Equivariant one-parameter deformations of associative algebra morphisms

TL;DR

This paper develops an equivariant deformation theory for associative algebra morphisms, introducing a new cohomology framework to study how these morphisms deform under symmetry constraints.

Contribution

It presents the first equivariant deformation cohomology for associative algebra morphisms, extending classical deformation theory to incorporate symmetry considerations.

Findings

Defined equivariant deformation cohomology for algebra morphisms

Analyzed the deformation theory within the equivariant setting

Provided tools for studying symmetric deformations of algebraic structures

Abstract

In this article, we introduce equivariant formal deformation theory of associative algebra morphisms. We introduce an equivariant deformation cohomology of associative algebra morphisms and using this we study the equivariant formal deformation theory of associative algebra morphisms.

Equations

$$\mu (a,\mu (b,c))=\mu (\mu (a,b),c),$$

$$\varphi (\mu (a,b))=\mu (\varphi a,\varphi b),$$

$$T(V)=k\oplus V\oplus \cdots \oplus V^{\otimes n}\oplus \cdots$$

$$v_1\cdots v_p\otimes v_{p+1}\cdots v_{p+q}=v_1\cdots v_p{v}_{p+1}\cdots v_{p+q}.$$

$${\delta }^{n}f(x_1,\cdots ,x_{n+1})$$

$$\varphi :G\times A\to A(g,a)\mapsto \varphi (g,a)=ga$$

$$C_G^n(A;M)=\{c\in C^n(A;M):c(g{x}_1,\cdots ,g{x}_n)=gc(x_1,\cdots ,x_n),\text{ for all}g\in G\}$$

$$c\in C_G^n(A;M)\Rightarrow {\delta }^n(c)\in C_G^{n+1}(A;M)$$

$$C_G^n(\varphi ;T)=C_G^n(A;M)\oplus C_G^n(B;N)\oplus C_G^{n-1}(A;N),$$

$$d^n(u,v,w)=({\delta }^{n}u,{\delta }^{n}v,Tu-v\varphi -{\delta }^{n-1}w),$$

$$d^{n+1}{d}^n(u,v,w)$$

$$\sum _{i+j=r}{\mu }_i({\mu }_j(a,b),c)=\sum _{i+j=r}{\mu }_i(a,{\mu }_j(b,c)),\text{for all}a,b,c\in A,r\ge 0.$$

$$\sum _{i+j=r}{\nu }_i({\nu }_j(a,b),c)=\sum _{i+j=r}{\nu }_i(a,{\nu }_j(b,c)),\text{for all}a,b,c\in B,r\ge 0.$$

$$\sum _{i+j=r}{\varphi }_i({\mu }_j(a,b))=\sum _{i+j+k=r}{\nu }_i({\varphi }_j(a),{\varphi }_k(b));\text{for all}a,b\in A,r\ge 0.$$

$${\delta }^2{\mu }_{n+1}(a,b,c)=\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\mu }_i({\mu }_j(a,b),c)-{\mu }_i(a,{\mu }_j(b,c)),\text{for all}a,b,c\in A.$$

$${\delta }^2{\nu }_{n+1}(a,b,c)=\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\nu }_i({\nu }_j(a,b),c)-{\nu }_i(a,{\nu }_j(b,c)),\text{for all}a,b,c\in B.$$

$$\varphi ({\mu }_{n+1}(a,b))$$

$$\sum ^{\prime }$$

$$d^2({\mu }_{n+1},{\nu }_{n+1},{\varphi }_{n+1})(a,b,c,x,y,z,p,q)$$

$$=(\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\mu }_i({\mu }_j(a,b),c)-{\mu }_i(a,{\mu }_j(b,c)),\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\nu }_i({\nu }_j(x,y),z)-{\nu }_i(x,{\nu }_j(y,z)),$$

$$\sum ^{\prime }{\nu }_i({\varphi }_j(p),{\varphi }_k(q))-\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\varphi }_i({\mu }_j(p,q))),$$

$$F_{n+1}(a,b,c,x,y,z,p,q)$$

$$=(\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\mu }_i({\mu }_j(a,b),c)-{\mu }_i(a,{\mu }_j(b,c)),\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\nu }_i({\nu }_j(x,y),z)-{\nu }_i(x,{\nu }_j(y,z)),$$

$$\sum ^{\prime }{\nu }_i({\varphi }_j(p),{\varphi }_k(q))-\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\varphi }_i({\mu }_j(p,q)))$$

$$F_{n+1}(ga,gb,gc,gx,gy,gz,gp,gq)=g{F}_{n+1}(a,b,c,x,y,z,p,q),$$

$${\mu }_i(ga,gb)=g{\mu }_i(a,b),{\nu }_i(gx,gy)=g{\nu }_i(x,y),{\varphi }_i(ga)=g{\varphi }_i(a),$$

