Gerstenhaber brackets for skew group algebras in positive characteristic

TL;DR

This paper develops new techniques using twisted product resolutions to evaluate Gerstenhaber brackets in skew group algebras over fields of positive characteristic, aiding deformation theory analysis.

Contribution

It introduces a method for computing Gerstenhaber brackets via twisted product resolutions, simplifying the study of deformation theory in positive characteristic settings.

Findings

Effective evaluation of Gerstenhaber brackets using twisted resolutions.

Application to graded Hecke algebras in positive characteristic.

Enhanced understanding of deformation theory for skew group algebras.

Abstract

The deformation theory of an algebra is controlled by the Gerstenhaber bracket, a Lie bracket on Hochschild cohomology. We develop techniques for evaluating Gerstenhaber brackets of semidirect product algebras recording actions of finite groups over fields of positive characteristic. The Hochschild cohomology and Gerstenhaber bracket of these skew group algebras can be complicated when the characteristic of the underlying field divides the group order. We show how to investigate Gerstenhaber brackets using twisted product resolutions, which are often smaller and more convenient than the cumbersome bar resolution typically used. These resolutions provide a concrete description of the Gerstenhaber bracket suitable for exploring questions in deformation theory. We demonstrate with the prototypical example of a graded Hecke algebra (rational Cherednik algebra) in positive characteristic.

Equations

$$(sg)\cdot (s^{\prime }{g}^{\prime })=s({}^g{s}^{\prime })g{g}^{\prime }\text{ for all }s,s^{\prime }\in S\text{ and }g,g^{\prime }\in G.$$

$$\text{ (i) }C:\dots \to$$

$$\text{ (ii) }D:\dots \to$$

$$g_{1}\big{(}(C_{i})_{g_{2}}\big{)}g_{3}=(C_{i})_{g_{1}g_{2}g_{3}}\quad\text{ for all }\ g_{1},g_{2},g_{3}\in G\ \text{and all degrees }i.$$

$$g\cdot (s\cdot d)={}^{g}s\cdot (g\cdot d)\text{ for all }g\in G,s\in S,d\in D$$

$$X=C{\otimes }^{G}D\text{ where }{X}_n=\underset{i+j=n}{⨁}{C}_i\otimes D_j$$

$$s^{\prime }{g}^{\prime }\cdot (c\otimes d)\cdot sg=g^{\prime }cg\otimes ({}^{(g^{\prime }hg{)}^{-1}}{s}^{\prime })({}^{g^{-1}}(ds))\text{ for }c\in C_h,d\in D,g,g^{\prime },h\in G,s,s^{\prime }\in S.$$

$$\dots \to X_2\to X_1\to X_0\to A\to 0.$$

$$\dots \to P_2\to P_1\to P_0\to 0.$$

$$\text{emcoassociative},$$

$$\text{emcounital},$$

$${\partial }_P{\varphi }_P+{\varphi }_P{\partial }_{P{\otimes }_{A}P}={\mu }_P\otimes 1_P-1_P\otimes {\mu }_P.$$

$${\Delta }_B(a_0\otimes \cdots \otimes a_{n+1})=\sum _{j=0}^n(a_0\otimes \cdots \otimes a_j\otimes 1){\otimes }_A(1\otimes a_{j+1}\otimes \cdots \otimes a_{n+1})$$

$${\varphi }_B$$

$$=(-1{)}^{p-1}{a}_0\otimes \cdots \otimes a_{p-1}\otimes a_p{a}_p^{\prime }\otimes a_{p+1}\otimes \cdots \otimes a_{n+1}\text{ for all }{a}_i,a_p^{\prime }\in A.$$

$$[,]:\text{mbox}{\mathrm{Hom}}_k(A^{\otimes n},A)\times \text{mbox}{\mathrm{Hom}}_k(A^{\otimes m},A)\to \text{mbox}{\mathrm{Hom}}_k(A^{\otimes (n+m-1)},A)$$

$$[f,f^{\prime }]=f\circ f^{\prime }-(-1{)}^{(n-1)(m-1)}{f}^{\prime }\circ f$$

$$\displaystyle\sum_{i=1}^{n}(-1)^{(m-1)(i-1)}\,f\Big{(}a_{1}\otimes\cdots\otimes a_{i-1}\otimes f^{\prime}(a_{i}\otimes\cdots\otimes a_{i+m-1})\otimes a_{i+m}\otimes\cdots\otimes a_{n+m-1}\Big{)}\,.$$

