# Symmetry criteria for Hochschild extensions

Yang Han

(KLMM, Academy of Mathematics and Systems Science, Chinese Academy of
Sciences,
Beijing 100190, China.
School of Mathematical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China.
E-mail<EMAIL_ADDRESS>
###### Abstract

We give two sufficient and necessary conditions for the Hochschild extension
of a finite dimensional algebra by its dual bimodule and a Hochschild
2-cocycle to be a symmetric algebra.

Mathematics Subject Classification (2020): 16G10, 16E40

Keywords : Hochschild extension, symmetric algebra, Hochschild (co)homology,
cyclic (co)homology, Connes boundary operator, contraction operator.

##  1 Introduction

Throughout this paper, $k$ is a fixed field, $\otimes:=\otimes_{k}$, and
$(-)^{*}:=\mathrm{Hom}_{k}(-,k)$. Let $A$ be a finite dimensional $k$-algebra.
Then $A^{*}$ is a natural $A$-bimodule whose $A$-bimodule structure is given
by $(cfa)(b):=f(abc)$ for all $f\in A^{*}$ and $a,b,c\in A$. The trivial
extension of $A$ is the finite dimensional $k$-algebra $T(A):=A\oplus A^{*}$
with multiplication $(a,f)(b,g):=(ab,ag+fb)$, which is always a symmetric
algebra, i.e., $T(A)\cong T(A)^{*}$ as $T(A)$-bimodules, and plays a quite
important role in the representation theory of finite dimensional algebras
[5]. Let $\alpha:A\otimes A\to A^{*}$ be a Hochschild 2-cocycle, i.e., a
$k$-linear map satisfying Hochschild 2-cocycle condition $a\alpha(b\otimes
c)-\alpha(ab\otimes c)+\alpha(a\otimes bc)-\alpha(a\otimes b)c=0$ for all
$a,b,c\in A$. Then the Hochschild extension $T(A,\alpha):=A\oplus A^{*}$ with
multiplication $(a,f)(b,g):=(ab,ag+fb+\alpha(a\otimes b))$ is always a
selfinjective algebra [10, Proposition 1.2], but unlike trivial extension it
is unnecessary to be a symmetric algebra [8, Proposition 2.8]. In [8, Theorem
2.2], Ohnuki-Takeda-Yamagata gave a sufficient condition for a Hochschild
extension $T(A,\alpha)$ to be a symmetric algebra, but the condition is not
necessary [8, Proposition 2.9]. In [6, Theorem 3.1], Itagaki gave a more
general sufficient condition than Ohnuki-Takeda-Yamagata’s, but the condition
is not necessary either. Moreover, in [4, Theorem 2] we provided a sufficient
condition for a Hochschild extension of a finite dimensional dg $k$-algebra to
be an $n$-symmetric $A_{\infty}$-algebra.

In this paper, we will give two sufficient and necessary conditions for a
Hochschild extension $T(A,\alpha)$ to be a symmetric algebra. Denote by $Z(A)$
the center of a $k$-algebra $A$ and $U(A)$ the group of units (=invertible
elements) in $A$. Let $\widetilde{}:\mathrm{Hom}_{k}(A^{\otimes
2},A^{*})\to(A^{\otimes 3})^{*},\alpha\mapsto\widetilde{\alpha}$, be the
canonical adjunction isomorphism where $\widetilde{\alpha}$ is given by
$\widetilde{\alpha}(a\otimes b\otimes c):=\alpha(b\otimes c)(a)$ for all
$a,b,c\in A$. The main result of this paper is the following theorem.

######  Theorem.

Let $A$ be a finite dimensional $k$-algebra and
$\alpha\in\mathrm{Hom}_{k}(A\otimes A,A^{*})$ a Hochschild 2-cocycle. Then the
following three statements are equivalent:

(1) The Hochschild extension $T(A,\alpha)$ is a symmetric algebra.

(2) There exist $c\in Z(A)\cap U(A)$ and $h\in A^{*}$ such that

$\alpha(a\otimes b)(c)-\alpha(b\otimes a)(c)+h(ab-ba)=0$

for all $a,b\in A$.