$$F_{n+1}(ga,gb,gc,gx,gy,gz,gp,gq)$$

$$=(\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\mu }_i({\mu }_j(ga,gb),gc)-{\mu }_i(ga,{\mu }_j(gb,gc)),\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\nu }_i({\nu }_j(gx,gy),gz)-{\nu }_i(gx,{\nu }_j(gy,gz)),$$

$$\sum ^{\prime }{\nu }_i({\varphi }_j(gp),{\varphi }_k(gq))-\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\varphi }_i({\mu }_j(gp,gq)))$$

$$=(\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\mu }_i(g{\mu }_j(a,b),gc)-{\mu }_i(ga,g{\mu }_j(b,c)),\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\nu }_i(g{\nu }_j(x,y),gz)-{\nu }_i(gx,g{\nu }_j(y,z)),$$

$$\sum ^{\prime }{\nu }_i(g{\varphi }_j(p),g{\varphi }_k(q))-\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}{\varphi }_i(g{\mu }_j(p,q)))$$

$$=(\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}g{\mu }_i({\mu }_j(a,b),c)-g{\mu }_i(a,{\mu }_j(b,c)),\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}g{\nu }_i({\nu }_j(x,y),z)-g{\nu }_i(x,{\nu }_j(y,z)),$$

$$\sum ^{\prime }g{\nu }_i({\varphi }_j(p),{\varphi }_k(q))-\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}g{\varphi }_i({\mu }_j(p,q)))$$

$$=g{F}_{n+1}(a,b,c,x,y,z,p,q).$$

$$d^{3}O{b}_{n+1}=({\delta }^3(O_1),{\delta }^3(O_2),\varphi O_1-O_2\varphi -{\delta }^2(O_3)),$$

$$O_1(a,b,c)=\sum _{\begin{array}{c}i+j=n+1\\ i,j>0\end{array}}\left\{{\mu }_i({\mu }_j(a,b),c)-{\mu }_i(a,{\mu }_j(b,c))\right\},$$

Full text

††journal: nowhere

1 Introduction

Origin of the idea of deformation theory goes back to a paper of Riemann on abelian functions published in 1857. Kodaira and Spencer initiated deformation theory of complex analytic structures [15], [16], [17]. M. Gerstenhaber introduced algebraic deformation theory in a series of papers [7],[8],[9], [10], [11]. He studied deformation theory of associative algebras. Deformation theory of associative algebra morphisms was introduced by M. Gerstenhaber and S.D. Schack [12], [13], [14]. Nijenhuis and Richardson introduced deformation theory of Lie algebras [1], [2]. Deformation theory of dialgebras has been studied by Majumdar and Mukherjee [3]. Recently, deformation theory of dialgebra morphisms and Leibniz algebra morphisms have been studied by Donald Yau and Ashis Mondal respectively [6], [5]. Equivariant deformation theory of associative algebras has been studied in [4].

Organization of the paper is as follows. In Section 2, we recall some definitions and results. In Section 3, we introduce equivariant deformation complex and equivariant deformation cohomology of an associative algebra morphism. In Section 4, we introduce equivariant deformation of an associative algebra morphism. In this section we prove that obstructions to equivariant deformations are cocycles. In Section 5, we study equivalence of two equivariant deformations and rigidity of an equivariant associative algebra morphism.

2 Preliminaries

In this section, we recall definitions of associative algebra, associative algebra morphisms, Hochschild cohomology and equivariant deformation cohomology of an associative algebra. Also, we recall definitions of a module over an associative algebra and module over an associative algebra morphism. Throughout the paper we denote a fixed field by k and a finite group by G.

Definition 2.1**.**

An associative algebra A is a k-module equipped with a k-bilinear map $\mu$ satisfying

[TABLE]

for all $a,b,c\in A$

Let A be an associative k-algebra. A bimodule M over A is a k-module M with two actions (left and right) of A, $\mu:A\times M\to M$ and $\mu:M\times A\to M$ (for simplicity we denote both the actions by same symbol, one can differentiate both of them from the context) such that $\mu(x,\mu(y,z))=\mu(\mu(x,y),z),$ whenever one of x,y,z is from M and others are from A.

Let A and B be associative k-algebras. An associative algebra morphism $\phi:A\to B$ is a k-linear map satisfying

[TABLE]

for all $a,b\in A.$

Example 2.1**.**

Let $M_{n}(\mathbb{R})$ be the collection of all $n\times n$ matrices with entries in $\mathbb{C}$. Then $M_{n}(\mathbb{C})$ is an associative algebra over $\mathbb{C}$ with respect to matrix multiplication.