$$[f,f^{\prime }{]}_P=f{\circ }_P{f}^{\prime }-(-1{)}^{(n-1)(m-1)}{f}^{\prime }{\circ }_{P}f$$

$$f{\circ }_P{f}^{\prime }:P\stackrel{{\Delta }_P^{(2)}}{\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}⟶}P{\otimes }_{A}P{\otimes }_{A}P\stackrel{1_P\otimes f^{\prime }\otimes 1_P}{\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}⟶}P{\otimes }_{A}P\stackrel{{\varphi }_P}{⟶}P\stackrel{f}{⟶}A.$$

$$(1_P\otimes f^{\prime }\otimes 1_P)(x\otimes y\otimes z)=(-1{)}^{im}x\otimes f^{\prime }(y)\otimes z$$

$${\tau }_{i,j}(c\otimes d)=(-1{)}^{ij}({}^{g}d)\otimes c\text{ for all }c\in (C_i{)}_g\text{ and }d\in D_j.$$

$$1\otimes \tau \otimes 1:(C{\otimes }_{kG}C)\otimes (D{\otimes }_{S}D)⟶(C{\otimes }^{G}D){\otimes }_{S⋊G}(C{\otimes }^{G}D).$$

$$C\otimes C\otimes D\otimes D\stackrel{1\otimes \tau \otimes 1}{\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}⟶}C\otimes D\otimes C\otimes D⟶(C{\otimes }^{G}D){\otimes }_{S⋊G}(C{\otimes }^{G}D),$$

$${\Delta }_X=(1\otimes \tau \otimes 1)({\Delta }_C\otimes {\Delta }_D).$$

$$({\Delta }_X\otimes 1_X){\Delta }_X$$

$$\displaystyle=\big{(}(1\otimes\tau\otimes 1)(\Delta_{C}\otimes\Delta_{D})\otimes 1\otimes 1\big{)}(1\otimes\tau\otimes 1)(\Delta_{C}\otimes\Delta_{D})$$

$$=(1\otimes \tau \otimes 1\otimes 1\otimes 1)({\Delta }_C\otimes {\Delta }_D\otimes 1\otimes 1)(1\otimes \tau \otimes 1)({\Delta }_C\otimes {\Delta }_D)$$

$$=(1\otimes \tau \otimes 1\otimes 1\otimes 1)(1\otimes 1\otimes 1\otimes \tau \otimes 1)(1\otimes 1\otimes \tau \otimes 1\otimes 1)({\Delta }_C\otimes 1\otimes {\Delta }_D\otimes 1)({\Delta }_C\otimes {\Delta }_D)$$

$$=(1\otimes \tau \otimes 1\otimes 1\otimes 1)(1\otimes 1\otimes 1\otimes \tau \otimes 1)(1\otimes 1\otimes \tau \otimes 1\otimes 1)(1\otimes {\Delta }_C\otimes 1\otimes {\Delta }_D)({\Delta }_C\otimes {\Delta }_D)$$

$$=(1\otimes 1\otimes 1\otimes \tau \otimes 1)(1\otimes \tau \otimes 1\otimes 1\otimes 1)(1\otimes 1\otimes \tau \otimes 1\otimes 1)(1\otimes {\Delta }_C\otimes 1\otimes {\Delta }_D)({\Delta }_C\otimes {\Delta }_D)$$

$$=(1\otimes 1\otimes 1\otimes \tau \otimes 1)(1\otimes 1\otimes {\Delta }_C\otimes {\Delta }_D)(1\otimes \tau \otimes 1)({\Delta }_C\otimes {\Delta }_D)$$

$$=(1_X\otimes {\Delta }_X){\Delta }_X.$$

$$({\mu }_X\otimes 1_X){\Delta }_X$$

$$=({\mu }_C\otimes 1\otimes {\mu }_D\otimes 1)({\Delta }_C\otimes {\Delta }_D)=(({\mu }_C\otimes 1){\Delta }_C)\otimes (({\mu }_D\otimes 1){\Delta }_D)$$

$$=1\otimes 1=1_X,$$

Full text

The first author was partially supported by Simons grant 429539. The second author was partially supported by NSF grant DMS-1665286. Corresponding author: Anne Shepler.