(3) There exists $c\in Z(A)\cap U(A)$ such that the Hochschild cohomology
class $[i_{c}^{*}(\widetilde{\alpha})]\in H^{2}(C_{\bullet}(A)^{*})\cong
HH_{2}(A)^{*}\cong HH^{2}(A,A^{*})$ can be lifted to a cyclic cohomology class
in $HC^{2}(A)\cong HC_{2}(A)^{*}$ along the canonical map $I^{2}:HC^{2}(A)\to
H^{2}(C_{\bullet}(A)^{*})$ in the Connes periodicity exact sequence.

######  Remark.

Let $A=kQ/I$ be a bound quiver algebra where $Q$ is a finite quiver and $I$ is
an admissible ideal of the path algebra $kQ$ of $Q$ (Ref. [1]), and
$K:=kQ_{0}$ where $Q_{0}$ is the vertex set of $Q$. Since $K\cong k^{|Q_{0}|}$
is separable, the homology of the $K$-relative Hochschild chain complex of $A$

$\cdots\xrightarrow{b_{3}}A\otimes_{K^{e}}A^{\otimes_{K}2}\xrightarrow{b_{2}}A\otimes_{K^{e}}A\xrightarrow{b_{1}}A\otimes_{K^{e}}K\to
0$

is isomorphic to the Hochschild homology of $A$, and the cohomology of the
$K$-relative Hochschild cochain complex of $A$ with coefficients in
$A$-bimodule $A^{*}$

$0\to\mathrm{Hom}_{K^{e}}(K,A^{*})\xrightarrow{\delta^{0}}\mathrm{Hom}_{K^{e}}(A,A^{*})\xrightarrow{\delta^{1}}\mathrm{Hom}_{K^{e}}(A^{\otimes_{K}2},A^{*})\xrightarrow{\delta^{2}}\cdots$

is isomorphic to the Hochschild cohomology of $A$ with coefficients in
$A$-bimodule $A^{*}$ (Ref. [2, Section 1]). For any Hochschild 2-cocycle
$\alpha\in\mathrm{Hom}_{K^{e}}(A^{\otimes_{K}2},A^{*})$ and Hochschild
extension $T(A,\alpha)$, we can obtain an almost the same theorem as above in
which the condition (2) with $h=0$ is just Itagaki’s sufficient condition in
[6, Theorem 3.1] and the condition (2) with $h=0$ and $c=1$ is just Ohnuki-
Takeda-Yamagata’s sufficient condition in [8, Theorem 2.2].

##  2 Preliminaries

In this part, we fix some terminologies and notations.

Hochschild homology and cyclic homology. Let $A$ be a (unital associative)
$k$-algebra. The Hochschild chain complex of $A$ is the $k$-vector space chain
complex

$C_{\bullet}(A)=(A^{\otimes\bullet+1},b_{\bullet}):\ \cdots\to A^{\otimes
n+2}\xrightarrow{b_{n+1}}A^{\otimes n+1}\xrightarrow{b_{n}}A^{\otimes
n}\to\cdots\to A^{\otimes 2}\xrightarrow{b_{1}}A$

where the Hochschild boundary operator $b_{n}:A^{\otimes n+1}\to A^{\otimes
n}$ is given by

$\begin{array}[]{ll}b_{n}(a_{0}\otimes a_{1}\otimes\cdots\otimes
a_{n}):=&\sum\limits_{i=0}^{n-1}(-1)^{i}\ a_{0}\otimes
a_{1}\otimes\cdots\otimes a_{i}a_{i+1}\otimes\cdots\otimes
a_{n}\\\\[8.53581pt] &\quad\quad+(-1)^{n}\ a_{n}a_{0}\otimes
a_{1}\otimes\cdots\otimes a_{n-1}.\end{array}$

The $n$-th Hochschild homology $HH_{n}(A)$ of $A$ is the $n$-th homology of
$C_{\bullet}(A)$.

