# Bracket structure on Hochschild cohomology of Koszul quiver algebras using
homotopy liftings

Tolulope Oke  Department of Mathematics, Texas A&M University, College
Station, Texas 77843, USA<EMAIL_ADDRESS>
###### Abstract.

We present the Gerstenhaber algebra structure on the Hochschild cohomology of
Koszul algebras defined by quivers and relations using the idea of homotopy
liftings. E.L. Green, G. Hartman, E.N. Marcos and Ø. Solberg provided a
canonical way of constructing a minimal projective bimodule resolution of a
Koszul quiver algebra. The resolution has a comultiplicative structure which
we use to define homotopy lifting maps. We first present a short example that
demonstrates the theory. We then study the Gerstenhaber algebra structure on
Hochschild cohomology of a family of bound quiver algebras, some members of
which are counterexamples to the Snashall-Solberg finite generation
conjecture. We give examples of homotopy lifting maps for degree $2$ and
degree $1$ cocycles and draw connections to derivation operators. As an
application, we describe Hochschild 2-cocycles satisfying the Maurer-Cartan
equation.

††Key words and phrases:  Hochschild cohomology, Gerstenhaber brackets, Koszul
algebras, quiver algebras, homotopy liftings, Maurer-Cartan Equation
MSC: 16E40; 16S37; 16W50.††Partially supported by NSF grant 1665286 and
2001163 during the summer semester of 2020 and Spring 2021.

##  1\. Introduction

The Hochschild cohomology $\operatorname{HH}^{*}(\Lambda)$ of an associative
algebra $\Lambda$ possesses a multiplicative map making it a graded
commutative ring. This map is known as the cup product. M. Gerstenhaber
introduced a bracket on $\operatorname{HH}^{*}(\Lambda)$ giving it a second
multiplicative structure. This bracket, now known as the Gerstenhaber bracket,
together with the cup product makes $\operatorname{HH}^{*}(\Lambda)$ into a
Gerstenhaber algebra. The bracket plays an important role in the theory of
deformation of algebras.

The bracket structure introduced by M. Gerstenhaber was defined using the bar
resolution. This definition is useful theoretically but not easily accessible
for computational purposes. For instance to compute the bracket on
$\operatorname{HH}^{*}(\Lambda)$ using an arbitrary resolution, appropriate
chain maps between the resolution and the bar resolution would have to be
constructed. Morphisms defined on the resolution would then have to be carried
over to the bar resolution and vice versa. This process is not always easy.

Several works have been carried out in interpreting the original definition of
the bracket given by M. Gerstenhaber using arbitrary resolutions. For example,
in 2004, B. Keller realized Hochschild cohomology as the Lie algebra of the
derived Picard group. M. Suárez-Álvarez showed in [10] that the bracket of a
degree 1 cocycle and any cocycle can be realized from derivation operators
associated to the degree 1 cocycles expressed on arbitrary projective
resolutions. In [6], C. Negron and S. Witherspoon introduced the idea of a
contracting homotopy for resolutions that are differential graded coalgebras.
In [7], Y. Volkov generalized the method introduced by C. Negron and S.
Witherspoon to arbitrary resolutions. He introduced the idea of homotopy
lifting by defining the bracket in terms of homotopy lifting maps on any
projective bimodule resolution.

In this work, we present the Gerstenhaber algebra structure on Hochschild
cohomology of Koszul algebras defined by quivers and relations using homotopy
lifting technique introduced by Y. Volkov in [7]. We construct explicit
examples of homotopy lifting maps for some cocycles. Our cocycles and homotopy
lifting maps are defined using the resolution $\mathbb{K}$ introduced in [4]
by E. L. Green, G. Hartman, E. N. Marcos and Ø. Solberg. We discuss the
construction of $\mathbb{K}$ in details in the Preliminaries section. There
are two necessary and sufficient conditions that make a map a homotopy lifting
map. For the resolution $\mathbb{K}$, the second condition is satisfied
because $\mathbb{K}$ is a differential graded coalgebra and more generally
because the algebra $\Lambda$ is Koszul.

One of the examples we use to illustrate the theory is a family of quiver
algebras. Some members of this family are counterexamples to the Snashall-
Solberg finite generation conjecture. The Snashall-Solberg finite generation
conjecture asserts that for finite dimensional algebras, Hochschild cohomology
modulo nilpotents is finitely generated. R. Hermann asked whether or not
Hochschild cohomology modulo the weak Gerstenhaber ideal generated by
nilpotent elements is finitely generated. The theory provided in this work
gives the necessary tools neeeded to address these questions.

Furthermore, there is an algorithmic approach to computing the resolution
$\mathbb{K}$, for instance in [3]. This means that perhaps, it might be
possible to employ our method to develop algorithms that can compute the
bracket on Hochschild cohomology for Koszul algebras defined by quivers and
relations.

Through out this article, $k$ is taken to be a field and $\otimes=\otimes_{k}$
unless otherwise specified. $\Lambda=kQ/I$ is taken to be a graded Koszul
quiver algebra, where $Q$ is a finite quiver and $I$ is an admissible ideal of
the quiver algebra $kQ$. For $R=kQ$, and suppose $\Lambda=\oplus_{i\geq
0}\Lambda_{i}$ is a grading of $\Lambda$, it was shown in [4] that there are
integers $t_{n}$ and uniform elements $\\{f^{n}_{i}\\}_{i=0}^{t_{n}}\in R$
such that the minimal projective resolution $\mathbb{L}\rightarrow\Lambda_{0}$
can be given in terms of filtration of right ideals involving the $f^{n}_{i}$.
It was shown that the elements $\\{f^{n}_{i}\\}_{i=0}^{t_{n}}$ satisfy a
“comultiplicative structure”, i.e. there are scalars $c_{pq}(n,i,r)$ such that
$f^{n}_{i}=\sum_{p=0}^{t_{r}}\sum_{q=0}^{t_{n-r}}c_{pq}(n,i,r)f^{r}_{p}f^{n-r}_{q}.$
From this comultiplicative equation, one constructs
$\mathbb{K}_{\bullet}\rightarrow\Lambda,$ a minimal projective
$\Lambda^{e}$-resolution of $\Lambda$, where for each $n$, the
$\Lambda^{e}$-module $\mathbb{K}_{n}$ has free basis elements
$\\{\varepsilon^{n}_{i}\\}_{i=0}^{t_{n}}$ [4]. The main results of this work
are the following:

  1. (i).

Theorems 4.1, 4.10, and 4.19 which presents definitions of homotopy lifting
maps for cocycles taking free basis elements $\varepsilon^{m}_{r}$ to an
idempotent, a path of length 1 and a path of length 2 respectively.

  2. (ii).

Theorem 4.20 providing a combinatorial description of the Gerstenhaber bracket
of any two Hochschild cochains and Theorem 4.21 which gives a general
Gerstenhaber algebra structure on Hochschild cohomology of Koszul quiver
algebras.

  3. (iii).

Theorem 5.3 specifying Hochschild 2-cocycles that satisfy the Maurer-Cartan
equation.

  4. (iv).

Proposition 7.3 showing examples of homotopy lifting maps for degree
$1$-cocycles that are also derivation operators when regarded as $k$-linear
maps. We raised Question 7.17 which asks whether the observation in
Proposition 7.3 is generally true.

The results of item (i) above assume certain conditions on the scalars
$c_{pq}(n,i,r)$ appearing in the multiplicative equation as well as some
scalars $b_{m,r}(m-n+1,s)$ coming from homotopy lifting maps. We present these
conditions in Equation (4). This article is organized in the following way:

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Comparison morphisms vs homotopy liftings
  4. 4 General bracket structure
  5. 5 Application
  6. 6 Short Example
  7. 7 Long Example
    1. 7.1 Homotopy lifting maps from degree 2 cocycles
    2. 7.2 Homotopy liftings from degree 1 cocycles
    3. 7.3 Connections to derivation operators

##  2\. Preliminaries

The Hochschild cohomology of an associative $k$-algebra $\Lambda$ was
originally defined using the following projective resolution known as the bar
resolution.

(2.1)
$\mathbb{B}_{\bullet}:\qquad\cdots\rightarrow\Lambda^{\otimes(n+2)}\xrightarrow{\;\delta_{n}\;}\Lambda^{\otimes(n+1)}\xrightarrow{\delta_{n-1}}\cdots\xrightarrow{\;\delta_{2}\;}\Lambda^{\otimes
3}\xrightarrow{\;\delta_{1}\;}\Lambda^{\otimes
2}\;(\;\xrightarrow{\mu}\Lambda)$

where $\mu$ is multiplication and the differentials $\delta_{n}$ are given by

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

for all $a_{0},a_{1},\ldots,a_{n+1}\in\Lambda$. This resolution consists of
$\Lambda$-bimodules or left modules over the enveloping algebra
$\Lambda^{e}=\Lambda\otimes\Lambda^{op}$, where $\Lambda^{op}$ is the opposite
algebra. The resolution is sometimes written
$\mathbb{B}_{\bullet}\xrightarrow{\mu}\Lambda$ with $\mu$ referred to as the
augmentation map. Let $M$ be a finitely generated left $\Lambda^{e}$-module,
the space of Hochschild $n$-cochains with coefficients in $M$ is obtained by
applying the functor $\operatorname{Hom}_{\Lambda^{e}}(-,M)$ to the complex
$\mathbb{B}_{\bullet}$, and then taking the cohomology of the resulting
cochain complex. We define

$\operatorname{HH}^{*}(\Lambda,M):=\bigoplus_{n\geq
0}\operatorname{HH}^{n}(\Lambda,M)=\bigoplus_{n\geq
0}\text{H}^{n}(\operatorname{Hom}_{\Lambda^{e}}(\mathbb{B}_{n},M))$

to be the Hochschild cohomology ring of $\Lambda$ with coefficients in $M$.
Letting $M=\Lambda$, we define
$\operatorname{HH}^{*}(\Lambda):=\operatorname{HH}^{*}(\Lambda,\Lambda)$ to be
the Hochschild cohomology of $\Lambda$. An element
$\chi\in\operatorname{Hom}_{\Lambda^{e}}(\mathbb{B}_{m},\Lambda)$ is a cocycle
if $(\delta^{*}(\chi))(\cdot):=\chi\delta(\cdot)=0.$ There is an isomorphism
of the abelian groups
$\operatorname{Hom}_{\Lambda^{e}}(\mathbb{B}_{m},\Lambda)\cong\operatorname{Hom}_{k}(\Lambda^{\otimes
m},\Lambda),$ so we can also view $\chi$ as an element of
$\operatorname{Hom}_{k}(\Lambda^{\otimes m},\Lambda)$.

The Gerstenhaber bracket of two cocycles
$\chi\in\operatorname{Hom}_{k}(\Lambda^{\otimes m},\Lambda)$ and
$\theta\in\operatorname{Hom}_{k}(\Lambda^{\otimes n},\Lambda)$ at the chain
level is given by

(2.3) $[\chi,\theta]=\chi\circ\theta-(-1)^{(m-1)(n-1)}\theta\circ\chi$

where $\chi\circ\theta=\sum_{j=1}^{m}(-1)^{(n-1)(j-1)}\chi\circ_{j}\theta$
with

$(\chi\circ_{j}\theta)(a_{1}\otimes\cdots\otimes a_{m+n-1})\\\
=\chi(a_{1}\otimes\cdots\otimes a_{j-1}\otimes\theta(a_{j}\otimes\cdots\otimes
a_{j+n-1})\otimes a_{j+n}\otimes\cdots\otimes a_{m+n-1}).$

This induces a well defined map
$[\cdot\;,\cdot]:\operatorname{HH}^{m}(\Lambda)\times\operatorname{HH}^{n}(\Lambda)\rightarrow\operatorname{HH}^{m+n-1}(\Lambda)$
on cohomology.

Quiver algebras: A quiver is a directed graph with the allowance of loops and
multiple arrows. A quiver $Q$ is sometimes denoted as a quadruple
$(Q_{0},Q_{1},o,t)$ where $Q_{0}$ is the set of vertices in $Q$, $Q_{1}$ is
the set of arrows in $Q$, and $o,t:Q_{1}\longrightarrow Q_{0}$ are maps which
assign to each arrow $a\in Q_{1}$, its origin vertex $o(a)$ and terminal
vertex $t(a)$ in $Q_{0}$. A path in $Q$ is a sequence of arrows
$a=a_{1}a_{2}\cdots a_{n-1}a_{n}$ such that the terminal vertex of $a_{i}$ is
the same as the origin vertex of $a_{i+1}$, using the convention of
concatenating paths from left to right. The quiver algebra or path algebra
$kQ$ is defined as a vector space having all paths in $Q$ as a basis. Vertices
are regarded as paths of length $0$, an arrow is a path of length $1$, and so
on. We take multiplication on $kQ$ as concatenation of paths. Two paths $a$
and $b$ satisfy $ab=0$ if $t(a)\neq o(b)$. This multiplication defines an
associative algebra over $k$. By taking $kQ_{i}$ to be the $k$-vector subspace
of $kQ$ with paths of length $i$ as basis, $kQ=\bigoplus_{i\geq 0}kQ_{i}$ can
be viewed as an $\mathbb{N}$-graded vector space. A relation on a quiver $Q$
is a linear combination of paths from $Q$ having the same origin vertex and
terminal vertex. A quiver together with a set of relations is called a quiver
with relations. Letting $I$ be an ideal of the path algebra $kQ$, we denote by
$(Q,I)$ the quiver $Q$ with relations $I$. The quotient $\Lambda=kQ/I$ is
called the quiver algebra associated with $(Q,I)$. Let
$\Lambda=\bigoplus_{i\geq 0}\Lambda_{i}$ be a grading on $\Lambda$. If
$\Lambda$ is Koszul and $\Lambda_{0}$ is isomorphic to $k$ or copies of $k$,
$\Lambda_{0}$ has a linear graded projective resolution $\mathbb{L}$ as a
right $\Lambda$-module [5, 8].