Example 2.2**.**

Let $X$ be a nonempty set and $A=\{\alpha:\alpha:X\to\mathbb{C}\;\text{is a function}\}$. Then A is an associative algebra over $\mathbb{C}$ with the product $(\alpha\beta)(x)=\alpha(x)\beta(x).$

Example 2.3**.**

Let V be a vector space over k. Define tensor module by

[TABLE]

$T(V)$* is an associative algebra with the concatenation product $T(V)\otimes T(V)\to T(V)$ given by*

[TABLE]

Definition 2.2**.**

Let A be an associative k-algebra and M be a bimodule over A. Let $C^{n}(A;M)=\hom_{k}(A^{\otimes n},M),$ for all integers $n\geq 0.$ Also, define a k-linear map $\delta^{n}:C^{n}(A;M)\to C^{n+1}(A;M)$ given by

[TABLE]

*This gives a cochain complex $(C^{\ast}(A;M),\delta)$, cohomology of which is denoted by $H^{\ast}(A;M)$ and called as Hochschild cohomology of A with coeffiecients in M. A is a bimodule over itself in an obvious way. So we can consider Hochschild cohomology $H^{\ast}(A;A)$.

Let A be an associative k-algebra with product $\mu(a,b)=ab$ and G be a finite group. The group G is said to act on A from the left if there exists a function*

[TABLE]

satisfying the following conditions.

$ex=x$* for all $x\in A$, where $e\in G$ is the group identity.* 2. 2.

$g_{1}(g_{2}x)=(g_{1}g_{2})x$* for all $g_{1},g_{2}\in G$ and $x\in A$.* 3. 3.

For every $g\in G$, the left translation $\phi_{g}=\phi(g,):A\to A,$ $a\to ga$ is a linear map. 4. 4.

For all $g\in G$ and $a,b\in A$, $\mu(ga,gb)=g\mu(a,b)=g(ab)$, that is, $\mu$ is equivari- ant with respect to the diagonal action on $A\times A$.

An action as above is denoted by $(G,A)$ [4]. Define

[TABLE]

An element in $C_{G}^{n}(A;M)$ is called an invariant n-cochain. Clearly, $C_{G}^{n}(A;M)$ is a submodule of $C^{n}(A;M)$.

From [4] we have following lemma.

Lemma 2.1**.**

If an n-cochain c is invariant then $\delta^{n}(c)$ is also invariant. In other words,

[TABLE]

From [4], we have $(C_{G}^{\ast}(A;M),\delta)$ is a cochain complex. Cohomology of this complex is called equivariant deformation cohomology of A.

Definition 2.3**.**

Let $A$ and $B$ be associative k-algebras, and $\phi:A\to B$ be an associative algebra morphism. Let $M$ and $N$ be k-modules. A k-linear map $T:M\to N$ is said to be a left (or right) module over $\phi$ if following conditions are satisfied.

$M$* and $N$ are left ( or right) modules over $A$ and $B$ respectively.* 2. 2.

T is a left (or right) A-module morphism when N is viewed as a left (or right) A-module by virtue of the morphism $\phi:A\to B.$

A k-linear map $T:M\to N$ is said to be a bimodule over $\phi$ if $T$ is a left as well as right module over $\phi$.

From [4], we recall equivariant deformation of an associative algebra morphism.

Definition 2.4**.**

Let A be an associative k-algebra with a $(G,A)$ action. Define $A_{t}=\{\sum_{i=0}^{\infty}a_{i}t^{i}:a_{i}\in A\}.$ An equivariant formal one-parameter deformation of A is a k-bilinear multiplication $m_{t}:A_{t}\times A_{t}\to A_{t}$ satisfying the following properties:

$m_{t}(a,b)=\sum_{i=0}^{\infty}m_{i}(a,b)t^{i},$* for all $a,b\in A,$ where $m_{i}:A\times A\to A$ are k-bilinear and $m_{0}(a,b)=ab$ is the original multiplication on A, and $m_{t}$ is associative.* 2. 2.

For every $g\in G$, $m_{i}(ga,gb)=gm_{i}(a,b)$, for all $a,b\in A$, $i\geq 1,$ that is $m_{i}\in Hom_{k}^{G}(A\otimes A,A),$ for all $i\geq 1.$

3 Equivariant Deformation complex of an associative algebra morphism

In this section, we introduce equivariant deformation complex of an associative algebra morphism. In the subsequent sections we show that second and third cohomology of this complex controls deformation.

Definition 3.1**.**

Let $G$ be a finite group, A and B be associative algebras with actions $(G,A)$ and $(G,B)$ respectively. A G-equivariant associative algebra morphism $\phi:A\to B$ is defined to be an associative algebra morphism such that $\phi(ga)=g\phi(a)$, for all $a\in A$, $g\in G.$

Definition 3.2**.**

Let $\phi:A\to B$ be a G-equivariant associative algebra morphism.