The cyclic bicomplex of $A$ is the $k$-vector space bicomplex

$\textstyle{A^{\otimes
4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{3}}$$\textstyle{A^{\otimes
3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{2}}$$\scriptstyle{B_{2}}$$\textstyle{A^{\otimes
2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{1}}$$\scriptstyle{B_{1}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{0}}$$\textstyle{A^{\otimes
3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{2}}$$\textstyle{A^{\otimes
2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{1}}$$\scriptstyle{B_{1}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{0}}$$\textstyle{A^{\otimes
2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{1}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{0}}$$\textstyle{A}$

where $b_{n}$ is the Hochschild boundary operator and $B_{n}:A^{\otimes
n+1}\to A^{\otimes n+2}$ is the Connes boundary operator given by

$\begin{array}[]{ll}B_{n}(a_{0}\otimes\cdots\otimes
a_{n}):=&\sum\limits_{i=0}^{n}(-1)^{ni}\ (1\otimes a_{i}\otimes\cdots\otimes
a_{n}\otimes a_{0}\otimes\cdots\otimes a_{i-1}\\\\[5.69054pt]
&\quad\quad\quad\quad-a_{i}\otimes 1\otimes a_{i+1}\otimes\cdots\otimes
a_{n}\otimes a_{0}\otimes\cdots\otimes a_{i-1}).\end{array}$

In particular, $B_{1}(a_{0}\otimes a_{1})=1\otimes a_{0}\otimes a_{1}-1\otimes
a_{1}\otimes a_{0}-a_{0}\otimes 1\otimes a_{1}+a_{1}\otimes 1\otimes a_{0}$.
The $n$-th cyclic homology $HC_{n}(A)$ of $A$ is the $n$-th homology of the
total complex of the cyclic bicomplex of $A$. The Connes periodicity exact
sequence is the long exact sequence

$\begin{array}[]{l}\cdots\xrightarrow{B_{n-1}}HH_{n}(A)\xrightarrow{I_{n}}HC_{n}(A)\xrightarrow{S_{n}}HC_{n-2}(A)\xrightarrow{B_{n-2}}HH_{n-1}(A)\xrightarrow{I_{n-1}}\\\\[5.69054pt]
\cdots\xrightarrow{B_{0}}HH_{1}(A)\xrightarrow{I_{1}}HC_{1}(A)\xrightarrow{S_{1}}0\xrightarrow{}HH_{0}(A)\xrightarrow{I_{0}}HC_{0}(A)\xrightarrow{}0\end{array}$

where $I_{n}$ is induced by the injection of Hochschild chain complex in the
first column of cyclic bicomplex, $B_{n}$ is induced by Connes boundary
operator, and $S_{n}$ is the periodicity map induced by shifting (Ref. [7,
2.2.1]).

Hochschild cohomology and cyclic cohomology. Let $A$ be a $k$-algebra and $M$
an $A$-bimodule. The Hochschild cochain complex of $A$ with coefficients in
$M$ is the $k$-vector space cochain complex
$C^{\bullet}(A,M)=(\mathrm{Hom}_{k}(A^{\otimes\bullet},M),\delta^{\bullet})$:

$0\to
M\xrightarrow{\delta^{0}}\mathrm{Hom}_{k}(A,M)\xrightarrow{\delta^{1}}\mathrm{Hom}_{k}(A^{\otimes
2},M)\xrightarrow{\delta^{2}}\mathrm{Hom}_{k}(A^{\otimes 3},M)\to\cdots$

where the Hochschild coboundary operators $\delta^{0}$ is given by
$\delta^{0}(m)(a):=am-ma$ and $\delta^{n}:\mathrm{Hom}_{k}(A^{\otimes
n},M)\to\mathrm{Hom}_{k}(A^{\otimes n+1},M)$ is given by
$\delta^{n}(\alpha)(a_{1}\otimes\cdots\otimes
a_{n+1}):=a_{1}\alpha(a_{2}\otimes\cdots\otimes
a_{n+1})+\sum\limits_{i=1}^{n}(-1)^{i}\ \alpha(a_{1}\otimes\cdots\otimes
a_{i}a_{i+1}\otimes\cdots\otimes a_{n+1})+(-1)^{n+1}\
\alpha(a_{1}\otimes\cdots\otimes a_{n})a_{n+1}$ for all $n\geq 1$. In
particular, $\delta^{2}(\alpha)(a_{1}\otimes a_{2}\otimes
a_{3})=a_{1}\alpha(a_{2}\otimes a_{3})-\alpha(a_{1}a_{2}\otimes
a_{3})+\alpha(a_{1}\otimes a_{2}a_{3})-\alpha(a_{1}\otimes a_{2})a_{3}$ for
all $a_{1},a_{2},a_{3}\in A$. The $n$-th Hochschild cohomology $HH^{n}(A,M)$
of $A$ with coefficients in $M$ is the $n$-th cohomology of the Hochschild
cochain complex $C^{\bullet}(A,M)$ of $A$ with coefficients in $M$. The $n$-th
Hochschild cohomology $HH^{n}(A)$ of $A$ is just $HH^{n}(A,A)$.