An algorithmic approach to finding such a minimal projective resolution
$\mathbb{L}$ of $\Lambda_{0}$ was given in [3]. The modules $\mathbb{L}_{n}$
are right $\Lambda$-modules for each $n$. There is a “comultiplicative
structure” on $\mathbb{L}$ and this structure was used to find a minimal
projective resolution $\mathbb{K}\rightarrow\Lambda$ of modules over the
enveloping algebra of $\Lambda$ in [4]. A non-zero element $x\in kQ$ is called
uniform if it is a linear combination of paths each having the same origin
vertex and the same terminal vertex: In other words, $x=\sum_{j}c_{j}w_{j}$
with scalars $c_{j}\neq 0$ for all $j$ and each path $w_{j}$ are of equal
length having the same origin vertex and the same terminal vertex. For $R=kQ$,
it was shown in [3] that there are integers $t_{n}$ and uniform elements
$\\{f^{n}_{i}\\}_{i=0}^{t_{n}}$ such that the right projective resolution
$\mathbb{L}\rightarrow\Lambda_{0}$ is obtained from a filtration of $R$. This
filtration is given by the following nested family of right ideals:

$\cdots\subseteq\bigoplus_{i=0}^{t_{n}}f^{n}_{i}R\subseteq\bigoplus_{i=0}^{t_{n-1}}f^{n-1}_{i}R\subseteq\cdots\subseteq\bigoplus_{i=0}^{t_{1}}f^{1}_{i}R\subseteq\bigoplus_{i=0}^{t_{0}}f^{0}_{i}R=R$

where for each $n$,
$\mathbb{L}_{n}=\bigoplus_{i=0}^{t_{n}}f^{n}_{i}R/\bigoplus_{i=0}^{t_{n}}f^{n}_{i}I$
and the differentials on $\mathbb{L}$ are induced by the inclusions
$\bigoplus_{i=0}^{t_{n}}f^{n}_{i}R\subseteq\bigoplus_{i=0}^{t_{n-1}}f^{n-1}_{i}R$.
Furthermore, it was shown in [3] that with some choice of scalars, the
$\\{f^{n}_{i}\\}_{i=0}^{t_{n}}$ satisfying the comultiplicative equation of
(2.4) make $\mathbb{L}$ minimal. In other words, for $0\leq i\leq t_{n}$,
there are scalars $c_{pq}(n,i,r)$ such that

(2.4)
$f^{n}_{i}=\sum_{p=0}^{t_{r}}\sum_{q=0}^{t_{n-r}}c_{pq}(n,i,r)f^{r}_{p}f^{n-r}_{q}$

holds and $\mathbb{L}$ is a minimal resolution. To construct the above
multiplicative equation for example, we can take
$\\{f^{0}_{i}\\}_{i=0}^{t_{0}}$ to be the set of vertices,
$\\{f^{1}_{i}\\}_{i=0}^{t_{1}}$ to be the set of arrows,
$\\{f^{2}_{i}\\}_{i=0}^{t_{2}}$ to be the set of uniform relations generating
the ideal $I$, and define $\\{f^{n}_{i}\\}_{i=0}^{t_{n}}(n\geq 3)$
recursively, that is in terms of $f^{n-1}_{i}$ and $f^{1}_{j}$. We present the
comultiplicative structure of a family of quiver algebras in Section (7). It
was first presented in [9] where we used the ideas in this work to show that
for certain algebras from the family, the Hochschild cohomology ring modulo
the weak Gerstenhaber ideal generated by homogeneous nilpotent elements is not
finitely generated.

The resolution $\mathbb{L}$ and the comultiplicative structure (2.4) were used
to construct a minimal projective resolution $\mathbb{K}\rightarrow\Lambda$ of
modules over the enveloping algebra $\Lambda^{e}=\Lambda\otimes\Lambda^{op}$
on which we now define Hochschild cohomology. This minimal projective
resolution $\mathbb{K}$ of $\Lambda^{e}$-modules associated to $\Lambda$ was
given in [4] and now restated with slight notational changes below.

######  Theorem 2.5.

[4, Theorem 2.1] Let $\Lambda=KQ/I$ be a Koszul algebra, and let
$\\{f^{n}_{i}\\}_{i=0}^{t_{n}}$ define a minimal resolution of $\Lambda_{0}$
as a right $\Lambda$-module. A minimal projective resolution $(\mathbb{K},d)$
of $\Lambda$ over $\Lambda^{e}$ is given by

$\mathbb{K}_{n}=\bigoplus_{i=0}^{t_{n}}\Lambda
o(f_{i}^{n})\otimes_{k}t(f_{i}^{n})\Lambda$

for $n\geq 0$, where the differential
$d_{n}:\mathbb{K}_{n}\xrightarrow{}\mathbb{K}_{n-1}$ applied to
$\varepsilon^{n}_{i}=(0,\ldots,0,o(f^{n}_{i})\otimes_{k}t(f^{n}_{i}),0,\ldots,0),$
$0\leq i\leq t_{n}$ where $o(f^{n}_{i})\otimes_{k}t(f^{n}_{i})$ is in the
$i$-th position is given by

(2.6)
$d_{n}(\varepsilon^{n}_{i})=\sum_{j=0}^{t_{n-1}}\Big{(}\sum_{p=0}^{t_{1}}c_{p,j}(n,i,1)f_{p}^{1}\varepsilon^{n-1}_{j}+(-1)^{n}\sum_{q=0}^{t_{1}}c_{j,q}(n,i,n-1)\varepsilon^{n-1}_{j}f_{q}^{1}\Big{)}$

and $d_{0}:K_{0}\xrightarrow{}\Lambda$ is the multiplication map. In
particular, $\Lambda$ is a linear module over $\Lambda^{e}$.

We note that for each $n$ and $i$, $\\{\varepsilon^{n}_{i}\\}_{i=0}^{t_{n}}$
is a free basis of $\mathbb{K}_{n}$ as a $\Lambda^{e}$-module. The scalars
$c_{p,j}(n,i,r)$ are those appearing in (2.4) and
$f^{1}_{*}:=\overline{f^{1}_{*}}$ is the residue class of $f^{1}_{*}$ in
$\bigoplus_{i=0}^{t_{1}}f^{1}_{i}R/\bigoplus_{i=0}^{t_{n}}f^{1}_{i}I$. Using
the comultiplicative structure of Equation (2.4), it was shown in [2] that the
cup product on the Hochschild cohomology ring of a Koszul quiver algebra has
the following description.

######  Theorem (See [2], Theorem 2.3).

Let $\Lambda=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a finite
quiver and $I\subseteq J^{2}$. Suppose that
$\eta:\mathbb{K}_{n}\rightarrow\Lambda$ and
$\theta:\mathbb{K}_{m}\rightarrow\Lambda$ represent elements in
$\operatorname{HH}^{*}(\Lambda)$ and are given by
$\eta(\varepsilon^{n}_{i})=\lambda_{i}$ for $i=0,1,\dotsc,t_{n}$ and
$\theta(\varepsilon^{m}_{i})=\lambda_{i}^{\prime}$ for $i=0,1,\dotsc,t_{m}.$
Then $\quad\eta\smile\theta:\mathbb{K}_{n+m}\rightarrow\Lambda$ can be
expressed as

$(\eta\smile\theta)(\varepsilon^{n+m}_{j})=\sum_{p=0}^{t_{n}}\sum_{q=0}^{t_{m}}c_{pq}(n+m,i,n)\lambda_{p}\lambda_{q}^{\prime},$

for $j=0,1,2,\dotsc,t_{n+m}.$

In Section 4, we present in Theorem 4.21 a similar generalized Gerstenhaber
algebra structure on Hochschild cohomology of Koszul algebras defined by
quivers and relations under certain conditions. Our presentation uses the
scalars coming from the comulplicative structure of Equation (2.4) as well as
scalars coming from homotopy lifting maps.

The reduced bar resolution of $\Lambda=kQ/I$ as presented in [2, Section 1]:
We recall the definition of the reduced bar resolution of algebras defined by
quivers and relations. If $\Lambda_{0}$ is isomorphic to $m$ copies of $k$,
take $\\{e_{1},e_{2},\ldots,e_{m}\\}$ to be a complete set of primitive
orthogonal central idempotents of $\Lambda$. In this case $\Lambda$ is not
necessarily an algebra over $\Lambda_{0}$. If $\Lambda_{0}$ is isomorphic to
$k$, then $\Lambda$ is an algebra over $\Lambda_{0}$. The reduced bar
resolution $(\mathcal{B},\delta)$, where
$\mathcal{B}_{n}:=\Lambda^{\otimes_{\Lambda_{0}}(n+2)}$ is the $(n+2)$-fold
tensor product of $\Lambda$ over $\Lambda_{0}$ and uses the same differential
on the usual bar resolution presented in Equation (2.2). The resolution
$\mathbb{K}$ can be embedded naturally into the reduced bar resolution
$\mathcal{B}$. There is a map $\iota:\mathbb{K}\rightarrow\mathcal{B}$ defined
by $\iota(\varepsilon^{n}_{r})=1\otimes\widetilde{f^{n}_{r}}\otimes 1$ such
that $\delta\iota=\iota d$, where

(2.7) $\widetilde{f^{n}_{j}}=\sum c_{j_{1}j_{2}\cdots
j_{n}}f^{1}_{j_{1}}\otimes f^{1}_{j_{2}}\otimes\cdots\otimes
f^{1}_{j_{n}}\quad\text{ if }\quad f^{n}_{j}=\sum c_{j_{1}j_{2}\cdots
j_{n}}f^{1}_{j_{1}}f^{1}_{j_{2}}\cdots f^{1}_{j_{n}}$

for some scalar $c_{j_{1}j_{2}\cdots j_{n}}$. It was shown in [2, Proposition
2.1] that $\iota$ is indeed an embedding. By taking
$\Delta:\mathcal{B}\rightarrow\mathcal{B}\otimes_{\Lambda}\mathcal{B}$ to be
the following comultiplicative map (or diagonal map) on the bar resolution,

(2.8) $\Delta(a_{0}\otimes\cdots\otimes
a_{n+1})=\sum_{i=0}^{n}(a_{0}\otimes\cdots\otimes a_{i}\otimes
1)\otimes_{\Lambda}(1\otimes a_{i+1}\otimes\cdots\otimes a_{n+1}).$

it was also shown in [2, Proposition 2.2] that the comultiplicative map
$\Delta_{\mathbb{K}}:\mathbb{K}\rightarrow\mathbb{K}\otimes_{\Lambda}\mathbb{K}$
on the complex $\mathbb{K}$ has the following form.

(2.9)
$\Delta_{\mathbb{K}}(\varepsilon^{n}_{r})=\sum_{v=0}^{n}\sum_{p=0}^{t_{v}}\sum_{q=0}^{t_{n-v}}c_{p,q}(n,r,v)\varepsilon^{v}_{p}\otimes_{\Lambda}\varepsilon^{n-v}_{q}.$

The compatibility of $\Delta_{\mathbb{K}},\Delta$ and $\iota$ means that
$(\iota\otimes\iota)\Delta_{\mathbb{K}}=\Delta\iota$ where
$(\iota\otimes\iota)(\mathbb{K}\otimes_{\Lambda}\mathbb{K})=\iota(\mathbb{K})\otimes_{\Lambda}\iota(\mathbb{K})\subseteq\mathcal{B}\otimes_{\Lambda}\mathcal{B}.$

##  3\. Comparison morphisms vs homotopy liftings

Comparisom morphisms: Let $(\mathbb{P},\mu_{P})$ and $(\mathbb{Q},\mu_{Q})$ be
two $A^{e}$-projective resolutions of a $k$-algebra $A$, where $\mu_{P}$ and
$\mu_{Q}$ are the augmentation maps. Since $\mathbb{P}$ is exact at $i$ for
all $i>0,$ there are chain maps
$\Phi_{P}^{Q}:\mathbb{P}\xrightarrow{}\mathbb{Q}$ such that
$\mu_{Q}\Phi_{P}^{Q}=\mu_{P}$. A similar argument implies that there are chain
maps $\Phi_{Q}^{P}:\mathbb{Q}\xrightarrow{}\mathbb{P}$ such that
$\mu_{P}\Phi_{Q}^{P}=\mu_{Q}$. So $\Phi_{P}^{Q}$ and $\Phi_{Q}^{P}$ are maps
between chain complexes. Let
$(\Phi_{Q}^{P})^{*}:\text{H}^{*}(\mathbb{P},A)\xrightarrow{}\text{H}^{*}(\mathbb{Q},A)$
and
$(\Phi_{P}^{Q})^{*}:\text{H}^{*}(\mathbb{Q},A)\xrightarrow{}\text{H}^{*}(\mathbb{P},A)$
be the induced map on cohomology. It is easy to see that
$(\Phi_{P}^{Q})^{*}(\Phi_{Q}^{P})^{*}=(\Phi_{Q}^{P}\Phi_{P}^{Q})^{*}=1_{\text{H}^{*}(\mathbb{P},A)}$.
This is because for any $F\in\text{H}^{*}(\mathbb{P},A)$,
$(\Phi_{Q}^{P}\Phi_{P}^{Q})^{*}F=F\Phi_{Q}^{P}\Phi_{P}^{Q}\sim F1_{P}=F$ since
$\Phi_{Q}^{P}\Phi_{P}^{Q}$ is homotopic to the identity map on the complex
$\mathbb{P}$. Analogously, we have
$(\Phi_{Q}^{P})^{*}(\Phi_{P}^{Q})^{*}=(\Phi_{P}^{Q}\Phi_{Q}^{P})^{*}=1_{\text{H}^{*}(\mathbb{Q},A)}.$
So Hochschild cohomology does not depend on the choice of projective
$A^{e}$-resolution.