1. Introduction

The Hochschild cohomology space of an associative algebra is a Gerstenhaber algebra under two binary operations, the cup product and the Gerstenhaber bracket. The Gerstenhaber bracket is a Lie bracket controlling the deformation theory of the algebra. Historically, it has been more difficult to compute than the cup product: The bracket is defined in terms of the cumbersome bar resolution and notoriously resists transfer to more convenient resolutions. In general, we lack user-friendly formulas giving the Gerstenhaber bracket explicitly.

We consider the Hochschild cohomology of a skew group algebra (semidirect product algebra) arising from the action of a finite group $G$ on an algebra $S$. We work in the modular setting, i.e., over a field $k$ of positive characteristic that may divide the group order $|G|$. In this setting, the Hochschild cohomology of $S\rtimes G$ is complicated by the potentially onerous cohomology of $kG$, in contrast to the characteristic zero case where it is always trivial.

Computations of the Gerstenhaber bracket on $S\rtimes G$ directly using the bar resolution often yield little useful information—the bar resolution itself is too large and unwieldy. It can be a struggle even to describe adequately the Hochschild cohomology using the bar resolution. Thus one seeks a description of the Gerstenhaber bracket in terms of smaller resolutions used to compute Hochschild cohomology, a description that is concrete and straightforward to apply in specific examples.

In this note, we consider the flexible twisted product resolution of a skew group algebra: one chooses a convenient resolution for $S$ and another for $G$ and then combines them to create a resolution of $S\rtimes G$. We show how to apply new techniques from [4] on Gerstenhaber brackets to twisted product resolutions for skew group algebras from [8, 9]. This approach provides advantages over employing the often unmanageable but traditional bar resolution. We produce an explicit description of the Gerstenhaber bracket that should prove user-friendly and we illustrate with an example from deformation theory. This quintessential example using a small transvection group captures the difference between the modular and nonmodular settings, both in the theory of reflection groups and in the theory of graded Hecke algebras (and rational Cherednik algebras, see [3]).

In Section 2, we recall the twisted product resolution from [8, 9] obtained by twisting a resolution of $S$ with one for $G$. We recall methods of [4] analyzing Gerstenhaber brackets in Section 3 and show how they apply to twisted product resolutions for skew group algebras. We illustrate these techniques by showing how to compute some Gerstenhaber brackets concretely for a small transvection group example from [8] in Section 5. Throughout, $k$ is a field of arbitrary characteristic and $\otimes=\otimes_{k}$.

2. Twisted product resolutions

We recall the twisted product resolution from [8, 9]. Consider a finite group $G$ acting on a $k$-algebra $S$ by automorphisms. Let $A=S\rtimes G$ be the corresponding skew group algebra: As a vector space, $S\rtimes G=S\otimes kG$, and we abbreviate the element $s\otimes g$ by $sg$ ($s\in S$, $g\in G$) when no confusion can arise. Multiplication is defined by

[TABLE]

The action of $g$ on $s^{\prime}$ here is denoted by ${}^{g}s^{\prime}$. We use the enveloping algebra $S^{e}=S\otimes S^{op}$ of any algebra $S$ to express bimodule actions as left actions.

The twisted product resolution

We consider projective resolutions

[TABLE]

We assume the resolution $C$ is $G$-graded, with compatible group action:

[TABLE]

We also assume D_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.5,2.5){\circle*{2.0}}\end{picture}} carries a compatible action of $G$: Each $D_{i}$ is left $kG$-module with

[TABLE]

and the differentials are $kG$-module homomorphisms. This ensures D_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.5,2.5){\circle*{2.0}}\end{picture}} is compatible with the twisting map $g\otimes s\mapsto{}^{g}s\otimes g$ given by the group action (see [9, Definition 2.17]). This is the setting, for example, when C_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.5,2.5){\circle*{2.0}}\end{picture}} is the bar or reduced bar resolution of $kG$ and when D_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.5,2.5){\circle*{2.0}}\end{picture}} is the Koszul resolution of a Koszul algebra $S$ (see [9, Prop 2.20(ii)]).