Applying the $k$-dual functor $(-)^{*}$ to the Hochschild chain complex
$C_{\bullet}(A)=(A^{\otimes\bullet+1},b_{\bullet})$ of $A$, we obtain the
cochain complex
$C_{\bullet}(A)^{*}=((A^{\otimes\bullet+1})^{*},b_{\bullet}^{*})$. According
to the canonical adjunction isomorphism $(A^{\otimes
n+1})^{*}\cong\mathrm{Hom}_{k}(A^{\otimes n},A^{*})$, the Hochschild cochain
complex
$C^{\bullet}(A,A^{*})=(\mathrm{Hom}_{k}(A^{\otimes\bullet},A^{*}),\delta^{\bullet})$
of $A$ with coefficients in $A^{*}$ is isomorphic to the cochain complex
$C_{\bullet}(A)^{*}=((A^{\otimes\bullet+1})^{*},b_{\bullet}^{*})$. In
particular, $H^{n}(C_{\bullet}(A)^{*})\cong HH_{n}(A)^{*}\cong
HH^{n}(A,A^{*})$ for all $n\in\mathbb{N}_{0}$.

Applying the $k$-dual functor $(-)^{*}$ to the cyclic bicomplex of $A$, we
obtain the cyclic cochain bicomplex of $A$

$\textstyle{(A^{\otimes
4})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{2}^{*}}$$\textstyle{(A^{\otimes
3})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{1}^{*}}$$\textstyle{(A^{\otimes
2})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{0}^{*}}$$\textstyle{A^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(A^{\otimes
3})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{3}^{*}}$$\scriptstyle{B_{1}^{*}}$$\textstyle{(A^{\otimes
2})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{2}^{*}}$$\scriptstyle{B_{0}^{*}}$$\textstyle{A^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{1}^{*}}$$\textstyle{(A^{\otimes
2})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{2}^{*}}$$\scriptstyle{B_{0}^{*}}$$\textstyle{A^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{1}^{*}}$$\textstyle{A^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{1}^{*}}$

The $n$-th cyclic cohomology $HC^{n}(A)$ of $A$ is the $n$-th cohomology of
the total complex of the cyclic cochain bicomplex of $A$ (Ref. [7, 2.4]).
Clearly, $HC^{n}(A)\cong HC_{n}(A)^{*}$ for all $n\in\mathbb{N}_{0}$. From the
cyclic cochain bicomplex of $A$ we can obtain the Connes periodicity exact
sequence (cohomology form)

$\begin{array}[]{l}0\xrightarrow{}HC^{0}(A)\xrightarrow{I^{0}}H^{0}(C_{\bullet}(A)^{*})\xrightarrow{}0\xrightarrow{S^{1}}HC^{1}(A)\xrightarrow{I^{1}}H^{1}(C_{\bullet}(A)^{*})\xrightarrow{B^{0}}\cdots\\\\[5.69054pt]
\xrightarrow{I^{n-1}}H^{n-1}(C_{\bullet}(A)^{*})\xrightarrow{B^{n-2}}HC^{n-2}(A)\xrightarrow{S^{n}}HC^{n}(A)\xrightarrow{I^{n}}H^{n}(C_{\bullet}(A)^{*})\xrightarrow{}\cdots\end{array}$

where $I^{n}$ is induced by the projection of the cyclic cochain bicomplex to
the first column, $B^{n}$ is induced by the dual $B_{n}^{*}$ of the Connes
boundary operator $B_{n}$, and $S^{n}$ is the periodicity map induced by
shifting. In particular, $I^{2}([(\beta,\gamma)])=[\beta]$ where
$(\beta,\gamma)\in(A^{\otimes 3})^{*}\oplus A^{*}$ is any cyclic 2-cocycle and
$\beta\in(A^{\otimes 3})^{*}$ is a Hochschild 2-cocycle.