Now suppose the cup product and the Gerstenhaber bracket on
$(\operatorname{HH}^{*}(A),\smallsmile,[\;,\;])$ are defined using the
resolution $\mathbb{Q},$ we may define the cup product $\smallsmile_{\Phi}$
with respect to the resolution $\mathbb{P}$ on two representatives
$F:\mathbb{P}_{m}\xrightarrow{}A$ and $G:\mathbb{P}_{n}\xrightarrow{}A$ by
$F\smallsmile_{\Phi}G=(F\Phi_{Q}^{P}\smallsmile G\Phi_{Q}^{P})\Phi_{P}^{Q}.$
Similarly, we can define the Gerstenhaber bracket by
$[F,G]_{\Phi}=[F\Phi_{Q}^{P},G\Phi_{Q}^{P}]\Phi_{P}^{Q}.$

Let $\mathbb{Q}$ be the reduced bar resolution $\mathcal{B}$ and let
$\mathbb{P}$ be the resolution $\mathbb{K}$. We can take
$\Phi_{K}^{B}:=\iota,$ the map defined just before Equation (2.7). It is not
obvious how to define $\Phi_{B}^{K}$, the chain map from the reduced bar
resolution to the resolution $\mathbb{K}$. The method of homotopy lifting
overcomes this barrier because it does not require definining the chain
$\Phi_{B}^{K}$. We now recall important notions about the homotopy lifting
technique given in [5] and [7].

Homotopy lifting: Let $\mathbb{P}\xrightarrow{\mu_{P}}A$ be a projective
resolution of $A$ as an $A^{e}$-module with differential $d^{P}$ and
augmentation map $\mu_{P}.$ We take d to be the differential on the Hom
complex $\operatorname{Hom}_{\Lambda^{e}}(\mathbb{P},\mathbb{P})$ defined for
any degree $n$ map $g:\mathbb{P}\rightarrow\mathbb{P}[-n]$ as

$\textbf{d}(g):=d^{P}g-(-1)^{n}gd^{P}$

where $\mathbb{P}[-n]$ is a shift in homological dimension with
$(\mathbb{P}[-n])_{m}=\mathbb{P}_{m-n}$. In the following definition, the
notation $\sim$ is used for two cocycles that are cohomologous, that is, they
differ by a coboundary.

######  Definition 3.1.

Let $\Delta_{\mathbb{P}}$ be a chain map (also called the diagonal map)
lifting the identity map on $A\cong A\otimes_{A}A$ and suppose that
$\eta\in\operatorname{Hom}_{A^{e}}(\mathbb{P}_{n},A)$ is a cocycle. A module
homomorphism $\psi_{\eta}:\mathbb{P}\rightarrow\mathbb{P}[1-n]$ is called a
homotopy lifting map of $\eta$ with respect to $\Delta_{\mathbb{P}}$ if

(3.2) $\displaystyle\textbf{d}(\psi_{\eta})$ $\displaystyle=(\eta\otimes
1_{P}-1_{P}\otimes\eta)\Delta_{\mathbb{P}}\qquad\text{ and }$
$\displaystyle\mu_{P}\psi_{\eta}$ $\displaystyle\sim\;(-1)^{n-1}\eta\psi$

for some $\psi:\mathbb{P}\rightarrow\mathbb{P}[1]$ for which
$\textbf{d}(\psi)=(\mu_{P}\otimes
1_{P}-1_{P}\otimes\mu_{P})\Delta_{\mathbb{P}}$.

######  Example 3.3.

Let $\mathbb{P}$ be the bar or reduced resolution $\mathcal{B}$. Suppose that
$g\in\operatorname{Hom}_{\Lambda^{e}}(\mathcal{B}_{n},A)\cong\operatorname{Hom}_{k}(A^{\otimes
n},A)$. Then one way to define a homotopy lifting map
$\psi_{g}:\mathcal{B}_{m+n-1}\longrightarrow\mathcal{B}_{m}$ for the cocycle
$g$ is the following:

$\psi_{g}(1\otimes a_{1}\otimes\cdots\otimes a_{m+n-1}\otimes 1)\\\
=\sum_{i=1}^{m}(-1)^{(m-1)(i-1)}1\otimes a_{1}\otimes\cdots\otimes
a_{i-1}\otimes g(a_{i}\otimes\cdots\otimes a_{i+n-1})\otimes
a_{i+n}\otimes\cdots\otimes a_{m+n-1}\otimes 1.$

######  Remark 3.4.

The resolution $\mathbb{K}$ for Koszul algebras is a differential graded
coalgebra i.e. $(\Delta_{\mathbb{K}}\otimes
1_{\mathbb{K}})\Delta_{\mathbb{K}}=(1_{\mathbb{K}}\otimes\Delta_{\mathbb{K}})\Delta_{\mathbb{K}}$
and $(d\otimes 1+1\otimes d)\Delta_{\mathbb{K}}=\Delta_{\mathbb{K}}d$.
Furthermore, the augmentation map $\mu:\mathbb{K}\rightarrow\Lambda$, which
can be thought of as a counit makes $(\mu\otimes
1_{\mathbb{K}})\Delta_{\mathbb{K}}-(1_{\mathbb{K}}\otimes\mu)\Delta_{\mathbb{K}}=0.$
We can therefore take $\psi=0$ in Equation (3.2), so that we have
$\mu\psi_{\eta}\sim 0.$ Next, we set $\psi_{\eta}(\mathbb{K}_{n-1})=0$ and the
second hypothesis of Definition 3.1 is satisfied. Later in Section 4, when we
define homotopy lifting maps using $\mathbb{K}$, it will be sufficient to only
verify the first Equation in (3.2). We now give a theorem of Y. Volkov which
is equivalent to the definition of the bracket presented earlier in Equation
(2.3).

######  Theorem 3.5.

[7, Theorem 4] Let $(\mathbb{P},\mu_{P})$ be a $\Lambda^{e}$-projective
resolution of $\Lambda$, and let
$\Delta_{\mathbb{P}}:\mathbb{P}\xrightarrow{}\mathbb{P}\otimes_{\Lambda}\mathbb{P}$
be a diagonal map. Let $\eta:\mathbb{P}_{n}\xrightarrow{}\Lambda$ and
$\theta:\mathbb{P}_{m}\xrightarrow{}\Lambda$ represent some cocycles. Suppose
that $\psi_{\eta}$ and $\psi_{\theta}$ are homotopy liftings for $\eta$ and
$\theta$ respectively. Then the Gerstenhaber bracket of the classes of $\eta$
and $\theta$ can be represented by the class of the element

$[\eta,\theta]_{\Delta_{\mathbb{P}}}=\eta\psi_{\theta}-(-1)^{(m-1)(n-1)}\theta\psi_{\eta}.$

###### Proof.

See [5, Section 6.3] or [7]. ∎

##  4\. General bracket structure

In this section, we define homotopy lifting maps for Koszul algebras defined
by quivers and relations and prove that under certain conditions on some
scalars, they are indeed homotopy lifting maps. We present examples of these
general definitions of homotopy lifting maps in Sections 6 and 7. The main
proofs are presented in Theorems 4.1,4.10 and 4.19. We build upon the results
presented in the preliminaries in Section 2. In particular, we refer to the
comultiplicative structure of Equation (2.4), the minimal projective
resolution $\mathbb{K}$ given in (2.5) and the differentials $d$ on
$\mathbb{K}$ given by Equation (2.6).

Notation: We will use the following standard notation. Since the set
$\\{\varepsilon^{n}_{i}\\}_{i=0}^{t_{n}},$ forms a basis for $\mathbb{K}_{n}$,
for any module homomorphism $\theta:\mathbb{K}_{n}\rightarrow\Lambda_{q}$
taking $\varepsilon^{n}_{i}$ to $\lambda_{i}$, $i=0,1,\ldots,t_{n}$, we use
the notation
$\theta=\begin{pmatrix}\lambda_{0}&\lambda_{1}&\cdots&\lambda_{t_{n}}\end{pmatrix}$
to encode this information. If $\theta$ takes $\varepsilon^{n}_{i}$ to
$\lambda$, and every other basis elements to $0$, we write
$\theta=\begin{pmatrix}0&\cdots&0&(\lambda)^{(i)}&0&\cdots&0\end{pmatrix}$.
Our results in this section can be summarized as considering cases where
$\lambda$ is an idempotent, a path of length 1, or a path of length 2. These
results are presented in Theorems 4.1, 4.10 and 4.19. We start with the case
where $\lambda$ is an idempotent.

######  Theorem 4.1.

Let $\Lambda=kQ/I$ be a Koszul quiver algebra. Suppose that
$\eta:\mathbb{K}_{n}\rightarrow\Lambda$ is a map such that for some $i,j$,
$\eta=\begin{pmatrix}0&\cdots&0&(e_{j})^{(i)}&0&\cdots&0\end{pmatrix}$ and for
all $0\leq p\leq t_{m-n},o(f^{m-n}_{p})\neq e_{j}$ and $t(f^{m-n}_{p})\neq
e_{j}.$ The associated map
$\psi_{\eta}:\mathbb{K}_{m}\xrightarrow{}\mathbb{K}_{m-n+1}$ satisfies

$(d\psi_{\eta}-(-1)^{n-1}\psi_{\eta}d)(\varepsilon^{m}_{r})=0.$

If there are $p^{\prime}$ such that $o(f^{m-n}_{p^{\prime}})=e_{j}$ and
$t(f^{m-n}_{p^{\prime}})=e_{j},$ the above equation holds provided
$c_{i,p^{\prime}}(m,r,n)=(-1)^{n(m-n)}c_{p^{\prime},i}(m,r,m-n)$

###### Proof.

The comultiplication on the resolution $\mathbb{K}$ is given by
$\displaystyle\Delta_{\mathbb{K}}(\varepsilon^{m}_{r})=\sum_{v=0}^{m}\sum_{p=0}^{t_{v}}\sum_{q=0}^{t_{m-v}}c_{p,q}(m,r,v)\varepsilon^{v}_{p}\otimes_{\Lambda}\varepsilon^{m-v}_{q}.$
The right hand side of Equation (3.2) which is $(\eta\otimes
1-1\otimes\eta)\Delta_{\mathbb{K}}(\varepsilon^{m}_{r})$ therefore becomes

$\displaystyle(\eta\otimes
1-1\otimes\eta)\sum_{v=0}^{m}\sum_{p=0}^{t_{v}}\sum_{q=0}^{t_{m-v}}c_{p,q}(m,r,v)\varepsilon^{v}_{p}\otimes_{\Lambda}\varepsilon^{m-v}_{q}$
$\displaystyle=\sum_{v=0}^{m}\sum_{p=0}^{t_{v}}\sum_{q=0}^{t_{m-v}}c_{p,q}(m,r,v)(\eta\otimes
1)(\varepsilon^{v}_{p}\otimes_{\Lambda}\varepsilon^{m-v}_{q})-\sum_{v=0}^{m}\sum_{p=0}^{t_{v}}\sum_{q=0}^{t_{m-v}}c_{p,q}(m,r,v)(1\otimes\eta)(\varepsilon^{v}_{p}\otimes_{\Lambda}\varepsilon^{m-v}_{q}).$

Whenever $v=n,p=i$ in the first part and $m-v=n,q=i$ in the second part, the
above expression yields

