Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

Gerstenhaber brackets were originally defined on Hochschild cohomology by M. Gerstenhaber himself [3, Section 1.1]. In 2002, A. Farinati and A. Solotar showed that for any Hopf algebra $A$, Hopf algebra cohomology H${}^{*}(A):=\operatorname{Ext}^{*}_{A}(k,k)$ is a Gerstenhaber algebra [2]. Hence, we can define a Gerstenhaber bracket on Hopf algebra cohomology. In the same year, R. Taillefer used a different approach and found a bracket on Hopf algebra cohomology [11] which is equivalent to the bracket constructed by A. Farinati and A. Solotar. The category of $A$-modules and the category of $A^{e}$-modules are examples of strong exact monoidal categories. In 2016, Reiner Hermann [5, Theorem 6.3.12, Corollary 6.3.15] proved that if the strong exact monoidal category is lax braided, then the bracket is constantly zero. Therefore, the Gerstenhaber bracket on the Hopf algebra cohomology of a quasi-triangular Hopf algebra is trivial. However, we do not know the bracket structure for a nonquasi-triangular Hopf algebra. Taft algebras are nice examples of nonquasi-triangular Hopf algebras. In this paper, we show that the Gerstenhaber bracket on the Hochschild cohomology of a Taft Algebra is nontrivial. However, the bracket structure on Hopf algebra cohomology of a Taft algebra is constantly zero. Also, we take the Gerstenhaber bracket formula on Hochschild comology and find a general formula for Gerstenhaber bracket on Hopf algebra cohomology.

We start by giving some basic definitions and some tools to calculate the bracket on Hochschild cohomology in Section 2. Then, we compute the Gerstenhaber bracket on the Hochschild cohomology of $A=k[x]/(x^{p})$ where the field $k$ has characteristic 0 and the integer $p>2$ in Section 3. We use the technique introduced by C. Negron and S. Witherspoon [7] and note that they computed the bracket on Hochschild cohomology of $A$ for the case that $k$ has positive characteristic $p$ [7, Section 5].

In Section 4, we compute the Gerstenhaber bracket for the Taft algebra $T_{p}$ which is a nonquasi-triangular Hopf algebra. We use a similar technique as in [7] to calculate the bracket on Hochschild cohomology of $T_{p}$. It is also known that the Hopf algebra cohomology of any Hopf algebra with a bijective antipode can be embedded in the Hochschild cohomology of the algebra [14, Theorem 9.4.5 and Corollary 9.4.7]. Since all finite dimensional Hopf algebras (also most of known infinite dimensional Hopf algebras) have bijective antipode, we can embed the Hopf algebra cohomology of $T_{p}$ into the Hochschild cohomology of $T_{p}$. Then, we use this explicit embedding and find the bracket on the Hopf algebra cohomology of $T_{p}$. As a result of our calculation, we obtain that the bracket on Hopf algebra cohomology of $T_{p}$ is also trivial.

In the last section, we derive a general expression for the bracket on Hopf algebra cohomology of any Hopf algebra $A$ with bijective antipode. We first consider a specific resolution that agrees with the bar resolution of $A$ and find a bracket formula for it. Then, we use the composition of various isomorphisms and an embedding from Hopf algebra cohomology into Hochschild cohomology in order to discover the bracket formula on Hopf algebra cohomology.

Let $k$ be a field, $A$ be a $k$-algebra, and $A^{e}=A\otimes_{k}A^{op}$ where $A^{op}$ is the opposite algebra with reverse multiplication. For simplicity, we write $\otimes$ instead of $\otimes_{k}$. The following resolution $B(A)$ is a free resolution of the $A^{e}$-module $A$, called the bar resolution,

$$B(A):\cdots\stackrel{{\scriptstyle d_{3}}}{{\longrightarrow}}A^{\otimes 4}\stackrel{{\scriptstyle d_{2}}}{{\longrightarrow}}A^{\otimes 3}\stackrel{{\scriptstyle d_{1}}}{{\longrightarrow}}A^{\otimes 2}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}A\longrightarrow 0,$$

(2.1)

where

$$d_{n}(a_{0}\otimes a_{1}\otimes\cdots\otimes a_{n+1})=\sum_{i=0}^{n}(-1)^{i}a_{0}\otimes a_{1}\otimes\cdots\otimes a_{i}a_{i+1}\otimes\cdots\otimes a_{n+1}$$

and $\pi$ is multiplication.