Hochschild extensions. Let $A$ be a $k$-algebra, $M$ an $A$-bimodule, and
$\alpha:A\otimes A\to M$ a Hochschild 2-cocycle, i.e., a $k$-linear map
satisfying the Hochschild 2-cocycle condition $a\alpha(b\otimes
c)-\alpha(ab\otimes c)+\alpha(a\otimes bc)-\alpha(a\otimes b)c=0$ for all
$a,b,c\in A$. The Hochschild extension $T(A,M,\alpha)$ of $A$ by $M$ and
$\alpha$ is the $k$-algebra $A\oplus M$ with multiplication
$(a,m)(a^{\prime},m^{\prime}):=(aa^{\prime},am^{\prime}+ma^{\prime}+\alpha(a\otimes
a^{\prime}))$ (Ref. [7, 1.5.3]). In the case of $M=A^{*}$, we denote by
$T(A,\alpha)$ the Hochschild extension $T(A,A^{*},\alpha)$ for short.

Contraction operator. Let $A$ be a $k$-algebra. For any Hochschild $m$-cochain
${\alpha}\in C^{m}(A):=C^{m}(A,A)$, the contraction operator $i_{\alpha}$ on
the Hochschild chain complex $C_{\bullet}(A)$ is the graded morphism
$i_{\alpha}:C_{\bullet}(A)\to C_{\bullet}(A)$ of degree $-m$ given by

$i_{\alpha}(a_{0}\otimes a_{1}\otimes\cdots\otimes
a_{n}):=a_{0}{\alpha}(a_{1}\otimes\cdots\otimes
a_{m})\otimes(a_{m+1}\otimes\cdots\otimes a_{n}).$

Note that $bi_{\alpha}-i_{\alpha}b=i_{\delta\alpha}$ where $b$ is the
Hochschild boundary operator and $\delta$ is the Hochschild coboundary
operator. Consequently, if $\alpha$ is a Hochschild cocycle then the
contraction operator $i_{\alpha}$ on $C_{\bullet}(A)$ induces an operator
$i_{\alpha}$ on $HH_{\bullet}(A)$. In particular, for any $c\in
Z(A)=HH^{0}(A)\subseteq C^{0}(A)=A$, $i_{c}$ is an operator on
$C_{\bullet}(A)$ which is given by $i_{c}(a_{0}\otimes
a_{1}\otimes\cdots\otimes a_{n}):=a_{0}c\otimes a_{1}\otimes\cdots\otimes
a_{n}$. Moreover, $b_{n}i_{c}=i_{c}b_{n}$ for all $n\geq 1$. For more
properties of contraction operators in noncommutative differential calculus
theory, we refer to [9, 3].

##  3 The proof of Theorem

Now we start the proof of Theorem. Write $T(A,\alpha)$ as $T$ for short. Any
element in $T^{*}=\mathrm{Hom}_{k}(A\oplus A^{*},k)\cong A^{*}\oplus
A^{**}\cong A^{*}\oplus A$ can be written as $(f,a)$ where $f\in A^{*},a\in
A$, and the $k$-linear form $(f,a):T\to k$ is given by
$(f,a)((b,g)):=f(b)+g(a)$ for all $(b,g)\in T$.

(1)$\Rightarrow$(2): Assume that $T$ is a symmetric algebra. Then there exists
a $T$-bimodule isomorphism $\phi:T\to T^{*}$. Let $\phi(1_{T})=(h,c)\in T^{*}$
where $h\in A^{*}$ and $c\in A$.

Claim 1. For all $(a,f),(b,g)\in T$, we have

$\alpha(a\otimes b)(c)-\alpha(b\otimes a)(c)+h(ab-ba)=0,\ f(bc)=f(cb),\
g(ac)=g(ca).$

Indeed, since $\phi$ is a $T$-bimodule morphism, for any $(a,f)\in T$, we have

$\phi(a,f)=(a,f)\phi(1_{T})=\phi(1_{T})(a,f)$

which maps any $(b,g)\in T$ to

$\phi(a,f)((b,g))=((a,f)\phi(1_{T}))((b,g))=(\phi(1_{T})(a,f))((b,g)).$

By the canonical $T$-bimodule structure of $T^{*}$, we get

$\phi(a,f)((b,g))=\phi(1_{T})((b,g)(a,f))=\phi(1_{T})((a,f)(b,g)).$

According to the multiplication of $T$, we have

$\phi(a,f)((b,g))=(h,c)((ba,bf+ga+\alpha(b\otimes
a)))=(h,c)((ab,ag+fb+\alpha(a\otimes b))).$

