):

[TABLE]

Theorem 4.4 implies that the following compositions (for $S=S(V)$) induce inverse isomorphisms on cohomology, as asserted in the next theorem:

[TABLE]

Theorem 6.6**.**

Let $A=S(V)\#G$. The maps

[TABLE]

\textstyle{\text{on cochains, }\operatorname{Hom}_{\mathbb{C}}(A^{\otimes p},A)\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma^{\prime}\hskip 11.38092pt}$$\textstyle{\ H^{p}\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces,\text{ induce inverse isomorphisms }}$$\scriptstyle{\Gamma\hskip 11.38092pt}

[TABLE]

The next theorem, a consequence of Theorem 4.7, describes the graded Lie bracket on (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G} induced by the Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G).

Theorem 6.7**.**

The Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) induces the following graded Lie bracket on (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G} under the isomorphism (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G}\cong{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) of Theorem 6.6: For $\alpha,\beta$ in (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G},

[TABLE]

where the bracket $[\ \ ,\ \,]$ on the right is the Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) given at the cochain level.

In the remainder of this section, we use the above theorem to give formulas for the Gerstenhaber bracket on (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G}\cong{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G). We introduce notation for a prebracket at the cochain level to aid computations and allow for an explicit, closed formula. Recall that we have fixed a basis $B_{g}$ of $V$ consisting of eigenvectors of $g$ for each $g$ in $G$. These choices are for computational convenience; our results do not depend on the choices. For a multi-index $I=(i_{1},\ldots,i_{m})$, we write $dv_{I}$ for $v_{i_{1}}^{*}\wedge\cdots\wedge v_{i_{m}}^{*}$. In formulas below, note that we sum over all multi-indices $I$ of a given length, not just those with indices of increasing order. We use the untagged version ${\Upsilon}\rule[3.01385pt]{3.44444pt}{0.43057pt}\hskip 0.43057pt$ of the map $\Upsilon$ with image in $S(V)$ (see Definition 5.9) to highlight the group elements appearing in various formulas.

Definition 6.8**.**

Define a prebracket (bilinear map) on the set of cochains C^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} defined in (5.3),

[TABLE]

for $g$ and $h$ in $G$, depending on a basis of eigenvectors $B_{1}$ for $g$ and a basis of eigenvectors $B_{2}$ for $h$ as follows. For $\alpha$ in $C_{g}^{p}$ and $\beta$ in $C_{h}^{q}$, define $\alpha\overline{\circ}\beta$ to be

[TABLE]

where $m=p+q-1$, $f_{h}^{(i_{k})}:={\Upsilon}\rule[3.01385pt]{3.44444pt}{0.43057pt}\hskip 0.43057pt_{h,B_{2}}(\beta)(v_{i_{k}}\otimes\cdots\otimes v_{i_{k+q-1}}),$ and $v_{1},\ldots,v_{n}$ is any basis of $V$. Define

[TABLE]

Remark 6.9**.**

When computing brackets, one may be tempted to seek results by working with just $\alpha\overline{\circ}\,\beta$ (or just $\beta\,\overline{\circ}\,\alpha$) and extending by symmetry. However the operation $\overline{\circ}$ is not defined on cohomology. Furthermore, one must exercise care in treating the operation $\overline{\circ}$ alone (e.g., see the proof of Theorem 9.2 below), as one risks covertly changing the bases used to apply ${\Upsilon}\rule[3.01385pt]{3.44444pt}{0.43057pt}\hskip 0.43057pt$ in the middle of a bracket calculation. Similarly, one must exercise care when examining the bracket of two cocycles summand by summand, although the bracket is linear: The bases used to apply ${\Upsilon}\rule[3.01385pt]{3.44444pt}{0.43057pt}\hskip 0.43057pt$ should not depend on the pair of summands considered. The maps $\Upsilon$ and ${\Upsilon}\rule[3.01385pt]{3.44444pt}{0.43057pt}\hskip 0.43057pt$ are independent of the choices of bases used when taking brackets of cohomology classes, but a choice should be made once and for all throughout the whole calculation of a Gerstenhaber bracket.