$\displaystyle\sum_{q=0}^{t_{m-n}}c_{i,q}(m,r,n)(\eta\otimes
1)(\varepsilon^{n}_{i}\otimes_{\Lambda}\varepsilon^{m-n}_{q})-\sum_{p=0}^{t_{m-n}}c_{p,i}(m,r,m-n)(1\otimes\eta)(\varepsilon^{m-n}_{p}\otimes_{\Lambda}\varepsilon^{n}_{i})$
$\displaystyle=\sum_{q=0}^{t_{m-n}}c_{i,q}(m,r,n)\eta(\varepsilon^{n}_{i})\varepsilon^{m-n}_{q}-(-1)^{n(m-n)}\sum_{p=0}^{t_{m-n}}c_{p,i}(m,r,m-n)\varepsilon^{m-n}_{p}\eta(\varepsilon^{n}_{i})$
$\displaystyle=\sum_{q=0}^{t_{m-n}}c_{i,q}(m,r,n)e_{j}\varepsilon^{m-n}_{q}-(-1)^{n(m-n)}\sum_{p=0}^{t_{m-n}}c_{p,i}(m,r,m-n)\varepsilon^{m-n}_{p}e_{j}.$

The above expression is equal to $0$ if for every $p$, $o(f^{m-n}_{q})\neq
e_{j}$ and $t(f^{m-n}_{p})\neq e_{j}$. Now let $q^{\prime}\;(0\leq
q^{\prime}\leq t_{m-n})$ be such that $o(f^{m-n}_{q^{\prime}})=e_{j}$, and
$p^{\prime}\;(0\leq p^{\prime}\leq t_{m-n})$ be such that
$t(f^{m-n}_{p^{\prime}})=e_{j}$ then $e_{j}\varepsilon^{m-n}_{q^{\prime}}$ is
equal to

$\displaystyle
e_{j}(0,\cdots,0,o(f^{m-n}_{q^{\prime}})\otimes_{k}t(f^{m-n}_{q^{\prime}}),0,\cdots,0)=(0,\cdots,0,e_{j}o(f^{m-n}_{q^{\prime}})\otimes_{k}t(f^{m-n}_{q^{\prime}}),0,\cdots,0)$
$\displaystyle=(0,\cdots,0,e_{j}^{2}\otimes_{k}t(f^{m-n}_{q^{\prime}}),0,\cdots,0)=(0,\cdots,0,e_{j}\otimes_{k}t(f^{m-n}_{q^{\prime}}),0,\cdots,0)$
$\displaystyle=(0,\cdots,0,o(f^{m-n}_{q^{\prime}})\otimes_{k}t(f^{m-n}_{q^{\prime}}),0,\cdots,0)=\varepsilon^{m-n}_{q^{\prime}}$

and similarly
$\varepsilon^{m-n}_{q^{\prime}}e_{j}=\varepsilon^{m-n}_{p^{\prime}}.$ The last
expression of the right hand side of Equation (LABEL:homolifting) therefore
becomes

$\displaystyle=\sum_{q^{\prime}}c_{i,q^{\prime}}(m,r,n)\varepsilon^{m-n}_{q^{\prime}}-(-1)^{n(m-n)}\sum_{p^{\prime}}c_{p^{\prime},i}(m,r,m-n)\varepsilon^{m-n}_{p^{\prime}}$
$\displaystyle=\sum_{p}[c_{i,p}(m,r,n)-(-1)^{n(m-n)}c_{p,i}(m,r,m-n)]\varepsilon^{m-n}_{p}=0$

for some $p$, $0\leq p\leq t_{m-n}$ for which $o(f^{m-n}_{p})=e_{j}$ and
$t(f^{m-n}_{p})=e_{j}.$ ∎

######  Remark 4.2.

A special case of Theorem 4.1 occurs if
$\eta:\mathbb{K}_{n}\rightarrow\Lambda$ is a cocycle and $m=n+1$. The
associated map $\psi_{\eta}:\mathbb{K}_{m}\xrightarrow{}\mathbb{K}_{m-n+1}$
satisfies $(d\psi_{\eta}-(-1)^{n-1}\psi_{\eta}d)(\varepsilon^{m}_{r})=0$ and
it is a homotopy lifting of $\eta.$ This means that
$\eta(\varepsilon^{n}_{i})=e_{j}$ and $\eta(\varepsilon^{n}_{r})=0$ for any
$r\neq i$ and as a cocycle $0=d^{*}\eta(\varepsilon^{n+1}_{r})=\eta
d(\varepsilon^{n+1}_{r})$ which is equal to

$\displaystyle\eta\sum_{j=0}^{t_{n}}\Big{(}\sum_{p=0}^{t_{1}}c_{p,j}(n+1,r,1)f^{1}_{p}\varepsilon^{n}_{j}+(-1)^{n+1}\sum_{q=0}^{t_{1}}c_{j,q}(n+1,r,n)\varepsilon^{n}_{j}f^{1}_{q}\Big{)}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{p,i}(n+1,r,1)f^{1}_{p}\eta(\varepsilon^{n}_{i})+(-1)^{n+1}\sum_{q=0}^{t_{1}}c_{i,q}(n+1,r,n)\eta(\varepsilon^{n}_{i})f^{1}_{q}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{p,i}(n+1,r,1)f^{1}_{p}e_{j}+(-1)^{n+1}\sum_{q=0}^{t_{1}}c_{i,q}(n+1,r,n)e_{j}f^{1}_{q}.$

There are some paths having origin and terminal vertex as $e_{j}$. Suppose
such paths are $f^{1}_{p^{\prime}}$, the above expression becomes
$\displaystyle{\sum_{p^{\prime}}[c_{p^{\prime},i}(n+1,r,1)+(-1)^{n+1}c_{i,p^{\prime}}(n+1,r,n)]f^{1}_{p^{\prime}}}$
and hence $c_{p^{\prime},i}(n+1,r,1)=(-1)^{n}c_{i,p^{\prime}}(n+1,r,n)$ for
some $0\leq p^{\prime}\leq t_{1}$.

We recall from Definition 3.1 that for Koszul algebras, we can take the first
homotopy lifting map
$(\psi_{\eta})_{n-1}:\mathbb{K}_{n-1}\xrightarrow{}\mathbb{K}_{0}$ to be the
zero map. From the result of Theorem 4.1,
$d(\psi_{\eta})_{n}=(-1)^{n-1}(\psi_{\eta})_{n-1}d=0,$ so we see that we can
define all homotopy lifting maps $\psi_{\eta}$ to be the zero map for all $n$.

######  Corollary 4.3.

Let $\Lambda=kQ/I$ be a Koszul algebra and suppose that
$\eta:\mathbb{K}_{n}\rightarrow\Lambda$ is a cocycle such that for some $i,j$,
$\eta=\begin{pmatrix}0&\cdots&0&(e_{j})^{(i)}&0&\cdots&0\end{pmatrix}.$ Then
the associated homotopy lifting maps to $\eta$ are all zero.

Moving on to the case where a free basis element is mapped to a path of length
1. We start with the following definition.

######  Definition 4.4.

For each fixed $n,r$, let
$<\;\;>_{n,r}:\mathbb{K}_{n-1}\rightarrow\mathbb{K}_{n-1}$ be a map defined on
$\varepsilon^{n-1}_{j}$ for each $j$ by

$<\varepsilon^{n-1}_{j}>_{n,r}=\sum_{v=0}^{t_{n-1}}\Big{(}\sum_{p=0}^{t_{1}}w^{(j)}_{pv}(n,r,1)f_{p}^{1}\varepsilon^{n-1}_{v}+\sum_{q=0}^{t_{1}}w^{(j)}_{vq}(n,r,n-1)\varepsilon^{n-1}_{v}f_{q}^{1}\Big{)}$

for scalars $w^{(j)}_{pv}(n,r,1)$ and $w^{(j)}_{vq}(n,r,n-1)$. Then extend to
all of $\mathbb{K}_{n-1}$ by requiring it to be a $\Lambda^{e}$-module
homomorphism.

######  Remark 4.5.

Whenever $w^{(j)}_{pv}(n,r,1)=0=w^{(j)}_{vq}(n,r,n-1)$ for all $v\neq j$ and
$w^{(j)}_{pj}(n,r,1)=c_{pj}(n,r,1),\;w^{(j)}_{jq}(n,r,n-1)=(-1)^{n}c_{jq}(n,r,n-1)$,
we obtain a special case of the module homomorphism which is defined as

(4.6)
$<\varepsilon^{n-1}_{j}>_{n,r}=\sum_{p=0}^{t_{1}}c_{pj}(n,r,1)f_{p}^{1}\varepsilon^{n-1}_{j}+(-1)^{n}\sum_{q=0}^{t_{1}}c_{jq}(n,r,n-1)\varepsilon^{n-1}_{j}f_{q}^{1}.$

The differential factors through this map i.e. there is a component of the
differential map $d^{j}_{n}$, $j=0,1,\ldots,t_{n-1}$ taking every basis
element $\varepsilon^{n}_{r}$ to a free basis element $\varepsilon^{n-1}_{j}$
such that in the special case defined above, the following diagram

$\textstyle{\mathbb{K}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;d^{j}_{n}\;}$$\scriptstyle{d_{n}}$$\textstyle{\mathbb{K}_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{<\;>_{n,r}}$$\textstyle{\mathbb{K}_{n-1}}$

commutes i.e.
$<d^{j}_{n}(\varepsilon^{n}_{r})>:=<\varepsilon^{n-1}_{j}>_{n,r}$ and
$d(\varepsilon^{n}_{r})=\sum_{j=0}^{t_{n-1}}<d^{j}_{n}(\varepsilon^{n}_{r})>$.
The subscript $(n,r)$ in $<\;\;\;>_{n,r}$ indicates that the scalars
$c_{pj}(n,r,1)$ and $(-1)^{n}c_{jq}(n,r,n-1)$ are coming from or associated
with the basis element $\varepsilon^{n}_{r}$. For example, in the expansion of
$<\varepsilon^{n-1}_{j}>_{n,r}$, $<\varepsilon^{n-1}_{j+1}>_{n,r}$ and
$<\varepsilon^{n-1}_{j}>_{n,r+1},$ the associated scalars to
$<\varepsilon^{n-1}_{j}>_{n,r}$ will be $c_{pj}(n,r,1)$ and
$(-1)^{n}c_{jq}(n,r,n-1)$ and they come from the expansion of
$d_{n}(\varepsilon^{n}_{r})$, the associated scalars to
$<\varepsilon^{n-1}_{j+1}>_{n,r}$ will be $c_{p,j+1}(n,r,1)$ and
$(-1)^{n}c_{j+1,q}(n,r,n-1)$ and they come from the expansion of
$d_{n}(\varepsilon^{n}_{r})$, while the associated scalars to
$<\varepsilon^{n-1}_{j}>_{n,r+1}$ will be $c_{pj}(n,r+1,1)$ and
$(-1)^{n}c_{jq}(n,r+1,n-1)$ and they come from the expansion of
$d_{n}(\varepsilon^{n}_{r+1}).$ We immediately see that under these
conditions,

$d_{n}(\varepsilon^{n}_{r})=\sum_{j=0}^{t_{n-1}}<\varepsilon^{n-1}_{j}>_{n,r}=\sum_{j=0}^{t_{n-1}}\Big{(}\sum_{p=0}^{t_{1}}c_{pj}(n,r,1)f_{p}^{1}\varepsilon^{n-1}_{j}+(-1)^{n}\sum_{q=0}^{t_{1}}c_{jq}(n,r,n-1)\varepsilon^{n-1}_{j}f_{q}^{1}\Big{)}.$

Henceforth, we will make use of the special case module homomorphism
$<\;\;>_{n,r}$ because of its connection to the differentials $d$.

We now give another main result about cocycles taking free basis elements to
paths of length 1. We assume in the proof of the next results, that the
scalars $c_{pj}(n,r,1),c_{jq}(n,r,n-1)$ of the comultiplicative structure
(2.4) satisfy Equation 4 below:

$\displaystyle c_{pj}(m,r,1)$
$\displaystyle=c_{pj^{\prime}}(m-n+1,r,1)=c_{p,j^{\prime}+1}(m-n+1,r,1)\quad\text{and}$
$\displaystyle(-1)^{m}c_{jq}(m,r,m-1)$
$\displaystyle=(-1)^{m-n+1}c_{j^{\prime}q}(m-n+1,r,m-n)$ (4.7)
$\displaystyle=(-1)^{m-n+1}c_{j^{\prime}+1,q}(m-n+1,r,m-n).$

In the next section where we will show several examples, these scalars are
mostly $\pm 1$, powers of $\pm q$ or 0. However, they can assume other values
different from these in general. For an $n$ cocycle $\eta$, a homotopy lifting
map of $\eta$ is a map
$\psi_{\eta}:\mathbb{K}_{m}\xrightarrow{}\mathbb{K}_{m-n+1}$ for any $m$. Our
goal is to define $\psi_{\eta}$ such that Equation (3.2) holds. Now define it
in the following way:

$\psi_{\eta}(\varepsilon^{m}_{r})=\sum_{j=0}^{t_{m-n+1}}b_{m,r}(m-n+1,j)\varepsilon^{m-n+1}_{j},$

where $b_{m,r}(m-n+1,j)$ are scalars and extend it to all of $\mathbb{K}_{m}$
by requiring it to be a $\Lambda^{e}$-module homomorphism. Now consider the
special case where $b_{m,r}(m-n+1,j)=0$ for all $j\neq s$ that is
$\psi_{\eta}(\varepsilon^{m}_{r})=b_{m,r}(m-n+1,s)\varepsilon^{m-n+1}_{s}$ for
some $s$. The next series of results shows that this special case is indeed a
homotopy lifting map under certain conditions on the scalars
$b_{m,r}(m-n+1,s)$ only. We start with the following lemma.