From the definition of $(h,c)\in T^{*}$, we obtain

$\begin{array}[]{ll}\phi(a,f)((b,g))&=h(ba)+(bf)(c)+(ga)(c)+\alpha(b\otimes
a)(c)\\\ &=h(ab)+(ag)(c)+(fb)(c)+\alpha(a\otimes b)(c).\end{array}$

By the canonical $T$-bimodule structure of $T^{*}$, we get

$\begin{array}[]{ll}\phi(a,f)((b,g))&=h(ba)+f(cb)+g(ac)+\alpha(b\otimes
a)(c)\\\ &=h(ab)+g(ca)+f(bc)+\alpha(a\otimes b)(c).\end{array}$

Taking $a=0$, we obtain $f(bc)=f(cb)$ for all $b\in A$ and $f\in A^{*}$.
Taking $b=0$, we get $g(ac)=g(ca)$ for all $a\in A$ and $g\in A^{*}$.
Furthermore, for all $(a,f),(b,g)\in T$, we have $\alpha(a\otimes
b)(c)-\alpha(b\otimes a)(c)+h(ab-ba)=0.$

Claim 2. $c\in Z(A)$.

By Claim 1, we have $f(bc-cb)=0$ for all $b\in A$ and $f\in A^{*}$. Thus $bc-
cb=0$ for all $b\in A$. Hence $c\in Z(A)$.

Claim 3. $c\in U(A)$.

Assume on the contrary $c\notin U(A)$. It follows from Claim 2 that the two-
sided ideal $AcA$ of $A$ generated by $c$ is equal to both $Ac$ and $cA$.
However, it is not equal to $A$ due to $c\notin U(A)$. Thus the complement
space $(Ac)^{\perp}$ of $Ac$ in $A$ is not zero. Since
$A=Ac\oplus(Ac)^{\perp}$, we have $A^{*}=(Ac)^{*}\oplus((Ac)^{\perp})^{*}$.
Thus there exists $f\in((Ac)^{\perp})^{*}\subseteq A^{*}$ such that $f\neq 0$
and $f(Ac)=0$. For any $(b,g)\in T$, as in the proof of Claim 1, we have
$\phi(0,f)((b,g))=h(0b)+g(c0)+f(bc)+\alpha(0\otimes b)(c)=f(bc)\in f(Ac)=0$.
So $\phi(0,f)=0$. Since $\phi$ is a $T$-bimodule isomorphism, we have
$(0,f)=(0,0)$, and further $f=0$. It is a contradiction.

Finally, it follows from Claim 1, 2 and 3 that there exist $c\in Z(A)\cap
U(A)$ and $h\in A^{*}$ such that $\alpha(a\otimes b)(c)-\alpha(b\otimes
a)(c)+h(ab-ba)=0$ for all $a,b\in A$.

(2)$\Rightarrow$(3): Assume that there exists $c\in Z(A)\cap U(A)$ such that
$\alpha(a\otimes b)(c)-\alpha(b\otimes a)(c)+h(ab-ba)=0$ for all $a,b\in A$.
From Hochschild 2-cocycle condition, we obtain $\alpha(1\otimes
ab)=\alpha(1\otimes a)b$ for all $a,b\in A$. Due to $c\in Z(A)$, we have