We are particularly interested in brackets and prebrackets of elements of cohomological degree $2$. For $\alpha$ in $C_{g}^{2}$ and $\beta$ in $C_{h}^{2}$, the above definition gives

[TABLE]

where $v_{1},\ldots,v_{n}$ is any basis of $V$ and $\Upsilon_{1}:=\Upsilon_{g,B_{1}}$ and $\Upsilon_{2}:=\Upsilon_{h,B_{2}}$.

Remark 6.11**.**

The theorem below articulates the Gerstenhaber bracket in terms of a fixed basis $B_{g}$ of $V$ for each $g$ in $G$. Although each individual prebracket (6.8) depends on these choices, the formula for the bracket given below is independent of these choices (by Remark 5.12).

Theorem 6.12**.**

Definition 6.8 gives a formula for the Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) as realized on (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G}: For $g,h$ in $G$ and $\alpha$ in H_{g}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} and $\beta$ in H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{h},

[TABLE]

Proof.

By Theorem 6.7, the Gerstenhaber bracket $[\ \ ,\ \,]$ on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) induces the following bracket on H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}:

[TABLE]

We show that this formula is exactly that claimed in the special case when $\alpha$ and $\beta$ have cohomological degree $2$; the general case follows analogously.

Let $v_{1},\ldots,v_{n}$ be any basis of $V$. We determine $\Phi^{*}[\Theta^{*}\,{}^{a}(\Upsilon\alpha),\Theta^{*}\,{}^{b}(\Upsilon\beta)]$ explicitly in the case $a=b=1_{G}$ as an element of

[TABLE]

by evaluating on input of the form $v_{i}\wedge v_{j}\wedge v_{k}$. The computation for general $a,b$ in $G$ is similar, using (for example) ${}^{a}(\Upsilon\alpha)=\Upsilon_{aga^{-1},{}^{a}B_{g}}({}^{a}\alpha)$ (see [18, Proposition 3.8]). For all $i,j,k$ (see Equation (5.4)),

[TABLE]

We expand the Gerstenhaber bracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) to obtain

[TABLE]

But for any $w_{1},w_{2},w_{3}$ in $V$,

[TABLE]

We obtain similar expressions for the other terms arising in the bracket, and hence the above sum is simply

[TABLE]

Similarly, we see that for all $a,b$ in $G$,

[TABLE]

and the result follows. ∎

Remark 6.14**.**

The theorem above actually gives a formula for a bracket on H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}},  not just on (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G}\cong{\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G). One can show that this extension to all of H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} agrees with the composition of the Reynolds operator on H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} with the Gerstenhaber bracket on (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G}: By Theorem 6.12 and Remark 4.6, $[\alpha,\beta]$ is $G$-invariant and cohomologous to $[\mathcal{R}(\alpha),\mathcal{R}(\beta)]$ for all $\alpha$, $\beta$ in H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}. Hence, the extension to H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} is artifical in some sense. Indeed, a natural Gerstenhaber bracket on all of H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} does not make sense, as this space does not present itself as the Hochschild cohomology of an algebra with coefficients in that same algebra. This idea may be used to explain the formula of Theorem 6.12: When applying Theorem 6.12 to $G$-invariant elements $\alpha$ and $\beta$, we might write $\alpha=\sum_{c\in[G/Z(g)]}{}^{c}\alpha^{\prime}$ and $\beta=\sum_{d\in[G/Z(h)]}{}^{d}\beta^{\prime}$, where \alpha^{\prime}\in(H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{g})^{Z(g)} and \beta^{\prime}\in(H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}_{h})^{Z(h)} are representative summands. (Here, $[G/A]$ is a set of representatives of the cosets $G/A$ for any subgroup $A$ of $G$.) Then

[TABLE]

However this more complicated expression is cohomologous to that of Theorem 6.12, since $\mathcal{R}\circ\Upsilon(\alpha)$ is cohomologous to $\mathcal{R}\circ\Upsilon\circ\mathcal{R}(\alpha)$ (see Remark 4.6, noting that $\Psi^{*}=\Upsilon\,$).

7. Explicit bracket formulas

Let $G$ be a finite group and let $V$ be a finite dimensional $\mathbb{C}G$-module. In this section, we give a closed formula for the prebracket on {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G). We shall use this formula in later sections to obtain new results on zero brackets and to locate noncommutative Poisson structures. We also recover the classical Schouten-Nijenhuis bracket in this section and we give an example.