Consider the following complex that is derived by applying Hom${}_{A^{e}}(-,A)$ to the bar resolution $B(A)$

$$0{\longrightarrow}\text{Hom}_{A^{e}}(A^{\otimes 2},A)\stackrel{{\scriptstyle d^{*}_{1}}}{{\longrightarrow}}\text{Hom}_{A^{e}}(A^{\otimes 3},A)\stackrel{{\scriptstyle d^{*}_{2}}}{{\longrightarrow}}\text{Hom}_{A^{e}}(A^{\otimes 4},A)\stackrel{{\scriptstyle d^{*}_{3}}}{{\longrightarrow}}\cdots$$

(2.2)

where $d_{n}^{*}(f)=fd_{n}$. The Hochshild cohomology of the algebra $A$ is the cohomology of the cochain complex (2.1), i.e.

$$\text{HH}^{*}(A,A)=\bigoplus_{n\geq 0}\operatorname{Ext}_{A^{e}}^{n}(A,A).$$

We also define the Hopf algebra cohomology of the Hopf algebra $A$ over the field $k$ as

$$\textup{H}^{*}(A,k)=\bigoplus_{n\geq 0}\textup{Ext}^{n}_{A}(k,k)$$

under the cup product.

Let $f\in\text{Hom}_{k}(A^{\otimes m},A)$ and $g\in\text{Hom}_{k}(A^{\otimes n},A)$. The Hochschild cohomology of $A$ is an algebra with the following cup product and the Gerstenhaber bracket structures. The cup product $f\smile g\in$Hom${}_{k}(A^{\otimes(m+n)},A)$ is defined by

$$(f\smile g)(a_{1}\otimes\cdots\otimes a_{m+n}):=(-1)^{mn}f(a_{1}\otimes\cdots\otimes a_{m})g(a_{m+1}\otimes\cdots a_{m+n})$$

for all $a_{1},\cdots,a_{m+n}\in A$, and the Gerstenhaber bracket $[f,g]$ is an element of
Hom${}_{k}(A^{\otimes(m+n-1)},A)$ given by

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

where the circle product $f\circ g$ is

	$\displaystyle(f\circ g)(a_{1}\otimes\cdots\otimes a_{m+n-1}):=$
	$\displaystyle\sum_{i=1}^{m}(-1)^{(n-1)(i-1)}f(a_{1}\otimes\cdots a_{i-1}\otimes g(a_{i}\otimes\cdots a_{i+n-1})\otimes a_{i+n}\otimes\cdots\otimes a_{m+n-1})$

for all $a_{1},\cdots,a_{m+n-1}\in A$. We note that these definitions directly come from the bar resolution.

There is an identity between cup product and bracket [3, Section 1]:

$$[f^{*}\smile g^{*},h^{*}]=[f^{*},h^{*}]\smile g^{*}+(-1)^{|f^{*}|(|h^{*}|-1)}f^{*}\smile[g^{*},h^{*}],$$

(2.3)

where $f^{*},g^{*},$ and $h^{*}$ are the images (in Hochschild cohomology) of the cocyles $f,g$, and $h$, respectively.

Computing the bracket on the bar resolution is not an ideal method. Instead, we can use another resolution, $\mathbb{A}\stackrel{{\scriptstyle\mu}}{{\rightarrow}}A$, satisfying the following hypotheses [7, (3.1) and Lemma 3.4.1]:

(a) $\mathbb{A}$ admits an embedding $\iota:\mathbb{A}\to B(A)$ of complexes of $A$-bimodules for which the following diagram commutes

(b) The embedding $\iota$ admits a section $\pi:B\to\mathbb{A}$, i.e. an $A^{e}$-chain map $\pi$ with $\pi\iota=id_{\mathbb{A}}$.