The twisted product resolution $X=C\otimes^{G}D$ of the algebra $S\rtimes G$ is the total complex of the double complex C_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.5,2.5){\circle*{2.0}}\end{picture}}\otimes D_{\begin{picture}(2.5,2.0)(1.0,1.0)\put(2.5,2.5){\circle*{2.0}}\end{picture}},

[TABLE]

with each $X_{n}$ suffused with the additional structure of a $(S\rtimes G)$-bimodule defined by

[TABLE]

The differential on $X$ is $\partial_{n}=\sum_{i+j=n}(\partial_{i}\otimes 1)+(-1)^{i}(1\otimes\partial_{j})$ as usual.

With this action, $X$ is a resolution of $A=S\rtimes G$, i.e., $X$ provides an exact sequence of $A$-bimodules (see [9] or  [7, §4]):

[TABLE]

When the $A$-bimodules $X_{n}$ are all projective as $A^{e}$-modules, $X$ is also a projective resolution of $A$. This occurs, for example, when $D$ is a Koszul resolution of a Koszul algebra and $C$ is the bar resolution of $kG$. (See [9, Proposition 2.20(ii)].)

3. Gerstenhaber brackets on differential graded coalgebras

In this section, we summarize some results of [4] and develop additional techniques for computing Gerstenhaber brackets in the modular setting. Contrast with [5, 6], where the characteristic of the underlying field was 0.

Resolutions as differential graded coalgebras

Consider a $k$-algebra $A$ and a projective resolution $P$ of $A$ as an $A$-bimodule:

[TABLE]

The resolution $P$ is a differential graded coalgebra when $P=\oplus_{i}P_{i}$ has a coalgebra structure compatible with its differential $\partial_{P}$. This means there is a (degree [math]) chain map $\Delta_{P}:P\rightarrow P\otimes_{A}P$ lifting the canonical isomorphism $A\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}A\otimes_{A}A$, called a diagonal map, that is required to be

[TABLE]

where $\mu_{P}:P_{0}\rightarrow A$ is augmentation of the complex (with $\mu_{P}$ zero on $P_{i}$ for $i\geq 1$). Throughout, we define $\mu_{P}\otimes 1_{P}:P\otimes P\rightarrow P$ as the map $p\otimes p^{\prime}\mapsto\mu_{P}(p)\cdot p^{\prime}$ (and similarly for $1_{P}\otimes\mu_{P}$). Recall that the differential on $P_{n}\otimes_{A}P_{m}$ is just $\partial_{P}\otimes 1_{P}+(-1)^{n}1_{P}\otimes\partial_{P}$.

Homotopy from right

to left

We may map the complex $P\otimes_{A}P$ to the complex $P$ using either $\mu_{P}\otimes 1_{P}$ or $1_{P}\otimes\mu_{P}$. When $P$ is a differential graded coalgebra, these mappings are chain homotopic by [4, Lemma 3.2.1]. (The hypotheses there are slightly stronger, but the same proof works under our hypotheses here.) Thus there exists a chain homotopy from $\mu_{P}\otimes 1_{P}$ to $1_{P}\otimes\mu_{P}$, i.e., a map $\phi_{P}:P\otimes_{A}P\rightarrow P$ with $P_{m}\otimes_{A}P_{n}\rightarrow P_{m+n+1}$ satisfying

[TABLE]

Example 3.2**.**

The bar resolution $B$ of the algebra $A$ is a differential graded coalgebra. Indeed, for $B_{n}=A\otimes A^{\otimes n}\otimes A$, a diagonal map $\Delta_{B}:B\rightarrow B\otimes_{A}B$ is defined by

[TABLE]

for $a_{0},\ldots,a_{n+1}$ in $A$. This map is coassociative and counital. One choice of homotopy $\phi_{B}:B\otimes_{A}B\rightarrow B$ from $\mu_{B}\otimes 1_{B}$ to $1_{B}\otimes\mu_{B}$ is defined by

[TABLE]

Koszul resolutions of Koszul algebras are also differential graded coalgebras [1]. The Koszul resolution $P$ of a Koszul algebra embeds into the bar resolution, however the above map $\phi_{B}$ does not preserve the image. Instead, a homotopy $\phi_{P}$ may be found directly in this case; see [4, §4], [5, §3.2], or [2, §4] for some examples. **

Definition of the

Gerstenhaber bracket

The Gerstenhaber bracket for $A$ is defined on cochains on the bar resolution $B$ of $A$. Identify each space of cochains $\mbox{\rm Hom\,}_{A^{e}}(B_{n},A)$ with $\mbox{\rm Hom\,}_{k}(A^{\otimes n},A)$ via the canonical isomorphism. Then the Gerstenhaber bracket