######  Lemma 4.8.

Suppose $\eta:\mathbb{K}_{n}\rightarrow\Lambda$ is a cocycle. The
$\Lambda^{e}$-module map defined by
$\psi_{\eta}(\varepsilon^{m}_{r})=b_{m,r}(m-n+1,s)\varepsilon^{m-n+1}_{s}$
satisfies
$\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r})=<\psi_{\eta}(\varepsilon^{m-1}_{j})>_{m,r}.$
Furthermore, whenever Equation (4) holds,

$\displaystyle
b_{m-1,j}(m-n,j^{\prime})<\varepsilon^{m-n}_{j^{\prime}}>_{m,r}=b_{m-1,j}(m-n,j^{\prime})<\varepsilon^{m-n}_{j^{\prime}}>_{m-n+1,r}.$

###### Proof.

We first observe that
$f^{1}_{w}\psi_{\eta}(\varepsilon^{m}_{r})=f^{1}_{w}b_{m,r}(m-n+1,s)\varepsilon^{m-n+1}_{s}$
for any $0\leq w\leq t_{1}$. This is the same as
$b_{,m,r}(m-n+1,s)f^{1}_{w}\varepsilon^{m-n+1}_{s}=\psi_{\eta}(f^{1}_{w}\varepsilon^{m-n+1}_{s})$
since $\psi_{\eta}$ is a $\Lambda^{e}$-module homomorphism.
$\psi_{\eta}(\varepsilon^{m}_{r})f^{1}_{w}=\psi_{\eta}(\varepsilon^{m}_{r}f^{1}_{w})$
holds similarly. Taking
$\psi_{\eta}(\varepsilon^{m-1}_{j})=b_{m-1,j}(m-n,j^{\prime})\varepsilon^{m-n}_{j^{\prime}}$,
we will have

$\displaystyle\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r})=\psi_{\eta}\Big{(}\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)f_{p}^{1}\varepsilon^{m-1}_{j}+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)\varepsilon^{m-1}_{j}f_{q}^{1}\Big{)}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)f_{p}^{1}\psi_{\eta}(\varepsilon^{m-1}_{j})+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)\psi_{\eta}(\varepsilon^{m-1}_{j})f_{q}^{1}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)b_{m-1,j}(m-n,j^{\prime})f_{p}^{1}\varepsilon^{m-n}_{j^{\prime}}$
$\displaystyle+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)b_{m-1,j}(m-n,j^{\prime})\varepsilon^{m-n}_{j^{\prime}}f_{q}^{1}.$

On the other hand

$\displaystyle<\psi_{\eta}(\varepsilon^{m-1}_{j})>_{m,r}$
$\displaystyle=<b_{m-1,j}(m-n,j^{\prime})\varepsilon^{m-n}_{j^{\prime}}>_{m,r}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)b_{m-1,j}(m-n,j^{\prime})f_{p}^{1}\varepsilon^{m-n}_{j^{\prime}}$
$\displaystyle+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)b_{m-1,j}(m-n,j^{\prime})\varepsilon^{m-n}_{j^{\prime}}f_{q}^{1}.$

It is now established that
$\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r})=<\psi_{\eta}(\varepsilon^{m-1}_{j})>_{m,r}$.
Also, you can factor out the scalars $b_{m-1,j}(m-n,j^{\prime})$ from the last
expansion so that
$\displaystyle{\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r})=b_{m-1,j}(m-n,j^{\prime})<\varepsilon^{m-n}_{j^{\prime}}>_{m,r}}$.
From the expansion of
$\displaystyle{(<\varepsilon^{m-n}_{j^{\prime}}>_{m-n+1,r})}$ the equality
$\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r})=b_{m-1,j}(m-n,j^{\prime})<\varepsilon^{m-n}_{j^{\prime}}>_{m,r}=b_{m-1,j}(m-n,j^{\prime})<\varepsilon^{m-n}_{j^{\prime}}>_{m-n+1,r}$
holds provided that Equation (4) is satisfied: that is whenever
$c_{pj}(m,r,1)=c_{pj^{\prime}}(m-n+1,r,1)$ and
$(-1)^{m}c_{jq}(m,r,m-1)=(-1)^{m-n+1}c_{j^{\prime}q}(m-n+1,r,m-n)$. Moreover,
since the map $\psi_{\eta}$ maps basis elements
$\\{\varepsilon^{m-1}_{r}\\}_{r=0}^{t_{m-1}}$ of $\mathbb{K}_{m-1}$ to basis
elements $\\{\varepsilon^{m-n}_{r}\\}_{r=0}^{t_{m-n}}$ of $\mathbb{K}_{m-n}$,
over a sum, the index $j$ shifts to $j^{\prime}$ where $0\leq j^{\prime}\leq
t_{m-n}$. This means that

(4.9)
$\sum_{j=0}^{t_{m-1}}\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r})=\sum_{j=0}^{t_{m-1}}<\psi_{\eta}(\varepsilon^{m-1}_{j})>_{m,r}=\sum_{j^{\prime}=0}^{t_{m-n}}b_{m-1,j}(m-n,j^{\prime})<\varepsilon^{m-n}_{j^{\prime}}>_{m-n+1,r}.$

∎

We now give the first main result.

######  Theorem 4.10.

Let $\Lambda=kQ/I$ be a Koszul algebra. Suppose that
$\eta:\mathbb{K}_{n}\rightarrow\Lambda$ is a cocycle such that
$\eta=\begin{pmatrix}0&\cdots&0&(f^{1}_{w})^{(i)}&0&\cdots&0\end{pmatrix}$ for
some $0\leq w\leq t_{1}$. A homotopy lifting map
$\psi_{\eta}:\mathbb{K}_{m}\rightarrow\mathbb{K}_{m-n+1}$ associated to $\eta$
can be defined by

$\psi_{\eta}(\varepsilon^{m}_{r})=b_{m,r}(m-n+1,s)\varepsilon^{m-n+1}_{s}$

for some scalars $b_{m,r}(m-n+1,s)$. Moreover the scalars $b_{m,r}(m-n+1,s)$
satisfy

  * (i).

$B=\begin{cases}c_{i,\alpha}(m,r,1)&{\rm when}\;p=w\\\ 0&{\rm when}\;p\neq
w\end{cases}$ and

  * (ii).

$B^{\prime}=\begin{cases}(-1)^{n(m-n)+1}c_{\alpha,i}(m,r,m-n)&{\rm
when}\;p=w\\\ 0&{\rm when}\;p\neq w,\end{cases}$

for all $\alpha$, where

$B=b_{m,r}(m-n+1,s)c_{p\alpha}(m-n+1,s,1)+(-1)^{n}b_{m-1,j}(m-n,\alpha)c_{p\alpha}(m-n+1,r,1),\text{
and }\\\ B^{\prime}=(-1)^{m+1}[(-1)^{n}b_{m,r}(m-n+1,s)c_{\alpha
q}(m-n+1,s,m-n)\\\ +b_{m-1,j}(m-n,\alpha)c_{\alpha q}(m-n+1,r,m-n)].$

###### Proof.

We have to show that under the stated conditions (i) and (ii), the equation

$(d\psi_{\eta}-(-1)^{n-1}\psi_{\eta}d)(\varepsilon^{m}_{r})=(\eta\otimes
1-1\otimes\eta)\Delta_{\mathbb{K}}(\varepsilon^{m}_{r})$

holds. Again, we use the Koszul sign convention in the expansion of
$(1\otimes\eta)(\varepsilon^{n}_{r}\otimes\varepsilon^{m}_{s})$ to obtain
$(-1)^{|\eta|n}\varepsilon^{n}_{r}\cdot\eta(\varepsilon^{m}_{s})$, where
$|\eta|$ is the degree of $\eta$. Next, we have that

$\displaystyle(d\psi_{\eta}-(-1)^{n-1}\psi_{\eta}d)(\varepsilon^{m}_{r})=d\psi_{\eta}(\varepsilon^{m}_{r})-(-1)^{n-1}\psi_{\eta}d(\varepsilon^{m}_{r})$
$\displaystyle=d(b_{m,r}(m-n+1,s)\varepsilon^{m-n+1}_{s})-(-1)^{n-1}\psi_{\eta}(\sum_{j=0}^{t_{m-1}}<\varepsilon^{m-1}_{j}>_{m,r})$
$\displaystyle=b_{m,r}(m-n+1,s)d(\varepsilon^{m-n+1}_{s})-(-1)^{n-1}\sum_{j=0}^{t_{m-1}}\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r}).$

Using Equation (4.9), and applying the definition of the differential, we get

$b_{m,r}(m-n+1,s)\sum_{\alpha=0}^{t_{m-n}}<\varepsilon^{m-n}_{\alpha}>_{m-n+1,s}-(-1)^{n-1}\sum_{j^{\prime}=0}^{t_{m-n}}b_{m-1,j}(m-n,j^{\prime})<\varepsilon^{m-n}_{j^{\prime}}>_{m-n+1,r},$

then applying the definition of the map $<\;\cdot\;>_{*,*}$

$\displaystyle=\sum_{\alpha=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}b_{m,r}(m-n+1,s)c_{p\alpha}(m-n+1,s,1)f_{p}^{1}\varepsilon^{m-n}_{\alpha}$
$\displaystyle+(-1)^{m-n+1}\sum_{\alpha=0}^{t_{m-n}}\sum_{q=0}^{t_{1}}b_{m,r}(m-n+1,s)c_{\alpha
q}(m-n+1,s,m-n)\varepsilon^{m-n}_{\alpha}f_{q}^{1}$
$\displaystyle+(-1)^{n}\sum_{j^{\prime}=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}b_{m-1,j}(m-n,j^{\prime})c_{pj^{\prime}}(m-n+1,r,1)f_{p}^{1}\varepsilon^{m-n}_{j^{\prime}}$
$\displaystyle+(-1)^{m+1}\sum_{j^{\prime}=0}^{t_{m-n}}\sum_{q=0}^{t_{1}}b_{m-1,j}(m-n,j^{\prime})c_{j^{\prime}q}(m-n+1,r,m-n)\varepsilon^{m-n}_{j^{\prime}}f_{q}^{1}.$

Collecting like terms and re-indexing, we get

$\displaystyle\sum_{\alpha=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}\Big{[}b_{m,r}(m-n+1,s)c_{p\alpha}(m-n+1,s,1)$
$\displaystyle+(-1)^{n}b_{m-1,j}(m-n,\alpha)c_{p\alpha}(m-n+1,r,1)\Big{]}f_{p}^{1}\varepsilon^{m-n}_{\alpha}$
$\displaystyle+(-1)^{m+1}\sum_{\alpha=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}\Big{[}(-1)^{n}b_{m,r}(m-n+1,s)c_{\alpha
p}(m-n+1,s,m-n)$ $\displaystyle+b_{m-1,j}(m-n,\alpha)c_{\alpha
p}(m-n+1,r,m-n)\Big{]}\varepsilon^{m-n}_{\alpha}f_{p}^{1}$

which is succintly expressed as

$\displaystyle=\sum_{\alpha=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}\Big{[}B\Big{]}f_{p}^{1}\varepsilon^{m-n}_{\alpha}+\sum_{\alpha=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}\Big{[}B^{\prime}\Big{]}\varepsilon^{m-n}_{\alpha}f_{p}^{1}.$

After applying the definitions of $B$ and $B^{\prime}$ given by (i) and (ii)
of the theorem, that is substitute $B=c_{i,\alpha}(m,r,1)$ when $p=w$ and
$B^{\prime}=-(-1)^{n(m-n)}c_{\alpha,i}(m,r,m-n)$ and $0$ otherwise, we get

$\displaystyle=\sum_{\alpha=0}^{t_{m-n}}c_{i,\alpha}(m,r,1)f^{1}_{w}\varepsilon^{m-n}_{\alpha}-(-1)^{n(m-n)}\sum_{\alpha=0}^{t_{m-n}}c_{\alpha,i}(m,r,m-n)\varepsilon^{m-n}_{\alpha}f^{1}_{w}.$

On the other hand, the comultiplication on the resolution $\mathbb{K}$ is
given by
$\displaystyle\Delta_{\mathbb{K}}(\varepsilon^{m}_{r})=\sum_{v=0}^{m}\sum_{p=0}^{t_{v}}\sum_{q=0}^{t_{m-v}}c_{p,q}(m,r,v)\varepsilon^{v}_{p}\otimes_{\Lambda}\varepsilon^{m-v}_{q}.$
Applying $(\eta\otimes 1-1\otimes\eta)$, we obtain