(c) There is a diagonal map that satisfies $\Delta_{\mathbb{A}}^{(2)}=(\pi\otimes_{A}\pi\otimes_{A}\pi)\Delta^{(2)}_{B(A)}\iota$ where $\Delta^{(2)}=(id\otimes\Delta)\Delta$.

We give the following theorem which is the combination of [7, Theorem 3.2.5] and [7, Lemma 3.4.1] that allows us to use a different resolution for the bracket calculation.

In general, it is not easy to calculate the map $\phi$ by the formula (2.5). We use alternative way to find $\phi$.

Let $h$ be any $k$-linear contracting homotopy for the identity map on the extended complex $\mathbb{A}\to A\to 0$ where $\mathbb{A}$ is free. A contracting homotopy $\phi_{i}:(\mathbb{A}\otimes_{A}\mathbb{A})_{i}\longrightarrow\mathbb{A}_{i+1}$ in Theorem 2.4 is constructed by the following formula [7, Lemma 3.3.1]:

$$\phi_{i}=h_{i}((F_{\mathbb{A}})_{i}-\phi_{i-1}d_{(\mathbb{A}\otimes_{A}\mathbb{A})_{i}}).$$

(2.8)

Let $A=k[x]/(x^{p})$ where $k$ is a field of characteristic 0 and $p>2$ is an integer. We compute the Lie bracket on Hochschild cohomology of $A$ by Theorem 2.4. We work on a smaller resolution of $A$ than the bar resolution of $A$. Consider the following $A^{e}$-module resolution of $A$:

$$\mathbb{A}:\cdots\stackrel{{\scriptstyle v.}}{{\longrightarrow}}A^{e}\stackrel{{\scriptstyle u.}}{{\longrightarrow}}A^{e}\stackrel{{\scriptstyle v.}}{{\longrightarrow}}A^{e}\stackrel{{\scriptstyle u.}}{{\longrightarrow}}A^{e}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}A\longrightarrow 0,$$

(3.1)

where $u=x\otimes 1-1\otimes x$, $v=x^{p-1}\otimes 1+x^{p-2}\otimes x+\cdots+x\otimes x^{p-2}+1\otimes x^{p-1}$, and $\pi$ is the multiplication.

The bracket on $A$ where $k$ is a field with positive characteristic, is calculated by C. Negron and S. Witherspoon [7, Section 5]. We adopt the contracting homotopy $h$ for the identity map from that calculation and obtain a new map $h$ for our setup. Let $\xi_{i}$ be the element $1\otimes 1$ of $\mathbb{A}_{i}$. The following maps $h_{n}:\mathbb{A}_{n}\longrightarrow\mathbb{A}_{n+1}$ form a contracting homotopy for identity map, as we can see by direct calculation:

$\displaystyle\begin{split}h_{-1}(x^{i})&=\xi_{0}x^{i},\\ h_{0}(x^{i}\xi_{0}x^{j})&=\sum_{l=0}^{i-1}x^{l}\xi_{1}x^{i+j-1-l},\\ h_{1}(x^{i}\xi_{1}x^{j})&=\delta_{i,p-1}x^{j}\xi_{2},\\ h_{2n}(x^{i}\xi_{2n}x^{j})&=-\sum_{l=0}^{j-1}x^{i+j-1-l}\xi_{2n+1}x^{l}\text{ $(n\geq 2)$ ,}\\ h_{2n+1}(x^{i}\xi_{2n+1}x^{j})&=\delta_{j,p-1}x^{i}\xi_{2n+2}\text{ $(n\geq 2)$ .}\end{split}$

(3.2)