[TABLE]

on cochains is defined by

[TABLE]

where, for $a_{i}$ in $A$, the circle product $(f\circ f^{\prime})(a_{1}\otimes\cdots\otimes a_{n+m-1})$ is

[TABLE]

Gerstenhaber brackets on

differential graded coalgebras

Although the Gerstenhaber bracket is defined using the bar resolution, we seek descriptions in terms of more convenient resolutions used to compute Hochschild cohomology. Suppose $P$ is a projective resolution of $A$ with a differential graded coalgebra structure. The Gerstenhaber bracket can be defined directly at the chain level on $P$ using [4, Theorem 3.2.5]; we recall how a homotopy $\phi_{P}$ (see (3.1)) gives the bracket explicitly.

Extend any cochain $f\in\mbox{\rm Hom\,}_{A^{e}}(P_{n},A)$ to all of $P$ by defining $f\equiv 0$ on $P_{m}$ with $m\neq n$. For $f\in\mbox{\rm Hom\,}_{A^{e}}(P_{n},A)$ and $f^{\prime}\in\mbox{\rm Hom\,}_{A^{e}}(P_{m},A)$, define

[TABLE]

where $f\circ_{P}f^{\prime}$ (similarly $f^{\prime}\circ_{P}f$) is the composition

[TABLE]

Here, $\Delta^{(2)}_{P}=(1_{P}\otimes\Delta_{P})\Delta_{P}=(\Delta_{P}\otimes 1_{P})\Delta_{P}$ and $1_{P}\otimes f^{\prime}\otimes 1_{P}$ has signs attached so that

[TABLE]

for $x\in P_{i}$, $y,z\in P$. Then [4, Theorem 3.2.5] implies that the Gerstenhaber bracket $[\ ,\ ]$ of any elements in cohomology is given at the cochain level on $P$ by the map $[\ ,\ ]_{P}$ on cocycles. (Note that [4, Theorem 3.2.5] has slightly stronger hypotheses, but the proof indeed holds for any resolution $P$ with the structure of a differential graded coalgebra.)

4. Twisted product resolution

as a differential graded coalgebra

We show in this section that a twisted product resolution $X$ of $S\rtimes G$ constructed from two differential graded coalgebras $C$ and $D$ is again a differential graded coalgebra. We then give the Gerstenhaber bracket for $X$ in terms of the maps describing the Gerstenhaber brackets of $C$ and $D$ individually.

Throughout this section, we fix

• •

a differential graded coalgebra bimodule resolution $(C,\Delta_{C},\mu_{C})$ of $G$ and

• •

a differential graded coalgebra bimodule resolution $(D,\Delta_{D},\mu_{D})$ of $S$, producing

• •

a twisted product resolution $X=C\otimes^{G}D$ of $A=S\rtimes G$.

We assume that $C$ is $G$-graded (as in (2.1)) with $\Delta_{C},\mu_{C}$ preserving the grading and also that $D$ carries a $G$-action (as in (2.2)) with $\Delta_{D},\mu_{D}$ both $kG$-module homomorphisms. This is the case, for example, if $C$ is the bar (or reduced bar) resolution of $kG$ and $D$ is the Koszul resolution of a Koszul algebra (see [9, Proposition 2.20(ii)]).

Twisted comultiplication

In the next lemmas, we use diagonal maps for $C$ and $D$ to produce a diagonal map $\Delta_{X}:X\rightarrow X\otimes_{A}X$.

Lemma 4.1**.**

Define a twisting map $\tau:C\otimes D\rightarrow D\otimes C$ by

[TABLE]

Then $\tau$ extends to a well-defined chain map

[TABLE]

Proof.

Consider the map

[TABLE]

where the latter map is the canonical surjection. Calculations show that the composition of these two maps is $kG$-middle linear in the first two arguments and $S$-middle linear in the last two arguments, and so it induces a well-defined map as claimed. A calculation shows that it is a chain map. ∎

Lemma 4.3**.**

Let $X=C\otimes^{G}D$ be a twisted product resolution of $S\rtimes G$ for differential graded coalgebras $C$ and $D$ resolving $kG$ and $S$, respectively, as above. Then $X$ is a differential graded coalgebra as well with comultiplication $\Delta_{X}:X\rightarrow X\otimes_{A}X$ given by

[TABLE]

Proof.