An equivariant left (or right) module over $\phi$ is defined to be a G-equivariant k-linear map $T:M\to N$ which is a left (or right) module over $\phi$. 2. 2.

An equivariant bimodule over $\phi$ is defined to be a G-equivariant k-linear map $T:M\to N$ which is a bimodule over $\phi$. In particular, $\phi$ is an equivariant bimodule over itself.

Definition 3.3**.**

Let $\phi:A\to B$ be a G-equivariant associative algebra morphism and $T:M\to N$ be an equivariant bimodule over $\phi$. We define

[TABLE]

for all $n\in\mathbb{N}$ and $C_{G}^{0}(\phi;T)=0$. Also, we define $d^{n}:C_{G}^{n}(\phi;T)\to C_{G}^{n+1}(\phi;T)$ by

[TABLE]

for all $(u,v,w)\in C_{G}^{n}(\phi;T).$ Here the $\delta^{n}$’s denote coboundaries of the cochain complexes $C_{G}^{\ast}(A;M)$, $C_{G}^{\ast}(B;N)$ and $C_{G}^{\ast}(A;N)$, the $Tu$ denotes the composition $T\circ u$ of $T$ and $u$ and the map $v\phi:A^{\otimes n}\to N$ is defined by $v\phi(x_{1},x_{2},\cdots,x_{n})=v(\phi(x_{1}),\phi(x_{2}),\cdots,\phi(x_{n})).$ From Lemma 2.1, one can easily verify that $d^{n}$ is well defined.

Proposition 3.1**.**

$(C_{G}^{\ast}(\phi;T),d)$* is a cochain complex.*

Proof.

We have

[TABLE]

One can easily see that $\delta^{n}(Tu-v\phi)=T(\delta^{n}u)-(\delta^{n}v)\phi$. So, since $\delta^{n+1}\delta^{n}u=\delta^{n+1}\delta^{n}v=\delta^{n+1}\delta^{n}w=0$, we have $d^{n+1}d^{n}=0$. Hence we conclude the result. ∎

We call the cochain complex $(C_{G}^{\ast}(\phi,\phi),d)$ as equivariant deformation complex of $\phi,$ and the corresponding cohomology as equivariant deformation cohomology of $\phi$. We denote the equivariant deformation cohomology by $H_{G}^{n}(\phi,\phi)$, that is $H_{G}^{n}(\phi,\phi)=H^{n}(C_{G}^{\ast}(\phi,\phi),d)$. Next proposition relates $H_{G}^{\ast}(\phi,\phi)$ to $H_{G}^{\ast}(A,A)$, $H_{G}^{\ast}(B,B)$ and $H_{G}^{\ast}(A,B)$.

Proposition 3.2**.**

If $H_{G}^{n}(A,A)=0$, $H_{G}^{n}(B,B)=0$ and $H_{G}^{n-1}(A,B)=0$, then $H_{G}^{n}(\phi,\phi)=0$.

Proof.

Let $(u,v,w)\in C_{G}^{n}(\phi,\phi)$ be a cocycle, that is $d^{n}(u,v,w)=(\delta^{n}u,\delta^{n}v,\phi u-v\phi-\delta^{n-1}w)=0$. This implies that $\delta^{n}u=0$, $\delta^{n}v=0$, $\phi u-v\phi-\delta^{n-1}w=0$. $H_{G}^{n}(A,A)=0\Rightarrow u=\delta^{n-1}u_{1}$ and $H_{G}^{n}(B,B)=0\Rightarrow\delta^{n-1}v_{1}=v$, for some $u_{1}\in C_{G}^{n-1}(\phi,\phi)$ and $v_{1}\in C_{G}^{n-1}(\phi,\phi)$. So $0=\phi u-v\phi-\delta^{n-1}w=\phi(\delta^{n-1}u_{1})-(\delta^{n-1}v_{1})\phi-\delta^{n-1}w=\delta^{n-1}(\phi u_{1})-\delta^{n-1}(v_{1}\phi)-\delta^{n-1}w=\delta^{n-1}(\phi u_{1}-v_{1}\phi-w)$. So $\phi u_{1}-v_{1}\phi-w\in C_{G}^{n-1}(A,B)$ is a cocycle. Now, $H_{G}^{n-1}(A,B)=0\Rightarrow\phi u_{1}-v_{1}\phi-w=\delta^{n-2}w_{1}\Rightarrow\phi u_{1}-v_{1}\phi-\delta^{n-2}w_{1}=w$. Thus $(u,v,w)=(\delta^{n-1}u_{1},\delta^{n-1}v_{1},\phi u_{1}-v_{1}\phi-\delta^{n-2}w_{1})=d^{n-1}(u_{1},v_{1},w_{1})$, for some $(u_{1},v_{1},w_{1})\in C_{G}^{n-1}(\phi,\phi).$ Thus every cocycle in $C_{G}^{n}(\phi,\phi)$ is a coboundary. Hence we conclude that $H_{G}^{n}(\phi,\phi)=0$. ∎

4 Equivariant deformation of an associative algebra morphism

Definition 4.1**.**

Let A and B be associative k-algebras with actions $(G,A)$ and $(G,B)$ respectively. An equivariant deformation of a G-equivariant associative algebra morphism $\phi:A\to B$ is a triple $(\mu_{t},\nu_{t},\phi_{t})$, in which:

$\mu_{t}=\sum_{i=0}^{\infty}\mu_{i}t^{i}$* is an equivariant formal one-parameter deformation for $A$.* 2. 2.