Then, we take $\phi_{-1}=0$ and construct the following $A^{e}$-linear maps $\phi_{i}:(\mathbb{A}\otimes_{A}\mathbb{A})_{i}\longrightarrow\mathbb{A}_{i+1}$ for degree 1 and 2 by (2.8):

$\displaystyle\begin{split}&\phi_{0}(\xi_{0}\otimes_{A}x^{i}\xi_{0})=\sum_{l=0}^{i-1}x^{l}\xi_{1}x^{i-1-l},\\ &\phi_{1}(\xi_{1}\otimes_{A}x^{i}\xi_{0})=-\delta_{i,p-1}\xi_{2},\\ &\phi_{1}(\xi_{0}\otimes_{A}x^{i}\xi_{1})=\delta_{i,p-1}\xi_{2}.\end{split}$

(3.3)

Lastly, we form the following diagonal map $\Delta:\mathbb{A}\longrightarrow\mathbb{A}\otimes_{A}\mathbb{A}$:

$\displaystyle\begin{split}&\Delta_{0}(\xi_{0})=\xi_{0}\otimes_{A}\xi_{0},\\ &\Delta_{1}(\xi_{1})=\xi_{1}\otimes_{A}\xi_{0}+\xi_{0}\otimes_{A}\xi_{1},\\ &\Delta_{2n}(\xi_{2n})=\sum_{i=0}^{n}\xi_{2i}\otimes_{A}\xi_{2n-2i}+\sum_{i=0}^{n-1}\sum_{a+b+c=p-2}x^{a}\xi_{2i+1}\otimes_{A}x^{b}\xi_{2n-2i-1}x^{c},\text{ for }n\geq 1\\ &\Delta_{2n+1}(\xi_{2n+1})=\sum_{i=0}^{2n+1}\xi_{i}\otimes_{A}\xi_{2n+1-i},\text{ for }n\geq 1.\end{split}$

(3.4)

It can be seen that the map $\Delta$ is a chain map lifting the canonical isomorphism $A\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A\otimes_{A}A$ by direct calculation.

Now, we are ready to calculate the brackets on cohomology in low degrees. By applying Hom${}_{A^{e}}(-,A)$ to $\mathbb{A}$, we see that the differentials are all 0 in odd degrees and $(px^{p-1})\cdot$ in even degrees. In each degree, the term in the Hom complex is the free $A$-module Hom${}_{A^{e}}(A^{e},A)\cong A$. Moreover, since $p$ is not divisible by the characteristic of $k$, we deduce HH${}^{0}(A)\cong A,\text{HH}^{2i+1}(A)\cong(x),\text{ and }\text{HH}^{2i}(A)\cong A/(x^{p-1})$ [14, Section 1.1].

Let $x^{j}\xi^{*}_{i}\in$ Hom${}_{A^{e}}(A^{e},A)$ denote the function that takes $\xi_{i}$ to $x^{j}$. Since the characteristic of $k$ does not divide $p$, the Hochschild cohomology as an $A$-algebra is generated by $\xi_{1}^{*}$ and $\xi_{2}^{*}$ [14, Example 2.2.2]. We only calculate the brackets of the elements of degrees 1 and 2 which can be extended to higher degrees by the formula (2.3). Hence, we have the following calculations:

The bracket of the elements of degrees 1 and 1:

	$\displaystyle(x^{i}\xi_{1}^{*}\circ_{\phi}x^{j}\xi_{1}^{*})(\xi_{1})$
	$\displaystyle=x^{i}\xi_{1}^{*}\phi(1\otimes_{A}x^{j}\xi_{1}^{*}\otimes_{A}1)\Delta^{(2)}(\xi_{1})$
	$\displaystyle=x^{i}\xi_{1}^{*}\phi(1\otimes_{A}x^{j}\xi_{1}^{*}\otimes_{A}1)(\xi_{1}\otimes_{A}\xi_{0}\otimes_{A}\xi_{0}+\xi_{0}\otimes_{A}\xi_{1}\otimes_{A}\xi_{0}+\xi_{0}\otimes_{A}\xi_{0}\otimes\xi_{1})$
	$\displaystyle=x^{i}\xi_{1}^{*}\phi(\xi_{0}\otimes_{A}x^{j}\xi_{0})$
	$\displaystyle=x^{i}\xi_{1}^{*}(\xi_{1}x^{j-1}+x\xi_{1}x^{j-2}+\cdots+x^{j-1}\xi_{1})$
	$\displaystyle=jx^{i+j-1}$