$\begin{array}[]{ll}&(B_{1}^{*}(i_{c}^{*}(\widetilde{\alpha})))(a\otimes b)\\\
=&(\widetilde{\alpha}i_{c}B_{1})(a\otimes b)\\\
=&(\widetilde{\alpha}i_{c})(1\otimes a\otimes b-1\otimes b\otimes a-a\otimes
1\otimes b+b\otimes 1\otimes a)\\\ =&\widetilde{\alpha}(c\otimes a\otimes
b-c\otimes b\otimes a-ac\otimes 1\otimes b+bc\otimes 1\otimes a)\\\
=&\widetilde{\alpha}(c\otimes a\otimes b)-\widetilde{\alpha}(c\otimes b\otimes
a)-\widetilde{\alpha}(ac\otimes 1\otimes b)+\widetilde{\alpha}(bc\otimes
1\otimes a)\\\ =&\alpha(a\otimes b)(c)-\alpha(b\otimes a)(c)-\alpha(1\otimes
b)(ac)+\alpha(1\otimes a)(bc)\\\ =&-h(ab-ba)+(\alpha(1\otimes 1)c)(ab-ba)\\\
=&-(h-\alpha(1\otimes 1)c)(ab-ba)\\\ =&-((h-\alpha(1\otimes
1)c)b_{1})(a\otimes b)\\\ =&-b_{1}^{*}(h-\alpha(1\otimes 1)c)(a\otimes
b).\end{array}$

Let $h^{\prime}=h-\alpha(1\otimes 1)c\in A^{*}$. Then
$B_{1}^{*}(i_{c}^{*}(\widetilde{\alpha}))+b_{1}^{*}(h^{\prime})=0$. Thus the
Hochschild cohomology class $[i_{c}^{*}(\widetilde{\alpha})]\in
H^{2}(C_{\bullet}(A)^{*})$ can be lifted to the cyclic cohomology class
$[(i_{c}^{*}(\widetilde{\alpha}),h^{\prime})]\in HC^{2}(A)$ along the
canonical map $I^{2}:HC^{2}(A)\to H^{2}(C_{\bullet}(A)^{*})$.

(3)$\Rightarrow$(2): Assume that there exists $c\in Z(A)\cap U(A)$ such that
the Hochschild cohomology class $[i_{c}^{*}(\widetilde{\alpha})]\in
H^{2}(C_{\bullet}(A)^{*})$ can be lifted to a cyclic cohomology class
$[(\alpha^{\prime},h^{\prime})]\in HC^{2}(A)$ along the canonical map
$I^{2}:HC^{2}(A)\to H^{2}(C_{\bullet}(A)^{*})$ where
$\alpha^{\prime}\in(A^{\otimes 3})^{*}$ and $h^{\prime}\in A^{*}$, i.e.,
$I^{2}([(\alpha^{\prime},h^{\prime})])=[\alpha^{\prime}]=[i_{c}^{*}(\widetilde{\alpha})]$.
Then $b_{3}^{*}(\alpha^{\prime})=0$,
$B_{1}^{*}(\alpha^{\prime})+b_{1}^{*}(h^{\prime})=0$, i.e.,
$\alpha^{\prime}B_{1}+h^{\prime}b_{1}=0$, and
$[\alpha^{\prime}]=[i_{c}^{*}(\widetilde{\alpha})]\in
H^{2}(C_{\bullet}(A)^{*})$. Thus there is $\beta\in(A^{\otimes 2})^{*}$ such
that $i_{c}^{*}(\widetilde{\alpha})=\alpha^{\prime}+b_{2}^{*}(\beta)$, i.e.,
$\widetilde{\alpha}i_{c}=\alpha^{\prime}+\beta b_{2}$. So
$\widetilde{\alpha}i_{c}B_{1}=\alpha^{\prime}B_{1}+\beta
b_{2}B_{1}=-h^{\prime}b_{1}-\beta B_{0}b_{1}=-(h^{\prime}+\beta B_{0})b_{1}$.
As in the proof of (2)$\Rightarrow$(3), we have $\alpha(a\otimes
b)(c)-\alpha(b\otimes a)(c)+(h^{\prime}+\beta B_{0}+\alpha(1\otimes 1)c)(ab-
ba)=0$ for all $a,b\in A$. Let $h=h^{\prime}+\beta B_{0}+\alpha(1\otimes
1)c\in A^{*}$. Then $\alpha(a\otimes b)(c)-\alpha(b\otimes a)(c)+h(ab-ba)=0$
for all $a,b\in A$.