$\nu_{t}=\sum_{i=0}^{\infty}\nu_{i}t^{i}$* is an equivariant formal one-parameter deformations for $B$.* 3. 3.

$\phi_{t}:A_{t}\to B_{t}$* is a G-equivariant associative algebra morphism of the form $\phi_{t}=\sum_{i=0}^{\infty}\phi_{i}t^{i}$, where $\phi_{i}:A\to B$ is G-equivariant associative algebra morphisms for all $i\geq 0$ and $\phi_{0}=\phi$.*

Therefore a triple $(\mu_{t},\nu_{t},\phi_{t})$, as given above, is an equivariant deformation of $\phi$ provided following properties are satisfied.

• (i)

$\mu_{t}(\mu_{t}(a,b),c)=\mu_{t}(a,\mu_{t}(b,c))$, for all $a,b,c\in A;$

• (ii)

$\mu_{i}(ga,gb)=g\mu_{i}(a,b)$, for all $a,b\in A$ and $g\in G;$

• (iii)

$\nu_{t}(\nu_{t}(a,b),c)=\nu_{t}(a,\nu_{t}(b,c))$, for all $a,b,c\in B;$

• (iv)

$\nu_{i}(ga,gb)=g\nu_{i}(a,b)$, for all $a,b\in B$ and $g\in G;$

• (v)

$\phi_{t}(\mu_{t}(a,b))=\nu_{t}(\phi_{t}(a),\phi_{t}(b)),$* for all $a,b\in A;$*

• (vi)

$\phi_{i}(ga)=g\phi_{i}(a),$* for all $a\in A$ and $g\in G.$*

The conditions $(i)$, $(iii)$ and $(v)$ are equivalent to following conditions respectively.

[TABLE]

[TABLE]

[TABLE]

Now we define equivariant deformations of finite order.

Definition 4.2**.**

Let A and B be associative k-algebras with actions $(G,A)$ and $(G,B)$ respectively. An equivariant deformation of order n of a G-equivariant associative algebra morphism $\phi:A\to B$ is a triple $(\mu_{t},\nu_{t},\phi_{t})$, in which:

$\mu_{t}=\sum_{i=0}^{n}\mu_{i}t^{i}$* is an equivariant formal one-parameter deformation of order n for $A$.* 2. 2.

$\nu_{t}=\sum_{i=0}^{n}\nu_{i}t^{i}$* is an equivariant formal one-parameter deformations of order n for $B$.* 3. 3.

$\phi_{t}:A_{t}\to B_{t}$* is a G-equivariant associative algebra morphism of the form $\phi_{t}=\sum_{i=0}^{n}\phi_{i}t^{i}$, where $\phi_{i}:A\to B$ is G-equivariant associative algebra morphisms for all $i\geq 0$ and $\phi_{0}=\phi$.*

Remark 4.1**.**

For $r=0$, conditions 1, 2 and 3 are equivalent to the fact that A and B are associative algebras and $\phi$ is an associative algebra morphism, respectively.

• 2.

For $r=1$, 1, 2 and 3 are equivalent to $\delta^{2}\mu_{1}=0,$ $\delta^{2}\nu_{1}=0$ and $\phi\mu_{1}-\nu_{1}\phi-\delta^{1}\phi_{1}=0$. Thus for $r=1$, 1, 2 and 3 are equivalent to saying that $(\mu_{1},\nu_{1},\phi_{1})\in C^{2}_{G}(\phi,\phi)$ is a cocycle. In general, for $r\geq 2$, $(\mu_{r},\nu_{r},\phi_{r})$ is just a 2-cochain in $C^{2}_{G}(\phi,\phi).$

Definition 4.3**.**

The 2-cochain $(\mu_{1},\nu_{1},\phi_{1})$ in $C^{2}_{G}(\phi,\phi)$ is called infinitesimal of the equivariant deformation $(\mu_{t},\nu_{t},\phi_{t})$. In general, if $(\mu_{i},\nu_{i},\phi_{i})=0,$ for $1\leq i\leq n-1$, and $(\mu_{n},\nu_{n},\phi_{n})$ is a nonzero cochain in $C^{2}_{G}(\phi,\phi)$, then $(\mu_{n},\nu_{n},\phi_{n})$ is called n-infinitesimal of the deformation $(\mu_{t},\nu_{t},\phi_{t})$.