and by symmetry $(x^{j}\xi_{1}^{*}\circ_{\phi}x^{i}\xi_{1}^{*})(\xi_{1})=ix^{i+j-1}$. Therefore, we have

$$[x^{i}\xi_{1}^{*},x^{j}\xi_{1}^{*}]=(j-i)x^{i+j-1}\xi_{1}^{*}.$$

The bracket of the elements of degrees 1 and 2:

	$\displaystyle(x^{i}\xi_{1}^{*}\circ_{\phi}x^{j}\xi_{2}^{*})(\xi_{2})$
	$\displaystyle=x^{i}\xi_{1}^{*}\phi(1\otimes_{A}x^{j}\xi_{2}^{*}\otimes_{A}1)\Delta^{(2)}(\xi_{2})$
	$\displaystyle=x^{i}\xi_{1}^{*}\phi(1\otimes_{A}x^{j}\xi_{2}^{*}\otimes_{A}1)(\xi_{0}\otimes_{A}\xi_{0}\otimes_{A}\xi_{2}+\xi_{0}\otimes_{A}\xi_{2}\otimes_{A}\xi_{0}+\xi_{2}\otimes_{A}\xi_{0}\otimes_{A}\xi_{0}$
	$\displaystyle+\xi_{0}\otimes_{A}\sum_{\begin{subarray}{c}a+b+c\\ =p-2\end{subarray}}(x^{a}\xi_{1}\otimes_{A}x^{b}\xi_{1}x^{c})+\sum_{\begin{subarray}{c}a+b+c\\ =p-2\end{subarray}}x^{a}\xi_{1}\otimes_{A}x^{b}(\xi_{0}\otimes_{A}\xi_{1}+\xi_{1}\otimes_{A}x_{0})x^{c})$
	$\displaystyle=x^{i}\xi_{1}^{*}\phi(\xi_{0}\otimes_{A}x^{j}\xi_{0})=x^{i}\xi_{1}^{*}(\xi_{1}x^{j-1}+x\xi_{1}x^{j-2}+\cdots+x^{j-1}\xi_{1})=jx^{i+j-1}.$