We first check that $\Delta_{X}$ is coassociative using the fact that $\Delta_{C}$ and $\Delta_{D}$ are each coassociative. We use the $G$-grading on $C$ and the compatible $G$-action on $D$:

[TABLE]

We next verify that $\Delta_{X}$ is counital using the fact that $\Delta_{C}$ and $\Delta_{D}$ are each counital. We use the extra assumption that $\mu_{C}$ preserves the $G$-grading and $\mu_{D}$ is a $kG$-module homomorphism as well as the definition of the $S\rtimes G$-bimodule structure on $C\otimes^{G}D$:

[TABLE]

and, similarly, $(1_{X}\otimes\mu_{X})\Delta_{X}=1_{X}$.

We now need only check that $\Delta_{X}$ is a chain map, i.e., $\Delta_{X}\,\partial=(\partial\otimes 1+1\otimes\partial)\Delta_{X}$, for $\partial$ the differential on $X$. This follows from the fact that $\tau,\Delta_{C},\Delta_{D}$ are all chain maps. ∎

Remark 4.4**.**

One may check that the map $1\otimes\tau\otimes 1$ of (4.2) interpolates between the maps of the form $\mu\otimes 1-1\otimes\mu$ for the various complexes, that is,

[TABLE]

We now give a theorem describing a homotopy from $\mu_{X}\otimes 1_{X}$ to $1_{X}\otimes\mu_{X}$ concretely in terms of homotopies from $\mu_{C}\otimes 1_{C}$ to $1_{C}\otimes\mu_{C}$ and from $\mu_{D}\otimes 1_{D}$ to $1_{D}\otimes\mu_{D}$ by adapting [2, Lemmas 3.3, 3.4, and 3.5] to our setting.

Theorem 4.6**.**

Let $X=C\otimes^{G}D$ as above with homotopies $\phi_{C}$ from $\mu_{C}\otimes 1_{C}$ to $1_{C}\otimes\mu_{C}$ and $\phi_{D}$ from $\mu_{D}\otimes 1_{D}$ to $1_{D}\otimes\mu_{D}$. Define $\phi_{X}:X\otimes_{A}X\rightarrow X$ by

[TABLE]

for $\epsilon_{C}:C\rightarrow C$ defined by $c\mapsto(-1)^{|c|}$ for homogeneous $c$. Then $\phi_{X}$ is a homotopy from $\mu_{X}\otimes 1_{X}$ to $1_{X}\otimes\mu_{X}$.

Proof.

Let $\phi_{X}^{\prime}:C\otimes C\otimes D\otimes D\rightarrow C\otimes D$ be the map $\phi_{C}\otimes\mu_{D}\otimes 1+\epsilon_{C}(1\otimes\mu_{C})\otimes\phi_{D}$ so that $\phi_{X}=\phi_{X}^{\prime}(1\otimes\tau^{-1}\otimes 1)$. Then on $(C\otimes C)\otimes(D\otimes D)$,

[TABLE]

and, since $1\otimes\tau\otimes 1$ is a chain map from $(C\otimes C)\otimes(D\otimes D)$ to $X\otimes_{A}X$,

[TABLE]

Here we used the fact that $\epsilon_{C}\phi_{C}=-\phi_{C}(\epsilon_{C}\otimes\epsilon_{C})$, $\epsilon_{C}(1\otimes\mu_{C})(\epsilon_{C}\otimes\epsilon_{C})=1\otimes\mu_{C}$, and $\partial_{C}\epsilon_{C}=-\epsilon_{C}\partial_{C}$. The second term of (4) cancels with the second term of (4.8) as $\mu_{D}\otimes 1$ is a chain map; likewise, the third terms cancel as $\mu_{C}\otimes 1$ is a chain map. Hence

[TABLE]

and, by equation (4.5),

[TABLE]

∎

Gerstenhaber bracket for skew group algebras

The next theorem gives the Gerstenhaber bracket on a twisted product resolution $X$. Note that the twisting map $\tau$ in the theorem is from Lemma 4.1, the map $1_{X}\otimes f^{\prime}\otimes 1_{X}$ has signs attached as in (3.7), and $\epsilon_{C}$ merely adjusts signs, $c\mapsto(-1)^{|c|}$ for homogeneous $c$ in $C$.