$\displaystyle(\eta\otimes
1-1\otimes\eta)\Delta_{\mathbb{K}}(\varepsilon^{m}_{r})=(\eta\otimes
1-1\otimes\eta)\sum_{v=0}^{m}\sum_{p=0}^{t_{v}}\sum_{q=0}^{t_{m-v}}c_{p,q}(m,r,v)\varepsilon^{v}_{p}\otimes_{\Lambda}\varepsilon^{m-v}_{q}$
$\displaystyle=\sum_{v=0}^{m}\sum_{p=0}^{t_{v}}\sum_{q=0}^{t_{m-v}}c_{p,q}(m,r,v)(\eta\otimes
1)(\varepsilon^{v}_{p}\otimes_{\Lambda}\varepsilon^{m-v}_{q})$
$\displaystyle-\sum_{v=0}^{m}\sum_{p=0}^{t_{v}}\sum_{q=0}^{t_{m-v}}c_{p,q}(m,r,v)(1\otimes\eta)(\varepsilon^{v}_{p}\otimes_{\Lambda}\varepsilon^{m-v}_{q}).$

Whenever $v=n,p=i$ in the first part and $m-v=n,q=i$ in the second part, and
changing the indices $p,q$ to $\alpha$ later on, the above expression yields

$\displaystyle\sum_{q=0}^{t_{m-n}}c_{i,q}(m,r,1)(\eta\otimes
1)(\varepsilon^{n}_{i}\otimes_{\Lambda}\varepsilon^{m-n}_{q})-\sum_{p=0}^{t_{m-n}}c_{p,i}(m,r,m-n)(1\otimes\eta)(\varepsilon^{m-n}_{p}\otimes_{\Lambda}\varepsilon^{n}_{i})$
$\displaystyle=\sum_{q=0}^{t_{m-n}}c_{i,q}(m,r,1)\eta(\varepsilon^{n}_{i})\varepsilon^{m-n}_{q}-(-1)^{n(m-n)}\sum_{p=0}^{t_{m-n}}c_{p,i}(m,r,m-n)\varepsilon^{m-n}_{p}\eta(\varepsilon^{n}_{i})$
$\displaystyle=\sum_{\alpha=0}^{t_{m-n}}c_{i,\alpha}(m,r,1)f^{1}_{w}\varepsilon^{m-n}_{\alpha}-(-1)^{n(m-n)}\sum_{\alpha=0}^{t_{m-n}}c_{\alpha,i}(m,r,m-n)\varepsilon^{m-n}_{\alpha}f^{1}_{w}$
$\displaystyle=(d\psi_{\eta}-\psi_{\eta}d)(\varepsilon^{n}_{r}).$

∎

We will next consider the case where free basis elements of $\mathbb{K}_{m}$
are mapped to paths of length 2. We start with the following definition.

######  Definition 4.11.

Let $f^{1}_{w},f^{1}_{w+1}$ be paths of length 1 in $\Lambda=kQ/I$. For any
$m,n$, let $0\leq r\leq t_{m}$, $0\leq s\leq t_{m-n+1},$ and define a map
$\psi:\mathbb{K}_{m}\rightarrow\mathbb{K}_{m-n+1},$ on the free basis elements
$\\{\varepsilon^{m}_{r}\\}_{r=0}^{t_{m}}$ of $\mathbb{K}_{m}$ by

(4.12)
$\psi(\varepsilon^{m}_{r})=\sum_{v=0}^{t_{m-n+1}-1}b_{m,r}(m-n+1,v+1)f^{1}_{w}\varepsilon^{m-n+1}_{v+1}+b_{m,r}(m-n+1,v)\varepsilon^{m-n+1}_{v}f^{1}_{w+1},$

for some scalars $b_{m,r}(m-n+1,s+1)$ and $b_{m,r}(m-n+1,s)$ and extend it to
all of $\mathbb{K}_{m}$ as a $\Lambda^{e}$-module homomorphism.

######  Remark 4.13.

For a cocycle $\eta:\mathbb{K}_{n}\rightarrow\Lambda$, Y. Volkov showed that
there are homotopy lifting maps
$\psi_{\eta}:\mathbb{K}\rightarrow\mathbb{K}[1-n],$ as presented in Definition
3.1. We will show that under certain conditions on the scalars, the above map
is a homotopy lifting map for some cocycle $\eta$ such that
$\eta(\varepsilon^{n}_{i})=f^{1}_{w}f^{1}_{w+1}$ for some $i,w$, and
$\eta(\varepsilon^{n}_{j})=0$ when $i\neq j$ (see Theorem 4.19 for instance).
For a fixed index $s$, If $b_{m,r}(m-n+1,v+1)=b_{m,r}(m-n+1,v)=0$ for all
$v\neq s$, we obtain the following special case:

$\psi_{\eta}(\varepsilon^{m}_{r})=b_{m,r}(m-n+1,s+1)f^{1}_{w}\varepsilon^{m-n+1}_{s+1}+b_{m,r}(m-n+1,s)\varepsilon^{m-n+1}_{s}f^{1}_{w+1}$

and show that under certain conditions on the scalars, this is a homotopy
lifting map for $\eta$.

Before presenting another major theorem (Theorem 4.19), we present two lemmas.
Since we will be expanding $\psi_{\eta}(<\varepsilon^{m}_{r}>_{m,r})$, the
first lemma gives information on how this expansion turns out. The second
lemma helps to give a succinct way to express the sum
$d\psi_{\eta}(\varepsilon^{m}_{r})+\psi_{\eta}d(\varepsilon^{m}_{r})$ in case
$\eta$ takes a basis element to a path of length 2.

######  Lemma 4.14.

Let $\Lambda=kQ/I$ be a Koszul algebra. Suppose
$\eta:\mathbb{K}_{n}\rightarrow\Lambda$ is a cocycle. The map defined by

$\psi_{\eta}(\varepsilon^{m}_{r})=b_{m,r}(m-n+1,s+1)f^{1}_{w}\varepsilon^{m-n+1}_{s+1}+b_{m,r}(m-n+1,s)\varepsilon^{m-n+1}_{s}f^{1}_{w+1}$

satisfies
$\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r})=<\psi_{\eta}(\varepsilon^{m-1}_{j})>_{m,r}$.
Moreover, the expansion

$\displaystyle<\psi_{\eta}(\varepsilon^{m-1}_{j})>_{m,r}$
$\displaystyle=b_{m-1,j}(m-n,j^{\prime}+1)<f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1}>_{m-n+1,r}$
$\displaystyle+b_{m-1,j}(m-n,j^{\prime})<\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1}>_{m-n+1,r}$

holds whenever Equation (4) holds.

###### Proof.

We use the fact that $\psi_{\eta}$ and $<\;\cdot\;>_{m,r}$ are
$\Lambda^{e}$-module homomorphism. Taking
$\psi_{\eta}(\varepsilon^{m-1}_{j})=b_{m-1,j}(m-n,j^{\prime}+1)f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1}+b_{m-1,j}(m-n,j^{\prime})\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1}$,
we will have

$\displaystyle\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r})$
$\displaystyle=\psi_{\eta}\Big{(}\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)f_{p}^{1}\varepsilon^{m-1}_{j}+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)\varepsilon^{m-1}_{j}f_{q}^{1}\Big{)}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)\psi_{\eta}(f_{p}^{1}\varepsilon^{m-1}_{j})+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)\psi_{\eta}(\varepsilon^{m-1}_{j}f_{q}^{1})$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)f_{p}^{1}\psi_{\eta}(\varepsilon^{m-1}_{j})+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)\psi_{\eta}(\varepsilon^{m-1}_{j})f_{q}^{1}=<\psi_{\eta}(\varepsilon^{m-1}_{j})>_{m,r}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)f_{p}^{1}\big{[}b_{m-1,j}(m-n,j^{\prime}+1)f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1}+b_{m-1,j}(m-n,j^{\prime})\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1}\big{]}$
$\displaystyle+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)\big{[}b_{m-1,j}(m-n,j^{\prime}+1)f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1}+b_{m-1,j}(m-n,j^{\prime})\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1}\big{]}f_{q}^{1}$
$\displaystyle=b_{m-1,j}(m-n,j^{\prime}+1)\Big{[}\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)f_{p}^{1}(f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1})+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)(f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1})f^{1}_{q}\Big{]}$
$\displaystyle+b_{m-1,j}(m-n,j^{\prime})\Big{[}\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)f_{p}^{1}(\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1})+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)(\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1})f^{1}_{q}\Big{]}.$

We now recall that the first equality of Equation (4) implies that

$\displaystyle<\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1}>_{m-n+1,r}=\sum_{p=0}^{t_{1}}c_{pj^{\prime}}(m-n+1,r,1)f_{p}^{1}(\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1})$
$\displaystyle+(-1)^{m-n+1}\sum_{q=0}^{t_{1}}c_{j^{\prime}q}(m-n+1,r,m-n)(\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1})f_{q}^{1}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)f_{p}^{1}(\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1})+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)(\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1})f_{q}^{1},$

and the second part of the equality of Equation (4) i.e.
$c_{pj}(m,r,1)=c_{p,j^{\prime}+1}(m-n+1,r,1)$ and
$(-1)^{m}c_{jq}(m,r,m-1)=(-1)^{m-n+1}c_{j^{\prime}+1,q}(m-n+1,r,m-n)$ implies
that

$\displaystyle<f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1}>_{m-n+1,r}=\sum_{p=0}^{t_{1}}c_{p,j^{\prime}+1}(m-n+1,r,1)f_{p}^{1}(f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1})$
$\displaystyle+(-1)^{m-n+1}\sum_{q=0}^{t_{1}}c_{j^{\prime}+1,q}(m-n+1,r,m-n)(f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1})f_{q}^{1}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{pj}(m,r,1)f_{p}^{1}(f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1})+(-1)^{m}\sum_{q=0}^{t_{1}}c_{jq}(m,r,m-1)(f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1})f_{q}^{1}.$

Putting all these together, we get the desired result:

$\displaystyle\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r})=<\psi_{\eta}(\varepsilon^{m-1}_{j})>_{m,r}$
$\displaystyle=<b_{m-1,j}(m-n,j^{\prime}+1)f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1}+b_{m-1,j}(m-n,j^{\prime})\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1}>_{m-n+1,r}$
$\displaystyle=b_{m-1,j}(m-n,j^{\prime}+1)<f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1}>_{m-n+1,r}+b_{m-1,j}(m-n,j^{\prime})<\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1}>_{m-n+1,r}.$

∎

The differentials map free basis elements $\varepsilon^{m}_{r}$ to a linear
combination of $f^{1}_{*}\varepsilon^{m-1}_{**}$ and
$\varepsilon^{m-1}_{**}f^{1}_{*}$ and the map $\psi$ of Definition 4.12 maps
free basis elements $\varepsilon^{m-1}_{*}$ to linear combination of
$f^{1}_{**}\varepsilon^{m-n}_{***}$ and $\varepsilon^{m-n}_{***}f^{1}_{**}.$
Combining these two maps means $\psi d$ and $d\psi$ will map free basis
elements $\varepsilon^{m}_{r}$ to a linear combination of
$f^{1}_{*}f^{1}_{**}\varepsilon^{m-n}_{***}$,$\displaystyle{f^{1}_{*}\varepsilon^{m-n}_{***}f^{1}_{**}}$,
$\displaystyle{f^{1}_{**}\varepsilon^{m-n}_{***}f^{1}_{*}}$ and
$\varepsilon^{m-n}_{***}f^{1}_{**}f^{1}_{*}.$ The map
$J:\mathbb{K}_{m}\rightarrow\mathbb{K}_{m-n+1}$ given in the next defiinition
describes all the possible $k$-linear combinations there are and the next
lemma shows that after suitable substitution of certain scalars,
$J=d\psi_{\eta}-(-1)^{n-1}\psi_{\eta}d.$

######  Definition 4.15.

For $0\leq r\leq t_{m-n}$, define a map
$J:\mathbb{K}_{m}\rightarrow\mathbb{K}_{m-n}$ on the basis elements
$\varepsilon^{m}_{s}$ of $\mathbb{K}_{m}$ by

$J(\varepsilon^{m}_{s})=\sum_{r=0}^{t_{m}}\sum_{i=0}^{t_{1}}\sum_{j=0}^{t_{1}}\Big{[}\sigma_{m,s}(i,j,r)f^{1}_{i}f^{1}_{j}\varepsilon^{m-n}_{r}+\sigma_{m,s}(i,r,j)f^{1}_{i}\varepsilon^{m-n}_{r}f^{1}_{j}+\sigma_{m,s}(r,i,j)\varepsilon^{m-n}_{r}f^{1}_{i}f^{1}_{j}\Big{]}$

for some scalars $\sigma_{m,s}(i,j,r)$ and extend to all of $\mathbb{K}_{m}$
by requiring it to be a $\Lambda^{e}$-module homomorphism.

######  Lemma 4.16.

Let $\Lambda=kQ/I$ be a Koszul algebra and
$\eta:\mathbb{K}_{n}\rightarrow\Lambda$ a cocycle. There are scalars
$\sigma_{*,*}(*,*,*)$ such that

(4.17)
$(d\psi_{\eta}-(-1)^{n-1}\psi_{\eta}d)(\varepsilon^{m}_{r})=J(\varepsilon^{m}_{r}).$

###### Proof.

We begin the proof with direct evaluation of these maps on the free basis
elements.