The circle product in the reverse order is

	$\displaystyle(x^{j}\xi_{2}^{*}\circ_{\phi}x^{p-1}\xi_{1}^{*})(\xi_{2})$
	$\displaystyle=x^{j}\xi_{2}^{*}\phi(1\otimes_{A}x^{p-1}\xi_{1}^{*}\otimes_{A}1)\Delta^{(2)}(\xi_{2})$
	$\displaystyle=x^{j}\xi_{2}^{*}\phi(1\otimes_{A}x^{p-1}\xi_{1}^{*}\otimes_{A}1)(\xi_{0}\otimes_{A}\xi_{0}\otimes_{A}\xi_{2}+\xi_{0}\otimes_{A}\xi_{2}\otimes_{A}\xi_{0}+\xi_{2}\otimes_{A}\xi_{0}\otimes_{A}\xi_{0}$
	$\displaystyle+\xi_{0}\otimes_{A}\sum_{\begin{subarray}{c}a+b+c\\ =p-2\end{subarray}}(x^{a}\xi_{1}\otimes_{A}x^{b}\xi_{1}x^{c})+\sum_{\begin{subarray}{c}a+b+c\\ =p-2\end{subarray}}x^{a}\xi_{1}\otimes_{A}x^{b}(\xi_{0}\otimes_{A}\xi_{1}+\xi_{1}\otimes_{A}x_{0})x^{c})$
	$\displaystyle=x^{j}\xi_{2}^{*}\phi(\sum_{\begin{subarray}{c}a+b+c\\ =p-2\end{subarray}}(\xi_{0}\otimes_{A}x^{a+b+i}\xi_{1}x^{c}-x^{a}\xi_{1}\otimes_{A}x^{b+i}\xi_{0}x^{c}))$
	$\displaystyle=x^{j}\xi_{2}^{*}(\sum_{\begin{subarray}{c}a+b+c\\ =p-2\end{subarray}}(\delta_{a+b+i,p-1}\xi_{2}x^{c}+x^{a}\delta_{b+i,p-1}\xi_{2}x^{c}))$
	$\displaystyle=x^{j}\xi_{2}^{*}((p-i)\xi_{2}x^{i-1}+\sum_{\begin{subarray}{c}a+c\\ =i-1\end{subarray}}x^{a}\xi_{2}x^{c})$
	$\displaystyle=(p-i)x^{i+j-1}+\sum_{\begin{subarray}{c}a+c\\ =i-1\end{subarray}}x^{a+c+j}=(p-i)x^{i+j-1}+ix^{i+j-1}=px^{i+j-1}.$

Therefore, we obtain

$$[x^{i}\xi_{1}^{*},x^{j}\xi_{2}^{*}]=(j-p)x^{i+j-1}\xi^{*}_{2}.$$

Lastly, the bracket of the elements of degrees 2 and 2:

	$\displaystyle(x^{i}\xi_{2}^{*}\circ_{\phi}x^{j}\xi_{2}^{*})(\xi_{3})$	$\displaystyle=x^{i}\xi_{2}^{*}\phi(1\otimes_{A}x^{j}\xi_{2}^{*}\otimes_{A}1)\Delta^{(2)}(\xi_{3})$
		$\displaystyle=x^{i}\xi_{2}^{*}\phi(\xi_{1}\otimes_{A}x^{j}\xi_{0}+\xi_{0}\otimes_{A}x^{j}\xi_{1})=x^{i}\xi_{2}^{*}(0)=0$

and by symmetry $(x^{j}\xi_{2}^{*}\circ_{\phi}x^{i}\xi_{2}^{*})(\xi_{3})=0$. Therefore, we have

$$[(x^{i}\xi_{2}^{*},x^{j}\xi_{2}^{*})]=0.$$

As a consequence, the brackets for the elements of degrees 1 and 2 are

	$\displaystyle[(x^{i}\xi_{1}^{*},x^{j}\xi_{1}^{*})]$	$\displaystyle=(j-i)x^{i+j-1}\xi_{1}^{*},$
	$\displaystyle[(x^{i}\xi_{1}^{*},x^{j}\xi_{2}^{*})]$	$\displaystyle=(j-p)x^{i+j-1}\xi^{*}_{2},$
	$\displaystyle[(x^{i}\xi_{2}^{*},x^{j}\xi_{2}^{*})]$	$\displaystyle=0.$

Brackets in higher degrees can be determined from these and the identity (2.3) since the Hochschild cohomology is generated as an $A$-algebra under the cup product in degrees 1 and 2.

L. Grimley, V. C. Nguyen, and S. Witherspoon [4] calculated Gerstenhaber brackets on Hochschild cohomology of a twisted tensor product of algebras. S. Sanchez-Flores [9] also calculated the bracket on group algebras of a cyclic group over a field of positive characteristic which is isomorphic to $A=k[x]/(x^{p})$. C. Negron and S. Witherspoon [7] calculated the bracket on group algebras of a cyclic group over a field of positive characteristic as well with the same $h,\phi$, and $\Delta$ maps. Our calculation agrees with those except slightly different $[(x^{i}\xi_{1}^{*},x^{j}\xi_{2}^{*})]$.