Theorem 4.9**.**

Let $X=C\otimes^{G}D$ be a twisted product resolution of $S\rtimes G$ for differential graded coalgebras $(C,\Delta_{C},\mu_{C})$ and $(D,\Delta_{D},\mu_{D})$ resolving $kG$ and $S$, respectively, as above. The Gerstenhaber bracket of elements of Hochschild cohomology represented by cocycles $f\in\mbox{\rm Hom\,}_{A^{e}}(X_{n},A)$ and $f^{\prime}\in\mbox{\rm Hom\,}_{A^{e}}(X_{m},A)$ is represented by the cocycle

[TABLE]

where $f\circ_{X}f^{\prime}$ (similarly $f^{\prime}\circ_{X}f$) is the composition

[TABLE]

with

[TABLE]

Proof.

We combine Lemmas 4.1, Lemma 4.3, and Theorem 4.6 with (3.6) and (3.5). ∎

Example 4.12**.**

In case $S=S(V)\cong k[x_{1},\ldots,x_{n}]$, the symmetric algebra on a finite dimensional vector space $V$, we take $D$ to be the Koszul resolution for which a choice of $\phi_{D}$ has been made in [4, §4] (see also [5, §3.2]). We may take $C$ to be the bar or reduced bar resolution of $kG$ for some applications, with homotopy $\phi_{C}$ as defined by equation (3.4). **

5. A small transvection group example

We end by demonstrating how to use a twisted product resolution to compute Gerstenhaber brackets explicitly via Theorem 4.9. We also see how computation of explicit brackets can shed light on questions in deformation theory (see [8]). We illustrate with the prototype example of a graded Hecke algebra (or rational Cherednik algebra) in positive characteristic (see [3] and [8]). In the nonmodular setting, these algebras have parameters supported only on the identity group element and on bireflections; in the modular setting, parameters can also be supported on reflections. All reflections in a finite linear group $G$ acting in the modular setting are either diagonalizable or act as in this example. We include some explicit details to illustrate how to evaluate the maps in Theorem 4.9 concretely. We find both a nonzero and a zero Gerstenhaber bracket.

5.1. Group action and twisted product resolution

Say $\text{char}(k)=p>0$ and consider the cyclic group $G\simeq\mathbb{Z}/p\mathbb{Z}$ acting on $V=k^{2}$ with basis $v,w$ generated by

[TABLE]

We work in the twisted product resolution $X=C\otimes^{G}D$ of $S(V)\rtimes G$ obtained from twisting the reduced bar resolution $C$ of $kG$ with the Koszul resolution $D$ of $S(V)$:

[TABLE]

Here, $C_{n}=kG\otimes(\overline{kG})^{\otimes n}\otimes kG$ with $\overline{kG}=kG/k1_{G}$ and $D_{n}=S(V)\otimes\textstyle\bigwedge^{n}V\otimes S(V)$. Then $C$ and $D$ satisfy the conditions specified in Section 3, and Theorem 4.9 applies.

5.2. Cochains

Consider cochains on the resolution $X$:

[TABLE]

defined by (with subscripts on the tensor signs suppressed for brevity)

[TABLE]

for $0\leq i\leq p-1$, with all other values determined by these. One can check directly that $\lambda$ and $\kappa$ are 2-cocycles and that $\delta$ is a $1$-cocycle for $X$. We will show that

[TABLE]

The diagonal maps

We give some values of the diagonal maps at play in finding the Gerstenhaber brackets. The diagonal map $\Delta_{C}$ on the reduced bar resolution of $kG$ is deduced from (3.3). For example, after identifying $g^{i}$ with its image in $\overline{kG}$,

[TABLE]

The diagonal map $\Delta_{D}$ is found from embedding the Koszul into the bar resolution and then using (3.3). For example, we identify $v\wedge w$ with $v\otimes w-w\otimes v$ and observe that

[TABLE]

Homotopies

Let $\phi_{C}:C\otimes_{kG}C\rightarrow C$ be the homotopy from $\mu_{C}\otimes 1$ to $1\otimes\mu_{C}$ from (3.4). We choose the homotopy $\phi_{D}:D\otimes_{S}D\rightarrow D$ from $\mu_{D}\otimes 1$ to $1\otimes\mu_{D}$ given in [4, Definition 4.1.3] and record a few values here for later use:

[TABLE]