$\displaystyle(d\psi_{\eta}-(-1)^{n-1}\psi_{\eta}d)(\varepsilon^{m}_{r})=d\psi_{\eta}(\varepsilon^{m}_{r})-(-1)^{n-1}\psi_{\eta}d(\varepsilon^{m}_{r})$
$\displaystyle=d\Big{(}b_{m,r}(m-n+1,s+1)f^{1}_{w}\varepsilon^{m-n+1}_{s+1}+b_{m,r}(m-n+1,s)\varepsilon^{m-n+1}_{s}f^{1}_{w+1}\Big{)}$
$\displaystyle-(-1)^{n-1}\psi_{\eta}(\sum_{j=0}^{t_{m-1}}<\varepsilon^{m-1}_{j}>_{m,r})$
$\displaystyle=b_{m,r}(m-n+1,s+1)f^{1}_{w}d(\varepsilon^{m-n+1}_{s+1})+b_{m,r}(m-n+1,s)d(\varepsilon^{m-n+1}_{s})f^{1}_{w+1}$
$\displaystyle-(-1)^{n-1}\sum_{j=0}^{t_{m-1}}\psi_{\eta}(<\varepsilon^{m-1}_{j}>_{m,r}).$

Since the map $\psi_{\eta}$ takes basis elements
$\\{\varepsilon^{m-1}_{j}\\}_{j=0}^{t_{m-1}}$ of $\mathbb{K}_{m-1}$ to basis
elements $\\{\varepsilon^{m-n}_{j^{\prime}}\\}_{j^{\prime}=0}^{t_{m-n}}$ of
$\mathbb{K}_{m-n}$, the index $j$ shifts to $j^{\prime}$ over the sum. Also
apply the result of Lemma 4.14 to obtain

$b_{m,r}(m-n+1,s+1)f^{1}_{w}\sum_{\alpha=0}^{t_{m-n}}<\varepsilon^{m-n}_{\alpha}>_{m-n+1,s+1}\\\
+b_{m,r}(m-n+1,s)\sum_{\beta=0}^{t_{m-n}}<\varepsilon^{m-n}_{\beta}>_{m-n+1,s}f^{1}_{w+1}\\\
-(-1)^{n-1}\sum_{j^{\prime}=0}^{t_{m-n}}b_{m-1,j}(m-n,j^{\prime}+1)<f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1}>_{m-n+1,r}\\\
+b_{m-1,j}(m-n,j^{\prime})<\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1}>_{m-n+1,r}$

Applying the definition of $<\varepsilon^{*}_{*}>_{*,*}$ at the appropriate
places, we obtain

$b_{m,r}(m-n+1,s+1)f^{1}_{w}\sum_{\alpha=0}^{t_{m-n}}\Big{[}\sum_{p=0}^{t_{1}}c_{p\alpha}(m-n+1,s+1,1)f_{p}^{1}\varepsilon^{m-n}_{\alpha}\\\
+(-1)^{m-n+1}\sum_{q=0}^{t_{1}}c_{\alpha
q}(m-n+1,s+1,m-n)\varepsilon^{m-n}_{\alpha}f_{q}^{1}\Big{]}\\\
+b_{m,r}(m-n+1,s)\sum_{\beta=0}^{t_{m-n}}\Big{[}\sum_{p=0}^{t_{1}}c_{p\beta}(m-n+1,s,1)f_{p}^{1}\varepsilon^{m-n}_{\beta}\\\
+(-1)^{m-n+1}\sum_{q=0}^{t_{1}}c_{\beta
q}(m-n+1,s,m-n)\varepsilon^{m-n}_{\beta}f_{q}^{1}\Big{]}f^{1}_{w+1}$
$-(-1)^{n-1}\sum_{j^{\prime}=0}^{t_{m-n}}b_{m-1,j}(m-n,j^{\prime}+1)f^{1}_{w}\Big{[}\sum_{p=0}^{t_{1}}c_{p,j^{\prime}+1}(m-n+1,r,1)f_{p}^{1}\varepsilon^{m-n}_{j^{\prime}+1}\\\
+(-1)^{m-n+1}\sum_{q=0}^{t_{1}}c_{j^{\prime}+1,q}(m-n+1,r,m-n)\varepsilon^{m-n}_{j^{\prime}+1}f_{q}^{1}\Big{]}\\\
-(-1)^{n-1}\sum_{j^{\prime}=0}^{t_{m-n}}b_{m-1,j}(m-n,j^{\prime})\Big{[}\sum_{p=0}^{t_{1}}c_{pj^{\prime}}(m-n+1,r,1)f_{p}^{1}\varepsilon^{m-n}_{j^{\prime}}\\\
+(-1)^{m-n+1}\sum_{q=0}^{t_{1}}c_{j^{\prime}q}(m-n+1,r,m-n)\varepsilon^{m-n}_{j^{\prime}}f_{q}^{1}\Big{]}f^{1}_{w+1}.$

After re-arranging, and bringing together like terms we get

(a1)
$\displaystyle=\sum_{\alpha=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}b_{m,r}(m-n+1,s+1)c_{p\alpha}(m-n+1,s+1,1)f^{1}_{w}f_{p}^{1}\varepsilon^{m-n}_{\alpha}$
(a2)
$\displaystyle-(-1)^{n-1}\sum_{j^{\prime}=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}b_{m-1,j}(m-n,j^{\prime}+1)c_{p,j^{\prime}+1}(m-n+1,r,1)f^{1}_{w}f_{p}^{1}\varepsilon^{m-n}_{j^{\prime}+1}$
(b1)
$\displaystyle+(-1)^{m-n+1}\sum_{\alpha=0}^{t_{m-n}}\sum_{q=0}^{t_{1}}b_{m,r}(m-n+1,s+1)c_{\alpha
q}(m-n+1,s+1,m-n)f^{1}_{w}\varepsilon^{m-n}_{\alpha}f_{q}^{1}$ (b2)
$\displaystyle-(-1)^{m}\sum_{j^{\prime}=0}^{t_{m-n}}\sum_{q=0}^{t_{1}}b_{m-1,j}(m-n,j^{\prime}+1)c_{j^{\prime}+1,q}(m-n+1,r,m-n)f^{1}_{w}\varepsilon^{m-n}_{j^{\prime}+1}f_{q}^{1}$
(c1)
$\displaystyle+\sum_{\beta=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}b_{m,r}(m-n+1,s)c_{p\beta}(m-n+1,s,1)f_{p}^{1}\varepsilon^{m-n}_{\beta}f^{1}_{w+1}$
(c2)
$\displaystyle-(-1)^{n-1}\sum_{j^{\prime}=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}b_{m-1,j}(m-n,j^{\prime})c_{pj^{\prime}}(m-n+1,r,1)f_{p}^{1}\varepsilon^{m-n}_{j^{\prime}}f^{1}_{w+1}$
(d1)
$\displaystyle+(-1)^{m-n+1}\sum_{\beta=0}^{t_{m-n}}\sum_{q=0}^{t_{1}}b_{m,r}(m-n+1,s)c_{\beta
q}(m-n+1,s,m-n)\varepsilon^{m-n}_{\beta}f_{q}^{1}f^{1}_{w+1}$ (d2)
$\displaystyle(-1)^{m}\sum_{j^{\prime}=0}^{t_{m-n}}\sum_{q=0}^{t_{1}}b_{m-1,j}(m-n,j^{\prime})c_{j^{\prime}q}(m-n+1,r,m-n)\varepsilon^{m-n}_{j^{\prime}}f_{q}^{1}f^{1}_{w+1}.$

Next we combine Equations (a1) and (a2) and re-index $\alpha=j^{\prime}+1$,
combine Equations (b1) and (b2) and re-index $\alpha=j^{\prime}+1$ and so on
to obtain

$\displaystyle\sum_{\alpha=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}\Big{[}A_{\alpha}\Big{]}f^{1}_{w}f_{p}^{1}\varepsilon^{m-n}_{\alpha}+\sum_{\alpha=0}^{t_{m-n}}\sum_{q=0}^{t_{1}}\Big{[}B_{\alpha}\Big{]}f^{1}_{w}\varepsilon^{m-n}_{\alpha}f^{1}_{q}$
(4.18)
$\displaystyle+\sum_{\beta=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}\Big{[}C_{\beta}\Big{]}f^{1}_{p}\varepsilon^{m-n}_{\beta}f^{1}_{w+1}+\sum_{\beta=0}^{t_{m-n}}\sum_{q=0}^{t_{1}}\Big{[}D_{\beta}\Big{]}\varepsilon^{m-n}_{\beta}f^{1}_{q}f^{1}_{w+1}$

where

$A_{\alpha}=b_{m,r}(m-n+1,s+1)c_{p\alpha}(m-n+1,s+1,1)\\\
+(-1)^{n}b_{m-1,j}(m-n,\alpha)c_{p,\alpha}(m-n+1,r,1)\\\
B_{\alpha}=(-1)^{m-n+1}b_{m,r}(m-n+1,s+1)c_{\alpha q}(m-n+1,s+1,m-n)\\\
-(-1)^{m}b_{m-1,j}(m-n,\alpha)c_{\alpha,q}(m-n+1,r,m-n)\\\
C_{\beta}=b_{m,r}(m-n+1,s)c_{p\beta}(m-n+1,s,1)\qquad\qquad\qquad\\\
+(-1)^{n}b_{m-1,j}(m-n,\beta)c_{p\alpha}(m-n+1,r,1)\\\
D_{\beta}=(-1)^{m-n+1}b_{m,r}(m-n+1,s)c_{\beta q}(m-n+1,s,m-n)\\\
+(-1)^{m}b_{m-1,j}(m-n,\beta)c_{\beta,q}(m-n+1,r,m-n).$

In the last expression that is Equation (4), substitute

$\begin{cases}\sigma_{m,r}(i,j,r)=A_{\alpha}&\;{\rm
when}\;i=w\;(fixed),j=p,r=\alpha,0\leq r\leq t_{m-n},\;0\leq j\leq t_{1}\\\
\sigma_{m,r}(i,r,j)=B_{\alpha}&\;{\rm when}\;i=w\;(fixed),j=q,r=\alpha,0\leq
r\leq t_{m-n},\;0\leq j\leq t_{1}\\\ \sigma_{m,r}(i,r,j)=C_{\beta}&\;{\rm
when}\;\;i=p,j=w+1\;(fixed),r=\beta,0\leq r\leq t_{m-n},\;0\leq i\leq t_{1}\\\
\sigma_{m,r}(r,i,j)=D_{\beta}&\;{\rm when}\;\;i=q,j=w+1\;(fixed),r=\beta,0\leq
r\leq t_{m-n},\;0\leq i\leq t_{1}\\\ \end{cases}$

and $0$ otherwise. We obtain the desired result

$(d\psi_{\eta}-(-1)^{n-1}\psi_{\eta}d)(\varepsilon^{m}_{r})=J(\varepsilon^{m}_{r}).$

∎

######  Theorem 4.19.

Let $\Lambda=kQ/I$ be a Koszul algebra. Suppose
$\eta:\mathbb{K}_{n}\rightarrow\Lambda$ is a cocycle such that
$\eta=\begin{pmatrix}0&\cdots&0&(f^{1}_{w}f^{1}_{w+1})^{(i)}&0&\cdots&0\end{pmatrix}$
for some $0\leq w\leq t_{1}$. A homotopy lifting map
$\psi_{\eta}:\mathbb{K}_{m}\rightarrow\mathbb{K}_{m-n+1}$ associated to $\eta$
can be defined by

$\psi_{\eta}(\varepsilon^{m}_{r})=b_{m,r}(m-n+1,s+1)f^{1}_{w}\varepsilon^{m-n+1}_{s+1}+b_{m,r}(m-n+1,s)\varepsilon^{m-n+1}_{s}f^{1}_{w+1}$

for some scalars $b_{m,r}(m-n+1,s).$ Moreover, the scalars $b_{m,r}(m-n+1,s)$
satisfy

  * (i)

$A_{\alpha}=\begin{cases}c_{i,\alpha}(m,r,1),&{\rm if}\;p=w+1\\\ 0,&{\rm
if}\;p\neq w+1,\end{cases},\\\
D_{\beta}=\begin{cases}(-1)^{n(m-n)+1}c_{\beta,i}(m,r,m-n),&{\rm if}\;q=w\\\
0,&{\rm if}\;q\neq w\end{cases}$

  * (ii).