The prebracket of Definition 6.8 simplifies enourmously when we work with a bases of simultaneous eigenvectors for $g$ and $h$ in $G$. Indeed, Remark 5.10 predicts that most terms of Definition 6.8 are zero. We capitalize on this idea in the next theorem and corollaries. When the actions of $g$ and $h$ are not simultaneously diagonalizable, we enact a change of basis at various points of the calculation of the prebracket to never-the-less take advantage of Remark 5.10. The following proof shows how to keep track of the effect on the map ${\Upsilon}\rule[3.01385pt]{3.44444pt}{0.43057pt}\hskip 0.43057pt$ (mindful of cautionary Remark 6.9).

First, some notation. For $g,h$ in $G$, let $M=M^{g,h}$ be the change of basis matrix between $B_{g}$ and $B_{h}$: For $B_{g}=\{w_{1},\ldots,w_{n}\}$ and $B_{h}=\{v_{1},\ldots,v_{n}\}$ (our fixed bases of eigenvectors of $g$ and $h$, respectively), set $M=(a_{ij})$ where, for $i=1,\ldots,n$,

[TABLE]

The formula below involves determinants of certain submatrices of $M$. If $I$ and $J$ are (ordered) subsets of $\{1,\ldots,n\}$, denote by $M_{I;J}$ the submatrix of $M$ obtained by deleting all rows except those indexed by $I$ and deleting all columns except those indexed by $J$. Recall that $dv_{I}:=v_{i_{1}}^{*}\wedge\cdots\wedge v_{i_{m}}^{*}$ for a multi-index $I=({i_{1}},\ldots,{i_{m}})$, not necessarily in increasing order. Below we use the operation $\overline{\circ}$ giving the prebracket of Definition 6.8. We also use the quantum partial differentiation operators $\partial_{v,\epsilon}$ (see (5.6)) after decomposing $g$ into reflections, $g=s_{1}\cdots s_{n}$, with respect to $B_{g}$, i.e., each $s_{i}$ in $\operatorname{GL}(V)$ is defined by $s_{i}(w_{i})={}^{g}w_{i}=\epsilon_{i}w_{i}$ and $s_{i}(w_{j})=w_{j}$ for $j\neq i$.

Theorem 7.2**.**

Let $g,h$ lie in $G$ with change of basis matrix $M=M^{g,h}$ as above. Decompose $g$ into reflections, $g=s_{1}\cdots s_{n}$, with respect to $B_{g}$ as above. Let

[TABLE]

Then $\alpha\overline{\circ}\beta$ is given as an element of $C^{p+q-1}_{gh}$ by the following formula: For $m=p+q-1$ and $I=(i_{1}<\ldots<i_{m})$,

[TABLE]

where $\det(M_{*;I-L})=0$ for $L\not\subset I$, $J_{k}:=(j_{1},\ldots,j_{k-1},j_{k+1},\ldots,j_{p})$, $\partial_{i}=\partial_{\epsilon_{i},w_{i}}$, and $\nu(k)=1-q-k+\lambda(1)+\cdots+\lambda(q)-\binom{q}{2}$ for $\lambda$ defined by $l_{s}=i_{\lambda(s)}$.

Proof.

By Definition 6.8, $(\alpha\overline{\circ}\,\beta)(v_{i_{1}}\wedge\cdots\wedge v_{i_{m}})$ is equal to

[TABLE]

where $(f_{h}^{\prime})^{(i_{\pi(k)})}=({\Upsilon}\rule[3.01385pt]{3.44444pt}{0.43057pt}\hskip 0.43057pt_{h}\beta)(v_{i_{\pi(k)}}\otimes\cdots\otimes v_{i_{\pi(k+q-1)}})$. Fix $k$ and note that by Remark 5.10, $(f_{h}^{\prime})^{(i_{\pi}(k))}$ is nonzero only if $L=(i_{\pi(k)},\ldots,i_{\pi({k+q-1})})$. Hence, we may restrict the sum to those permutations $\pi$ for which $i_{\pi(k)}=l_{1},\ldots,i_{\pi({k+q-1})}=l_{q}$. We identify this set of permutations with the symmetric group $\operatorname{Sym}_{p-1}$ (of permutations on the set $\{1,\ldots,k-1,k+q,\ldots,m\}$) in the standard way. Under this identification, the factor $\operatorname{sgn}(\pi)$ changes by