Nonzero bracket

We use Theorem 4.9 to show explicitly that $[\delta,\kappa]=\kappa$. First note that $[\delta,\kappa]$ is zero on all components of $X$ except possibly $X_{0,2}$. We consider the composition (4.11) with $f^{\prime}=\delta$ and $f=\kappa$ to find $\kappa\circ_{X}\delta$. As a first step, we apply the map $(\Delta_{X}\otimes 1_{X})\Delta_{X}$ to the element $(1_{G}\otimes 1_{G})\otimes(1_{S}\otimes v\wedge w\otimes 1_{S})$ of $X_{0,2}$, where, recall

[TABLE]

Direct calculation confirms that

[TABLE]

as an element of $X\otimes_{A}X\otimes_{A}X$. We have suppressed all subscripts for brevity; for example, the second summand may be written

[TABLE]

We next apply the map $1_{X}\otimes\delta\otimes 1_{X}$; it is nonzero on exactly two summands, the second and the penultimate, and we obtain (with the tensor products over $A$ indicated here)

[TABLE]

To apply $\phi_{X}$ next, we first rearrange terms with $1_{G}\otimes\tau^{-1}\otimes 1_{S}$, producing

[TABLE]

and then apply the map $\phi_{C}\otimes\mu_{D}\otimes 1_{D}+1_{C}\otimes\mu_{C}\otimes\phi_{D}$ to obtain

[TABLE]

Lastly, we apply $\kappa$ as the last step of (4.11) and obtain [math] from the first term and $-g$ from the second. Thus

[TABLE]

and $\kappa\circ_{X}\delta=-\kappa$. We inspect the above calculation with an eye toward switching the order of $\kappa$ and $\delta$ and deduce that $\delta\circ_{X}\kappa=0$. We conclude, as claimed,

[TABLE]

Zero brackets

We now use Theorem 4.9 to show that $[\lambda,f]=0$ when $f$ is $\lambda$ or $\kappa$. We evaluate composition (4.11) on $X_{1,2}$ with $f^{\prime}=\lambda$. Other calculations are similar. We first apply $\Delta_{X}=(1_{G}\otimes\tau\otimes 1_{S})(\Delta_{C}\otimes\Delta_{D})\,$ to sample input in $X_{1,2}$, noting that ${}^{g^{i}}w=iv+w$ (with subscripts suppressed again):

[TABLE]

[TABLE]

an element of $X\otimes_{A}X$. Next we apply $\Delta_{X}\otimes 1_{X}$: Evaluating ${\Delta_{C}\otimes\Delta_{D}\otimes 1_{X}}$ on the last expression yields 27 summands; the map $(1_{G}\otimes\tau\otimes 1_{S})\otimes 1_{X}$ transforms these to 27 summands in $X\otimes_{A}X\otimes_{A}X$. A quick check verifies that $1_{X}\otimes\lambda\otimes 1_{X}$ vanishes on all but two summands, namely

[TABLE]

and we obtain

[TABLE]

Applying $\phi_{X}$ followed by $f=\lambda$ or $f=\kappa$ gives [math] as $w$ does not appear in the input.

Remark 5.1**.**

The cocycles $\lambda$ and $\kappa$ above were not chosen randomly. These cocycles define a PBW deformation of $S\rtimes G$, and the zero brackets calculated above predict the PBW property. Indeed, in [8], we considered PBW deformations of $S\rtimes G$ given by analogs of Lusztig’s graded Hecke algebras and symplectic reflection algebras over fields of positive characteristic. These algebras $\mathcal{H}_{\lambda,\kappa}$ depend on two parameters $\lambda$ and $\kappa$ with $\lambda:kG\otimes V\rightarrow kG$ and $\kappa:V\otimes V\rightarrow kG$. The Hochschild 2-cocycles above of the same name $\lambda$ and $\kappa$ are these parameters converted into cocycles on the resolution $X$; see [8, Example 2.2] and also [10, Section 5]. A necessary condition for the parameters $\lambda$ and $\kappa$ to define a PBW deformation is that

[TABLE]

when the cochains $\kappa$ and $\lambda$ they define are cocycles. (More generally, we require that $\lambda$ is a cocycle, $[\lambda,\lambda]=0$, and $[\lambda,\lambda]=2\partial^{*}\kappa$.) Thus knowing explicit values for brackets is helpful for finding new deformations. The cocycle $\delta$ above is included merely for illustration purposes; it provides an example of a nonzero Gerstenhaber bracket. **