Proposition 4.1**.**

The infinitesimal $(\mu_{1},\nu_{1},\phi_{1})$ of the equivariant deformation $(\mu_{t},\nu_{t},\phi_{t})$ is a 2-cocycle in $C^{2}_{G}(\phi,\phi).$ In general, n-infinitesimal $(\mu_{n},\nu_{n},\phi_{n})$ is a cocycle in $C^{2}_{G}(\phi,\phi).$

Proof.

For n=1, proof is obvious from the Remark 4.1. For $n>1$, proof is similar. ∎

We can write equations 1, 2 and 3 for $r=n+1$ using the definition of coboundary $\delta$ as

[TABLE]

[TABLE]

[TABLE]

for all $a,b\in A$, where,

[TABLE]

By using equations 4, 5 and 6 we have

[TABLE]

for all $a,b,c,p,q\in A$ and $x,y,z\in B$.

Define a 3-cochain $F_{n+1}$ by

[TABLE]

Lemma 4.1**.**

The 3-cochain $F_{n+1}$ is invariant, that is $F_{n+1}\in C_{G}^{3}(\phi,\phi).$

Proof.

To prove that $F_{n+1}$ is invariant we show that

[TABLE]

for all $a,b,c,p,q\in A$ and $x,y,z\in B.$ From definition 4.1, we have

[TABLE]

for all $a,b\in A$ and $x,y\in B.$ So, we have, for all $a,b,c,p,q\in A$ and $x,y,z\in B,$

[TABLE]

So we conclude that $F_{n+1}\in C_{G}^{n}(\phi,\phi).$ ∎

Definition 4.4**.**

The 3-cochain $F_{n+1}\in C_{G}^{n}(\phi,\phi)$ is called $(n+1)th$ obstruction cochain for extending the given equivariant deformation of order n to an equivariant deformation of $\phi$ of order $(n+1)$. Now onwards we denote $F_{n+1}$ by $Ob_{n+1}(\phi_{t})$

We have the following result.

Theorem 4.1**.**

The (n+1)th obstruction cochain $Ob_{n+1}(\phi_{t})$ is a 3-cocycle.

Proof.

We have,

[TABLE]

where $O_{1}$, $O_{2}$ and $O_{3}$ are given by

[TABLE]

[TABLE]

[TABLE]

From [4], we have $\delta^{3}(O_{1})=0$, $\delta^{3}(O_{2})=0$. So, to prove that $d^{3}Ob_{n+1}=0$, it remains to show that $\phi O_{1}-O_{2}\phi-\delta^{2}(O_{3})=0.$ To prove that $\phi O_{1}-O_{2}\phi-\delta^{2}(O_{3})=0$ we use similar ideas as have been used in [5] and [6]. We have,

[TABLE]

and

[TABLE]

From 3, we have

[TABLE]

Substituting expression for $\phi_{j}\mu_{0}$ from 12, in the third sum on the right hand side of 11 we can rewrite it as

[TABLE]

Here the first sum of 13 is given by

[TABLE]

the second sum of 13 is given by

[TABLE]

The first sum of 12 splits into four sums. The first one of these four sums splits as

[TABLE]

The fist sum on the r.h.s. of 4 appears as third sum on the r.h.s. of 4. By applying a similar arguement to the fifth sum on the r.h.s. of 11, using 3 on $\phi_{k}\mu_{0}(y,z)$, one can rewrite it as

[TABLE]

As above first sum on r.h.s. of 17 is a sum of four sums, similar to 14 except that the roles of j and k are interchanged. One of these four terms splits as

[TABLE]

The fist sum on the r.h.s. of 4 appears as fourth sum on the r.h.s. of 4. In the fourth sum on the r.h.s. of 11, we use 1 to substitute $\mu_{j}(\mu_{0}(x,y),z)$ to obtain

[TABLE]

Fist sum on the r.h.s. of 19 cancels with the sixth sum on the r.h.s. of 11. Second sum on the r.h.s. 0f 19 splits as

[TABLE]

Second sum on the r.h.s. of 20 appears as second sum on the r.h.s. of 4. Also, by using 3 first sum on the r.h.s. of 20 splits as

[TABLE]

In the last line the two terms cancel with second terms on the r.h.s of 11 and 17, respectively. The third term on the r.h.s. of 20 splits as

[TABLE]

On the r.h.s. of 22, last term cancels with first sum on the r.h.s. of 4, first sum cancels with the last sum on the r.h.s. of 11 and second term cancels with the second sum on the r.h.s. of 13. From our previous arquements we have,

[TABLE]

We can write above equation more compactly as

[TABLE]

where

[TABLE]

It follows from 24 and 2 that the sum on the r.h.s. of 24 is 0, and hence $\phi O_{1}-O_{2}\phi-\delta^{2}(O_{3})=0$. This finishes the proof of the theorem. ∎

Theorem 4.2**.**

Let $(\mu_{t},\nu_{t},\phi_{t})$ be an equivariant deformation of $\phi$ of order n. Then $(\mu_{t},\nu_{t},\phi_{t})$ extends to an equivariant deformation of order $n+1$ if and only if cohomology class of $(n+1)$th obstruction $Ob_{n+1}(\phi_{t})$ vanishes.