[TABLE]

For such permutations $\pi$, the vectors $v_{i_{\pi({k+q})}},\ldots,v_{i_{\pi(m)}}$ lie in $V^{h}$ (as the exterior part of $\beta\in H_{h}^{q}$ includes a volume form on $(V^{h})^{\perp}$, and hence $v_{i}\in V^{h}$ for $i\notin L$). Therefore ${}^{h}v_{i_{\pi(k+q)}}=v_{i_{\pi(k+q)}},\ldots,{}^{h}v_{i_{\pi(m)}}=v_{i_{\pi(m)}}$. Note also that $(f_{h}^{\prime})^{(i_{\pi(k)})}=f_{h}^{\prime}$ for all such $\pi$. We now invoke the change of basis

[TABLE]

for $s=1,\ldots,k-1,k+q,\ldots,m$ to obtain

xxx $\Upsilon_{g}(\alpha)(v_{i_{\pi(1)}}\otimes\cdots\otimes v_{i_{\pi({k-1})}}\otimes f_{h}^{\prime}\otimes v_{i_{\pi({k+q})}}\otimes\cdots\otimes v_{i_{\pi(m)}})$

[TABLE]

Thus the $k$-th summand of $(\alpha\overline{\circ}\,\beta)(v_{i_{1}}\wedge\cdots\wedge v_{i_{m}})$ is a sum over $\pi\in\operatorname{Sym}_{p-1}$ of

[TABLE]

times

[TABLE]

By definition of the determinant and some sign simplifications, this is precisely the formula claimed. ∎

In the remainder of this section, we refine the formula of Theorem 7.2 in the special case that the actions of $g$ and $h$ commute. In this case, we may choose $B_{g}=B_{h}$ to be a simultaneous basis of eigenvectors for both $g$ and $h$, and the change of basis matrix $M$ in Theorem 7.2 is simply the identity matrix. We introduce some notation to capture the single summand that remains, for each $k$, after the prebracket formula in Definiton 6.8 collapses. For any two multi-indices $J=(j_{1},\ldots,j_{p})$ and $L=(l_{1},\ldots,l_{q})$, and $k\leq p$, set

[TABLE]

Corollary 7.3**.**

Suppose the actions of $g,h$ in $G$ on $V$ commute and $B_{g}=B_{h}=\{v_{1},\ldots,v_{n}\}$ is a simultaneous basis of eigenvectors for $g$ and $h$. Let $\alpha=f_{g}\overline{g}\otimes dv_{J}$ lie in $H^{p}_{g}$ and $\beta=f_{h}^{\prime}\overline{h}\otimes dv_{L}$ lie in $H^{q}_{h}$, for multi-indices $J=(j_{1}<\ldots<j_{p})$ and $L=(l_{1}<\ldots<l_{q})$. Then

[TABLE]

where $s_{i}$, $s_{i}^{\prime}$ are diagonal reflections (or identity maps) with ${}^{s_{i}}v_{i}=\,^{g}v_{i}=\epsilon_{i}v_{i}$ and ${}^{s^{\prime}_{i}}v_{i}=\,^{h}v_{i}=\epsilon^{\prime}_{i}v_{i}$, and where $\partial_{i}=\partial_{v_{i},\epsilon_{i}}$ and $\partial^{\prime}_{i}=\partial_{v_{i},\epsilon_{i}^{\prime}}$ .

If the group elements $g$ and $h$ act as the identity, we recover the classical Schouten-Nijenhuis bracket, as detailed in the next two corollaries. Indeed, we view Corollary 7.3 as giving a quantum version of the Schouten-Nijenhuis bracket.