The Taft algebra $T_{p}$ with $p>2$ is a $k$-algebra generated by $g$ and $x$ satisfying the relations : $g^{p}=1,x^{p}=0,\text{ and }xg=\omega gx$ where $\omega$ is a primitive $p$-th root of unity. It is a Hopf algebra with the structure:

• •

  $\Delta(g)=g\otimes g$, $\Delta(x)=1\otimes x+x\otimes g$

• •

  $\varepsilon(g)=1,\varepsilon(x)=0$

• •

  $S(g)=g^{-1},S(x)=-xg^{-1}.$

Note that as an algebra, $T_{p}$ is a skew group algebra $A\rtimes kG$ where $A=k[x]/(x^{p})$ and $G=<g\mid g^{p}=1>$. The action of $G$ on $A$ is given by ${}^{g}x=\omega x$.

In this section, our main goal is to calculate the bracket on Hochschild cohomology of $T_{p}$ with the same technique in Section 3 and find the bracket on Hopf algebra cohomology of $T_{p}$ by using the embedding of $\text{H}^{*}(T_{p},k)$ into $\text{HH}^{*}(T_{p},T_{p})$.

We first find the bracket on Hochschild cohomology of $T_{p}$. Let $\mathcal{D}$ be the skew group algebra $A^{e}\rtimes G$ where the action of $G$ on $A^{e}$ is diagonal, i.e. ${}^{g}(a\otimes b)=(^{g}a)\otimes(^{g}b)$. Then, there is the following isomorphism [1, Section 2]

$$\mathcal{D}=A^{e}\rtimes G\cong\underset{g\in G}{\bigoplus}Ag\otimes Ag^{-1}\subset T_{p}^{e}.$$

Hence $\mathcal{D}$ is isomorphic to a subalgebra of $T_{p}^{e}$ via $a_{1}\otimes a_{2}\otimes g\mapsto a_{1}g\otimes(^{g^{-1}}a_{2}g^{-1})$. Moreover, $A$ is a $\mathcal{D}$-module under the following left and right action [1, Section 4]:

$$(a_{1}g\otimes a_{2}g^{-1})a_{3}=a_{1}ga_{3}a_{2}g^{-1}=a_{1}(^{g}(a_{3}a_{2}))$$

$$a_{3}(a_{1}g\otimes a_{2}g^{-1})=a_{2}g^{-1}a_{3}a_{1}g=a_{2}(^{g^{-1}}(a_{3}a_{1})).$$

Remember the resolution (3.1)

$$\mathbb{A}:\cdots\stackrel{{\scriptstyle v.}}{{\longrightarrow}}A^{e}\stackrel{{\scriptstyle u.}}{{\longrightarrow}}A^{e}\stackrel{{\scriptstyle v.}}{{\longrightarrow}}A^{e}\stackrel{{\scriptstyle u.}}{{\longrightarrow}}A^{e}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}A\longrightarrow 0.$$

This is also a $\mathcal{D}$-projective resolution of A and the action of $G$ on $A^{e}$ is given by

• •

  $g\cdot(a_{1}\otimes a_{2})=(^{g}a_{1})\otimes(^{g}a_{2})$ in even degrees,

• •

  $g\cdot(a_{1}\otimes a_{2})=\omega(^{g}a_{1})\otimes(^{g}a_{2})$ in odd degrees.

From the resolution $\mathbb{A}$, we construct the following $T_{p}^{e}$ resolution of $T_{p}$:

$$T_{p}^{e}\otimes_{\mathcal{D}}\mathbb{A}:\cdots{\longrightarrow}T_{p}^{e}\otimes_{\mathcal{D}}{A^{e}}{\longrightarrow}T_{p}^{e}\otimes_{\mathcal{D}}{A^{e}}{\longrightarrow}T_{p}^{e}\otimes_{\mathcal{D}}{A^{e}}{\longrightarrow}T_{p}^{e}\otimes_{\mathcal{D}}{A}\longrightarrow 0.$$