Proof.

Suppose that an equivariant deformation $(\mu_{t},\nu_{t},\phi_{t})$ of $\phi$ of order n extends to an equivariant deformation of order $n+1$. This implies that 1,2 and 3 are satisfied for $r=n+1.$ Observe that this implies $Ob_{n+1}(\phi_{t})=d_{2}(\mu_{n+1},\nu_{n+1},\phi_{n+1})$. So cohomology class of $Ob_{n+1}(\phi_{t})$ vanishes. Conversely, suppose that cohomology class of $Ob_{n+1}(\phi_{t})$ vanishes, that is $Ob_{n+1}(\phi_{t})$ is a coboundary. Let

[TABLE]

for some 2-cochain $(\mu_{n+1},\nu_{n+1},\phi_{n+1})\in C^{2}_{G}(\phi,\phi).$ Take

[TABLE]

. Observe that $(\tilde{\mu_{t}},\tilde{\nu_{t}},\tilde{\phi_{t}})$ satisfies 1,2 and 3 for $0\leq r\leq n+1$. So $(\tilde{\mu_{t}},\tilde{\nu_{t}},\tilde{\phi_{t}})$ is an equivariant extension of $(\mu_{t},\nu_{t},\phi_{t})$ of order $n+1$.

∎

Corollary 4.1**.**

If $H^{3}_{G}(\phi,\phi)=0$, then every 2-cocycle in $C^{2}_{G}(\phi,\phi)$ is an infinitesimal of some equivariant deformation of $\phi.$

5 Equivalence of equivariant deformations, and rigidity

Let $(\mu_{t},\nu_{t},\phi_{t})$ and $(\tilde{\mu_{t}},\tilde{\nu_{t}},\tilde{\phi_{t}})$ be two equivariant deformations of $\phi$. Recall from [4] that an equivariant formal isomorphism between the equivariant deformations $\mu_{t}$ and $\tilde{\mu_{t}}$ of an associative algebra A is a $k[[t]]$-linear G-automorphism $\Psi_{t}:A[[t]]\to A[[t]]$ of the form $\Psi_{t}=\sum_{i\geq 0}\psi_{i}t^{i}$, where each $\psi_{i}$ is an equivariant $k$-linear map $A\to A$, $\psi_{0}(a)=a$, for all $a\in A$ and $\tilde{\mu_{t}}(\Psi_{t}(a),\Psi_{t}(b))=\Psi_{t}\mu_{t}(a,b),$ for all $a,b\in A$

Definition 5.1**.**

An equivariant formal isomorphism from $(\mu_{t},\nu_{t},\phi_{t})$ to $(\tilde{\mu_{t}},\tilde{\nu_{t}},\tilde{\phi_{t}})$ is a pair $(\Psi_{t},\Theta_{t})$, where $\Psi_{t}:A[[t]]\to A[[t]]$ and $\Theta_{t}:B[[t]]\to B[[t]]$ are equivariant formal isomorphisms from $\mu_{t}$ to $\tilde{\mu_{t}}$ and $\nu_{t}$ to $\tilde{\nu_{t}}$, respectively, such that

[TABLE]

Two equivariant deformations $(\mu_{t},\nu_{t},\phi_{t})$ and $(\tilde{\mu_{t}},\tilde{\nu_{t}},\tilde{\phi_{t}})$ are said to be equivalent if there exists an equivariant formal isomorphism $(\Psi_{t},\Theta_{t})$ from $(\mu_{t},\nu_{t},\phi_{t})$ to $(\tilde{\mu_{t}},\tilde{\nu_{t}},\tilde{\phi_{t}})$.

Definition 5.2**.**

Any equivariant deformation of $\phi:A\to B$ that is equialent to the deformation $(\mu_{0},\nu_{0},\phi)$ is said to be a trivial deformation.

Theorem 5.3**.**

The cohomology class of the infinitesimal of an equivariant deformation $(\mu_{t},\nu_{t},\phi_{t})$ of $\phi:A\to B$ is determined by the equivalence class of $(\mu_{t},\nu_{t},\phi_{t})$.

Proof.