Corollary 7.4**.**

Let $g$ and $h$ in $G$ both act trivially on $V$ (e.g., $g=h=1_{G}$). Let $\alpha=f_{g}\overline{g}\otimes dv_{J}$ lie in $H^{p}_{g}$ and $\beta=f_{h}^{\prime}\overline{h}\otimes dv_{L}$ lie in $H^{q}_{h}$, for multi-indices $J=(j_{1}<\ldots<j_{p})$ and $L=(l_{1}<\ldots<l_{q})$. If $B_{g}=\{v_{1},\ldots,v_{n}\}=B_{h}$, then

[TABLE]

If the group $G$ is trivial, then the bracket agrees with the prebracket:

Corollary 7.5**.**

If $G=\{1\}$, we recover the classical Schouten-Nijenhuis bracket: Let $\alpha=f_{1}\otimes dv_{J}$ in degree $p$ and $\beta=f_{2}\otimes dv_{L}$ in degree $q$. Then

[TABLE]

For abelian groups, brackets enjoy a combinatorial description:

Example 7.6**.**

Let $G$ be an abelian group acting on $V$ with $\dim V\geq 3$. Let $B=\{v_{1},\ldots,v_{n}\}$, a simultaneous basis of eigenvectors for $G$. For $i\in\{1,\ldots,n\}$, let $\chi_{i}$ be the character defined by ${}^{g}v_{i}=\chi_{i}(g)\,v_{i}$ for each $g$ in $G$. Fix $g,h\in G$ and

[TABLE]

elements of $H^{2}_{g}$, $H^{2}_{h}$, respectively. (Note that if $g$ acts nontrivially on $V$, then $c_{1}=c_{2}=0$, and if $h$ acts nontrivially on $V$, then $d_{2}=d_{3}=0$.) A calculation using Theorems 6.12 and 7.2 gives the bracket:

[TABLE]

where

[TABLE]

Here, $\epsilon={\chi_{2}(h)},\epsilon^{\prime}={\chi_{2}(g)}$, $[m]_{\lambda}$ is the quantum integer $1+\lambda+\ldots+\lambda^{m-1}$ (or zero when $m=0$), and $\langle\hskip 7.22743pt,\hskip 2.84544pt\rangle$ denotes the inner product of characters on $G$. The map ${\operatorname{Proj}}_{H}$ in particular projects the polynomial coefficient onto $S(V^{gh})$: $v_{i}\mapsto v_{i}$ if $\chi_{i}(gh)=1$, $v_{i}\mapsto 0$ otherwise. Thus $[\alpha,\beta]$ is usually [math], but can be nonzero as a consequence of the orthogonality relations on characters of finite groups (see Proposition 10.1 below).

8. Zero brackets

Let $G$ be a finite group and let $V$ be a finite dimensional $\mathbb{C}G$-module. Every deformation arises from a Hochschild $2$-cocycle $\alpha$ whose square bracket $[\alpha,\alpha]$ is a coboundary. We use the formulas of Section 7 to now determine some conditions under which brackets are zero. In the process, we take advantage of our depiction of cohomology automorphisms in terms of quantum partial derivatives. We begin with an easy corollary of our formulation of the Gerstenhaber bracket as a sum of prebrackets.

We say that a cochain $\alpha$ in C^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} is constant if the polynomial coefficient of $\alpha$ is constant, i.e., if $\alpha$ lies in the subspace $\bigoplus_{g\in G}\mathbb{C}\,\overline{g}\otimes\textstyle\bigwedge V^{*}$. We showed in [17] that any constant Hochschild $2$-cocycle lifts to a graded Hecke algebra. As graded Hecke algebras are deformations of $S(V)\#G$ (see Section 11), every constant Hochschild $2$-cocycle defines a noncommutative Poisson structure. The following consequence of Theorem 6.12 extends this result to arbitrary cohomological degree.

Theorem 8.1**.**

Suppose $\alpha$ and $\beta$ in H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} are constant. Then $[\alpha,\beta]=0$.

Proof.

Theorem 6.12 gives the Gerstenhaber bracket as a sum of prebrackets of cochains ${}^{a}\alpha$ and ${}^{b}\beta$ for $a,b$ in $G$. Definition 6.8 expresses the prebracket in terms of quantum partial differentiation of polynomial coefficients of cochains (see Definitions 5.8 and 5.9). As the cochains ${}^{a}\alpha$ and ${}^{b}\beta$ are constant, any partial derivative of their coefficients is zero. Hence, each prebracket is zero. ∎

As an immediate consequence of the theorem, we obtain:

Corollary 8.2**.**

Any constant Hochschild cocycle in {\rm HH}^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}}(S(V)\#G) defines a noncommutative Poisson structure.