(4.1)

It is known that, $T_{p}\cong T_{p}^{e}\otimes_{\mathcal{D}}{A}$ as $T_{p}$-bimodules via the map sending $x^{i}\otimes g^{k}$ to $(1\otimes g^{k})\otimes_{\mathcal{D}}x^{i}$ [14, Section 3.5]. Then we have $A\otimes T_{p}\cong T_{p}^{e}\otimes_{\mathcal{D}}{A^{e}}$ with the $T_{p}$-bimodule isomorphism given by

$$\kappa(x^{i}\otimes(x^{j}\otimes g^{k}))=(1\otimes g^{k})\otimes_{\mathcal{D}}(x^{i}\otimes x^{j}).$$

(4.2)

Then, we obtain the following resolution $\tilde{\mathbb{A}}$ which is isomorphic to the resolution (4.1), i.e.

$$\tilde{\mathbb{A}}:\cdots\stackrel{{\scriptstyle\tilde{u}.}}{{\longrightarrow}}A\otimes T_{p}\stackrel{{\scriptstyle\tilde{v}.}}{{\longrightarrow}}A\otimes T_{p}\stackrel{{\scriptstyle\tilde{u}.}}{{\longrightarrow}}A\otimes T_{p}\stackrel{{\scriptstyle\tilde{\pi}.}}{{\longrightarrow}}T_{p}\longrightarrow 0$$

(4.3)

where $\tilde{v}=v\otimes id_{kG},\tilde{u}=u\otimes id_{kG}$, and $\tilde{\pi}=\pi\otimes id_{kG}$.

The following lemma gives us a contracting homotopy for the identity map on the resolution $\tilde{\mathbb{A}}$.

We abbreviate $a_{1}\otimes a_{2}\otimes g\in A\otimes T_{p}$ by $a_{1}\otimes a_{2}g$. By the Lemma 4.4, we obtain

	$\displaystyle\tilde{h}_{-1}(x^{i}g)$	$\displaystyle=\xi_{0}x^{i}g,$
	$\displaystyle\tilde{h}_{0}(x^{i}\xi_{0}x^{j}g)$	$\displaystyle=\sum_{l=0}^{i-1}x^{l}\xi_{1}x^{i+j-1-l}g,$
	$\displaystyle\tilde{h}_{1}(x^{i}\xi_{1}x^{j}g)$	$\displaystyle=\delta_{i,p-1}x^{j}\xi_{2}g,$
	$\displaystyle\tilde{h}_{2n}(x^{i}\xi_{2n}x^{j}g)$	$\displaystyle=-\sum_{l=0}^{j-1}x^{i+j-1-l}\xi_{2n+1}x^{l}g,$
	$\displaystyle\tilde{h}_{2n+1}(x^{i}\xi_{2n+1}x^{j}g)$	$\displaystyle=\delta_{j,p-1}x^{i}\xi_{2n+2}g.$

We need a lemma to have the linear maps $\tilde{\phi}_{i}:(\tilde{\mathbb{A}}\otimes_{T_{p}}\tilde{\mathbb{A}})_{i}\longrightarrow\tilde{\mathbb{A}}_{i+1}$. However, we first mention that there is an isomorphism from $(A\otimes T_{p})\otimes_{T_{p}}(A\otimes T_{p})$ to $(A\otimes A)\otimes_{A}(A\otimes A)\otimes kG$ as $T_{p}^{e}$-modules given by

$$\psi((x^{i_{1}}\otimes x^{j_{1}}g^{k_{1}})\otimes_{T_{p}}(x^{i_{2}}\otimes x^{j_{2}}g^{k_{2}}))=\omega^{k_{1}(i_{2}+j_{2})}(x^{i_{1}}\otimes x^{j_{1}})\otimes_{A}(x^{i_{2}}\otimes x^{j_{2}})g^{(k_{1}+k_{2})}.$$

(4.5)