Let $(\Psi_{t},\Theta_{t})$ from $(\mu_{t},\nu_{t},\phi_{t})$ to $(\tilde{\mu_{t}},\tilde{\nu_{t}},\tilde{\phi_{t}})$ be an equivariant formal isomorphism. So, we have $\tilde{\mu_{t}}\Psi_{t}=\Psi_{t}\circ\mu_{t},$ $\tilde{\nu_{t}}\Theta_{t}=\Theta_{t}\circ\nu_{t},$ and $\tilde{\phi_{t}}\circ\Psi_{t}=\Theta_{t}\circ\phi_{t}.$ This implies that $\mu_{1}-\tilde{\mu_{1}}=\delta\psi_{1}$, $\nu_{1}-\tilde{\nu_{1}}=\delta\theta_{1}$ and $\phi_{1}-\tilde{\phi_{1}}=\phi\psi_{1}-\theta_{1}\phi$. So we have $d^{1}(\psi_{1},\theta_{1},0)=(\mu_{1},\nu_{1},\phi_{1})-(\tilde{\mu_{1}},\tilde{\nu_{1}},\tilde{\phi_{1}}).$ This finishes the proof. ∎

Definition 5.3**.**

An equivariant associative algebra morphism $\phi:A\to B$ is said to be rigid if every deformation of $\phi$ is trivial.

Theorem 5.4**.**

A non-trivial equivariant deformation of an associative algebra morphism is equivalent to an equivariant deformation whose n-infinitesimal is not a coboundary, for some $n\geq 1.$

Proof.

Let $(\mu_{t},\nu_{t},\phi_{t})$ be an equivariant deformation of $\phi$ with n-infinitesimal $(\mu_{n},\nu_{n},\phi_{n})$, for some $n\geq 1.$ Assume that there exists a 1-cochain $(\psi,\theta,m)\in C_{G}^{1}(\phi,\phi)$ with $d(\psi,\theta,m)=(\mu_{n},\nu_{n},\phi_{n}).$ Since $d(\psi,\theta,m)=d(\psi,\theta+\delta m,0)$, without any loss of generality we may assume $m=0.$ This gives $\mu_{n}=\delta\psi$, $\nu_{n}=\delta\theta$, $\phi_{n}=\phi\psi-\theta\phi$. Take $\Psi_{t}=Id_{A}+\psi t^{n}$, $\Theta_{t}=Id_{B}=\theta t^{n}$. Define $\tilde{\mu_{t}}=\Psi_{t}\circ\mu_{t}\circ\Psi_{t}^{-1},$ $\tilde{\nu_{t}}=\Theta_{t}\circ\nu_{t}\circ\Theta_{t}^{-1},$ and $\tilde{\phi_{t}}=\Theta_{t}\circ\phi_{t}\circ\Psi_{t}^{-1}$. Clearly, $(\tilde{\mu_{t}},\tilde{\nu_{t}},\tilde{\phi_{t}})$ is an equivariant deformation of $\phi$ and $(\Psi_{t},\Theta_{t})$ is an equivariant formal isomorphism from $(\mu_{t},\nu_{t},\phi_{t})$ to $(\tilde{\mu_{t}},\tilde{\nu_{t}},\tilde{\phi_{t}})$. For $u,v\in A,$ we have $\tilde{\mu_{t}}(\Psi_{t}u,\Psi_{t}v)=\Psi_{t}(\mu_{t}(u,v)),$ which implies $\tilde{\mu_{i}}=0,$ for $1\leq i\leq n.$ For $u,v\in B$, we have $\tilde{\nu_{t}}(\Theta_{t}u,\Theta v)=\Theta_{t}(\nu_{t}(u,v)),$ which implies $\tilde{\nu_{i}}=0,$ for $1\leq i\leq n.$ For $u\in A,$ we have $\tilde{\phi_{t}}(\Psi_{t}u)=\Theta_{t}(\phi_{t}u),$ which gives $\phi_{i}=0,$ for $1\leq i\leq n.$ So $(\tilde{\mu_{t}},\tilde{\nu_{t}},\tilde{\phi_{t}})$ is equivalent to the given deformation and $(\tilde{\mu_{i}},\tilde{\nu_{i}},\tilde{\phi_{i}})=0,$ for $1\leq i\leq n.$ We can repeat the arguement to get rid off any infinitesimal that is a coboundary. So the process must stop if the deformation is nontrivial. ∎

An immediate consequence of the Theorem 5.4 is following corollary.

Corollary 5.1**.**

If $H_{G}^{2}(\phi,\phi)=0,$ then $\phi:A\to B$ is rigid.

From Proposition 3.2 and Theorem 5.4, we conclude following Corollary.

Corollary 5.2**.**

If $H_{G}^{2}(A,A)=0,$ $H_{G}^{2}(B,B)=0,$ and $H_{G}^{1}(A,B)=0,$ then $\phi:A\to B$ is rigid.