We need a quick linear algebra lemma in order to give more results on zero brackets.

Lemma 8.3**.**

Let $v_{1},\ldots,v_{n}$ and $w_{1},\ldots,w_{n}$ be two bases of $V$ with change of basis matrix $M$ determined by Equation 7.1. If $J$ and $L$ are two multi-indices with

[TABLE]

then $\det(M_{J;L^{\prime}})=0$ for any $L^{\prime}$ with $L\cap L^{\prime}=\emptyset$.

Proof.

Write $\sum_{j\in J}c_{j}w_{j}=\sum_{l\in L}d_{l}v_{l}$ for some scalars $c_{j}$, $d_{l}$, not all 0. We substitute $v_{i}=a_{1i}w_{i}+\ldots+a_{ni}w_{n}$ to see that $\sum_{l\in L}d_{l}a_{il}=0$ for $i\notin J$. If we delete the rows of $M$ indexed by $J$, these equations give a linear dependence relation among the columns indexed by $L$. Thus $\det(M_{J;L^{\prime}})=0$. ∎

As a consequence, we have the following proposition.

Proposition 8.4**.**

Suppose $g,h$ lie in $G$ and $(V^{g})^{\perp}\cap(V^{h})^{\perp}$ is nontrivial and fixed setwise by $G$. Let $\alpha$ and $\beta$ be elements of (H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}})^{G} supported on the conjugacy classes of $g$ and $h$, respectively. Then

[TABLE]

Proof.

Let $\alpha^{\prime}$ be the summand of $\alpha$ supported on $g$ itself and let $\beta^{\prime}$ be the summand of $\beta$ supported on $h$. Then $|Z(g)|\,\alpha=\mathcal{R}\alpha^{\prime}$. (Indeed, since $\alpha$ is $G$-invariant, $\alpha^{\prime}$ is $Z(g)$-invariant and $\alpha=\sum_{c}\,^{c}\alpha^{\prime}$, a sum over coset representatives $c$ of $G/Z(g)$.) Similarly, $|Z(h)|\,\beta=\mathcal{R}\beta^{\prime}$. Remark 4.6 then implies that

[TABLE]

and hence we compute $[\alpha^{\prime},\beta^{\prime}\,]$. Set $W:=(V^{g})^{\perp}\cap(V^{h})^{\perp}$ and recall that the Hermitian form on $V$ is $G$-invariant. By Remark 6.11, we may assume that $B_{g}=\{w_{1},w_{2},\ldots,w_{n}\}$ and $B_{h}=\{v_{1},v_{2},\ldots,v_{n}\}$ are orthogonal bases with $w_{1},v_{1}$ in $W$. Without loss of generality, suppose

[TABLE]

in H^{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.0,3.5){\circle*{2.0}}\end{picture}} where $f_{g}\in S(V^{g})$, $f_{h}^{\prime}\in S(V^{h})$ and $j_{1}=l_{1}=1$. As $\beta^{\prime}$ lies in $H^{q}_{h}$, the linear span of $v_{l_{1}},\ldots,v_{l_{q}}$ contains $(V^{h})^{\perp}$ and thus $w_{1}$. Hence, for $k\neq 1$,

[TABLE]

Theorem 7.2 and Lemma 8.3 then imply that the $k$-th summand of $\alpha^{\prime}\,\overline{\circ}\,\beta^{\prime}$ in the prebracket $[[\,\alpha^{\prime},\,\beta^{\prime}\,]]_{(\rule[0.0pt]{0.0pt}{6.02777pt}\,B_{g},\,B_{h})}$ is zero for $k\neq 1$ (as $\det M_{J_{k};I-L}=0$). But the first ($k=1$) summand is also zero, since $\,w_{1}=w_{j_{1}}$ lies in $(V^{h})^{\perp}$ while $f_{h}^{\prime}$ lies in $V^{h}$, forcing the partial derivative of $f_{h}^{\prime}$ with respect to $w_{1}$ and $B_{g}$ to be zero: $\partial_{1}(f_{h}^{\prime})=0$. (Indeed, since $w_{1}\in(V^{h})^{\perp}$ and $B_{g}$ is orthogonal, $V^{h}\subset\operatorname{Span}_{\mathbb{C}}\{w_{2},\ldots,w_{n}\}$