(2)$\Rightarrow$(1): Assume that there exists $c\in Z(A)\cap U(A)$ and $h\in
A^{*}$ such that $\alpha(a\otimes b)(c)-\alpha(b\otimes a)(c)+h(ab-ba)=0$ for
all $a,b\in A$. Define a map $\phi:T\to T^{*}$ by
$\phi(a,f)((b,g)):=(h,c)((a,f)(b,g))$. Note that
$(h,c)((a,f)(b,g))=h(ab)+g(ca)+f(bc)+\alpha(a,b)(c)$ and
$(h,c)((b,g)(a,f))=h(ba)+f(cb)+g(ac)+\alpha(b,a)(c).$ Due to $c\in Z(A)$, we
get $f(bc)=f(cb)$ and $g(ac)=g(ca)$. Furthermore, for all $(a,f),(b,g)\in T$,
we have

$\phi(a,f)((b,g))=(h,c)((a,f)(b,g))=(h,c)((b,g)(a,f)),$

or equivalently,

$\phi(a,f)=(h,c)(a,f)=(a,f)(h,c).$

Now we prove that $\phi$ is a $T$-bimodule morphism. Indeed, for all
$(a^{\prime},f^{\prime}),\linebreak(a,f),(a^{\prime\prime},f^{\prime\prime})\in
T$, we have

$\begin{array}[]{ll}\phi((a^{\prime},f^{\prime})(a,f)(a^{\prime\prime},f^{\prime\prime}))&=(h,c)(a^{\prime},f^{\prime})(a,f)(a^{\prime\prime},f^{\prime\prime})\\\
&=(a^{\prime},f^{\prime})(h,c)(a,f)(a^{\prime\prime},f^{\prime\prime})\\\
&=(a^{\prime},f^{\prime})\phi(a,f)(a^{\prime\prime},f^{\prime\prime}).\end{array}$

Next we show that $\phi$ is a $T$-bimodule isomorphism. Since
$\mathrm{dim}_{k}T=\mathrm{dim}_{k}T^{*}<\infty$, it suffices to prove that
$\phi$ is injective. For this, suppose $\phi(a,f)=0$. We need to prove
$(a,f)=0$. Indeed, on one hand, for any $g\in A^{*}$, we have
$0=\phi(a,f)((0,g))=h(a0)+g(ca)+f(0c)+\alpha(a\otimes 0)(c)=g(ca)$. Thus
$ca=0$. Due to $c\in U(A)$, we get $a=0$. On the other hand, for any $b\in A$,
we have $0=\phi(0,f)((b,0))=h(0b)+0(c0)+f(bc)+\alpha(0\otimes b)(c)=f(bc)$.
Due to $c\in U(A)$, we obtain $f(A)=0$. Thus $f=0$.

Now we finish the proof of Theorem.

ACKNOWLEDGEMENT. The author is sponsored by Project 11971460 NSFC.

## References

  * [1] I. Assem, D. Simson and A. Skowroński, Elements of the representation theory of associative algebras, Vol. 1, Techniques of representation theory, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006.
  * [2] M. Gerstenhaber and S.D. Schack, Relative Hochschild cohomology, rigid algebras, and the Bockstein, J. Pure Appl. Algebra 43 (1986), 53–74.
  * [3] Y. Han, X. Liu and K. Wang, Hochschild (co)homologies of dg $K$-rings and their Koszul duals, arXiv:1810.05969v1 [math.KT].
  * [4] Y. Han, X. Liu and K. Wang, Exact Hochschild extensions and deformed Calabi-Yau completions, arXiv:1909.02200v1 [math.RA].
  * [5] D. Happel, Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lecture Notes Series 119, Cambridge University Press, 1988.
  * [6] T. Itagaki, Symmetric Hochschild extension algebras and normalized 2-cocycles, Arch. Math. 112 (2019), no. 3, 249–259.
  * [7] J.-L. Loday, Cyclic homology, The second edition, Grundlehren der mathematischen Wissenschaften 301, Springer-Verlag, Berlin Heidelberg, 1998.
  * [8] Y. Ohnuki, K. Takeda and K. Yamagata, Symmetric Hochschild extension algebras, Colloq. Math. 80 (1999), 155–174.
  * [9] D. Tamarkin and B. Tsygan, The ring of differential operators on forms in noncommutative calculus, In: Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math. 73, Amer. Math. Soc., Providence, RI 2005, 105–131.
  * [10] K. Yamagata, Extensions over hereditary artinian rings with self-dualities, I, J. Algebra 73 (1981), 386–433.