$B_{\alpha}=0$ and $C_{\beta}=0$ for all $\alpha$ and $\beta$ where

$A_{\alpha}=b_{m,r}(m-n+1,s+1)c_{p\alpha}(m-n+1,s+1,1)\\\
+(-1)^{n}b_{m-1,j}(m-n,\alpha)c_{p,\alpha}(m-n+1,r,1),\\\
B_{\alpha}=(-1)^{m-n+1}b_{m,r}(m-n+1,s+1)c_{\alpha q}(m-n+1,s+1,m-n)\\\
-(-1)^{m}b_{m-1,j}(m-n,\alpha)c_{\alpha,q}(m-n+1,r,m-n),\\\
C_{\beta}=b_{m,r}(m-n+1,s)c_{p\beta}(m-n+1,s,1)\qquad\qquad\qquad\\\
+(-1)^{n}b_{m-1,j}(m-n,\beta)c_{p\alpha}(m-n+1,r,1)\text{ and }\\\
D_{\beta}=(-1)^{m-n+1}b_{m,r}(m-n+1,s)c_{\beta q}(m-n+1,s,m-n)\\\
+(-1)^{m}b_{m-1,j}(m-n,\beta)c_{\beta,q}(m-n+1,r,m-n).$

###### Proof.

We have already established from Lemma 4.16 that

$(d\psi_{\eta}-(-1)^{n-1}\psi_{\eta}d)(\varepsilon^{m}_{r})=J(\varepsilon^{m}_{r})$

Applying definition (ii) of the theorem, that is substitute $B_{\alpha}=0$ and
$C_{\beta}=0$ into Equation 4 of Lemma 4.16. We get

$\displaystyle(d\psi_{\eta}-(-1)^{n-1}\psi_{\eta}d)(\varepsilon^{m}_{r})$
$\displaystyle=\sum_{\alpha=0}^{t_{m-n}}\sum_{p=0}^{t_{1}}\Big{[}A_{\alpha}\Big{]}f^{1}_{w}f_{p}^{1}\varepsilon^{m-n}_{\alpha}+\sum_{\beta=0}^{t_{m-n}}\sum_{q=0}^{t_{1}}\Big{[}D_{\beta}\Big{]}\varepsilon^{m-n}_{\beta}f^{1}_{q}f^{1}_{w+1}.$

After applying the definition of $A_{\alpha}$ and $D_{\beta}$ of (i) of the
theorem into the above expression, we obtain

$\displaystyle\sum_{\alpha=0}^{t_{m-n}}c_{i,\alpha}(m,r,1)f^{1}_{w}f^{1}_{w+1}\varepsilon^{m-n}_{\alpha}-(-1)^{n(m-n)}\sum_{\beta=0}^{t_{m-n}}c_{\beta,i}(m,r,m-n)\varepsilon^{m-n}_{\beta}f^{1}_{w}f^{1}_{w+1}.$

On the other hand, using the multiplicative structure on $\mathbb{K}$, we get
$\displaystyle{\Delta_{\mathbb{K}}(\varepsilon^{m}_{r})=\sum_{v=0}^{m}\sum_{x=0}^{t_{v}}\sum_{y=0}^{t_{m-v}}c_{x,y}(m,r,v)\varepsilon^{v}_{x}\otimes_{\Lambda}\varepsilon^{m-v}_{y}.}$
Applying $(\eta\otimes 1-1\otimes\eta)$, we obtain

$(\eta\otimes
1-1\otimes\eta)\Delta_{\mathbb{K}}(\varepsilon^{m}_{r})=\sum_{v=0}^{m}\sum_{x=0}^{t_{v}}\sum_{y=0}^{t_{m-v}}c_{x,y}(m,r,v)(\eta\otimes
1)(\varepsilon^{v}_{x}\otimes_{\Lambda}\varepsilon^{m-v}_{y})\\\
-\sum_{v=0}^{m}\sum_{x=0}^{t_{v}}\sum_{y=0}^{t_{m-v}}c_{x,y}(m,r,v)(1\otimes\eta)(\varepsilon^{v}_{x}\otimes_{\Lambda}\varepsilon^{m-v}_{y})$

whenever $v=n,x=i$ in the first part and $m-n=v,y=i$ in the second part, the
above expression will yield

$\displaystyle=\sum_{y=0}^{t_{m-n}}c_{i,y}(m,r,1)(\eta\otimes
1)(\varepsilon^{n}_{i}\otimes_{\Lambda}\varepsilon^{m-n}_{y})-\sum_{x=0}^{t_{m-n}}c_{x,i}(m,r,m-n)(1\otimes\eta)(\varepsilon^{m-n}_{x}\otimes_{\Lambda}\varepsilon^{n}_{i})$
$\displaystyle=\sum_{y=0}^{t_{m-n}}c_{i,y}(m,r,1)\eta(\varepsilon^{n}_{i})\varepsilon^{m-n}_{y}-(-1)^{n(m-n)}\sum_{x=0}^{t_{m-n}}c_{x,i}(m,r,m-n)\varepsilon^{m-n}_{x}\eta(\varepsilon^{n}_{i})$

which after applying the definition of $\eta$ and re-indexing, we get

$\displaystyle=\sum_{y=0}^{t_{m-n}}c_{i,y}(m,r,1)f^{1}_{w}f^{1}_{w+1}\varepsilon^{m-n}_{y}-(-1)^{n(m-n)}\sum_{x=0}^{t_{m-n}}c_{x,i}(m,r,m-n)\varepsilon^{m-n}_{x}f^{1}_{w}f^{1}_{w+1}$

∎

The following theorem gives a combinatorial description of what we obtain when
the Gerstenhaber bracket of any two Hochschild cochains is applied to free
basis elements.

######  Theorem 4.20.

Suppose that $\eta:\mathbb{K}_{n}\rightarrow\Lambda$ and
$\theta:\mathbb{K}_{m}\rightarrow\Lambda$ represent elements in
$\operatorname{HH}^{*}(\Lambda)$ and are given by
$\eta=\begin{pmatrix}0&\cdots&0&(\lambda_{i})^{(i)}&0&\cdots&0\end{pmatrix}$
and
$\theta=\begin{pmatrix}0&\cdots&0&(\lambda_{j})^{(j)}&0&\cdots&0\end{pmatrix}$
for fixed $i,j$ where $0\leq i\leq t_{n}$ and $0\leq j\leq t_{m}.$ Then the
bracket $[\eta,\theta]:\mathbb{K}_{n+m-1}\rightarrow\Lambda$ has the property
that

$\displaystyle[\eta,\theta](\varepsilon^{m+n-1}_{r})\in$
$\displaystyle\begin{cases}kQ_{1}&{\rm if}\;\lambda_{i}=f^{1}_{i}\;{\rm
and}\;\lambda_{j}=f^{1}_{j},\\\ kQ_{2}&{\rm
if}\;\lambda_{i}=f^{1}_{i}f^{1}_{i+1}\;{\rm and}\;\lambda_{j}=f^{1}_{j}\\\
kQ_{3}&{\rm if}\;\lambda_{i}=f^{1}_{i}f^{1}_{i+1}\;{\rm
and}\;\lambda_{j}=f^{1}_{j}f^{1}_{j+1}.\end{cases}$

###### Proof.

Using the definition of Gerstenhaber bracket of Definition 3.5, we get

$\displaystyle[\eta,\theta](\varepsilon^{m+n-1}_{r})$
$\displaystyle=(\eta\psi_{\theta}-(-1)^{(m-1)(n-1)}\theta\psi_{\eta})(\varepsilon^{m+n-1}_{r})$
$\displaystyle=\eta\psi_{\theta}(\varepsilon^{m+n-1}_{r})-(-1)^{(m-1)(n-1)}\theta\psi_{\eta}(\varepsilon^{m+n-1}_{r})$

We now apply the definition of a homotopy lifting map as given in Theorems
4.10 and 4.19.
(1) Suppose that both are paths of length 1, i.e.
$\lambda_{i}=f^{1}_{w},\lambda_{j}=f^{1}_{p}$. We will get
$\eta(b_{m+n-1,r}(n,r^{\prime\prime})\cdot\varepsilon^{n}_{r^{\prime\prime}})-(-1)^{(m-1)(n-1)}\theta(b_{m+n-1,r}(m,r^{\prime})\cdot\varepsilon^{m}_{r^{\prime}})$.
This expression will give $0$ or a non-zero path. We are interested in the
non-zero case i.e. when $r^{\prime\prime}=i$ (or
$\eta(\varepsilon^{n}_{r^{\prime\prime}})=\lambda_{i}$) and $r^{\prime}=j$ (or
$\theta(\varepsilon^{m}_{r^{\prime}})=\lambda_{j}$). This yields
$b_{m+n-1,r}(n,i)\lambda_{i}-(-1)^{(m-1)(n-1)}b_{m+n-1,r}(m,j)\lambda_{j}$
which is a combination of paths of length 1.
(2) Suppose that one of them is a path of length 2,
i.e.$\lambda_{i}=f^{1}_{w}f^{1}_{w+1},\;\lambda_{j}=f^{1}_{p}.$ The expression
will yield
$\eta(b_{m+n-1,r}(n,r^{\prime\prime})\cdot\varepsilon^{n}_{r^{\prime\prime}})-(-1)^{(m-1)(n-1)}\theta[b_{m+n-1,r}(m,r^{\prime}+1)f^{1}_{w}\cdot\varepsilon^{m}_{r^{\prime}+1}+b_{m+n-1,r}(m,r^{\prime})\cdot\varepsilon^{m}_{r^{\prime}}f^{1}_{w+1}].$
Now consider cases in which we get non-zero paths:

$\begin{cases}b_{m+n-1,r}(n,i)\lambda_{i}-(-1)^{(m-1)(n-1)}b_{m+n-1,r}(m,j)f^{1}_{w}\lambda_{j}&{\rm
if}\;r^{\prime\prime}=i,r^{\prime}+1=j\\\
b_{m+n-1,r}(n,i)\lambda_{i}-(-1)^{(m-1)(n-1)}b_{m+n-1,r}(m,j)\lambda_{j}f^{1}_{w+1}&{\rm
if}\;r^{\prime\prime}=i,r^{\prime}=j\end{cases}$

which are combination of paths of lenth 2.
(3) Suppose that both are paths of length 2. Let
$\lambda_{i}=f^{1}_{w}f^{1}_{w+1},\;\lambda_{j}=f^{1}_{p}f^{1}_{p+1},$ we
obtain for the bracket expression

$\eta[b_{m+n-1,r}(n,r^{\prime\prime}+1)f^{1}_{p}\cdot\varepsilon^{n}_{r^{\prime\prime}+1}+b_{m+n-1,r}(n,r^{\prime\prime})\cdot\varepsilon^{n}_{r^{\prime\prime}}f^{1}_{p+1}]\\\
-(-1)^{(m-1)(n-1)}\theta[b_{m+n-1,r}(m,r^{\prime}+1)f^{1}_{w}\cdot\varepsilon^{m}_{r^{\prime}+1}+b_{m+n-1,r}(m,r^{\prime})\cdot\varepsilon^{m}_{r^{\prime}}f^{1}_{w+1}].$

After eliminating the cases which becomes 0, what we get is

$\begin{cases}b_{m+n-1,r}(n,i)f^{1}_{p}\lambda_{i}-(-1)^{(m-1)(n-1)}b_{m+n-1,r}(m,j)f^{1}_{w}\lambda_{j}&{\rm
if}\;r^{\prime\prime}+1=i,r^{\prime}+1=j\\\
b_{m+n-1,r}(n,i)f^{1}_{p}\lambda_{i}-(-1)^{(m-1)(n-1)}b_{m+n-1,r}(m,j)\lambda_{j}f^{1}_{w+1}&{\rm
if}\;r^{\prime\prime}+1=i,r^{\prime}=j\\\
b_{m+n-1,r}(n,i)\lambda_{i}f^{1}_{p}-(-1)^{(m-1)(n-1)}b_{m+n-1,r}(m,j)f^{1}_{w}\lambda_{j}&{\rm
if}\;r^{\prime\prime}=i,r^{\prime}+1=j\\\
b_{m+n-1,r}(n,i)\lambda_{i}f^{1}_{p+1}-(-1)^{(m-1)(n-1)}b_{m+n-1,r}(m,j)\lambda_{j}f^{1}_{w}&{\rm
if}\;r^{\prime\prime}=i,r^{\prime}=j\end{cases}$

which are all linear combination of paths of length 3. ∎

Any cocycle $\eta$ of degree $n$ can be thought of as a sum of maps
$\eta=\sum_{i=0}^{t_{n}}\eta^{(i)}$ where $\eta^{(i)}$ is the map taking the
$i$-th basis element $\varepsilon^{n}_{i}$ to $\lambda,$ a non-zero element of
the algebra, and all other basis elements $\varepsilon^{n}_{s}$ to $0$, $i\neq
s$. Consistent with our notation, this map is written
$\eta^{(i)}=\begin{pmatrix}0&\cdots&0&(\lambda)^{(i)}&0&\cdots&0\end{pmatrix},$
and we use this notation in the following theorem.

######  Theorem 4.21.

Let $\eta:\mathbb{K}_{n}\rightarrow\Lambda$ and
$\theta:\mathbb{K}_{m}\rightarrow\Lambda$ represent elements in
$\operatorname{HH}^{*}(\Lambda)$ and are given by
$\eta(\varepsilon^{n}_{i})=\lambda_{i}$ for $i=0,1,\ldots,t_{n}$ and
$\theta(\varepsilon^{m}_{j})=\beta_{j}$ for $j=0,1,\ldots,t_{m}.$ Then the
$r$-component of the bracket
$[\eta,\theta]:\mathbb{K}_{n+m-1}\rightarrow\Lambda$ denoted by
$[\eta,\theta]^{(r)}$ can be expressed on the $r$-th basis element
$\varepsilon^{m+n-1}_{r}$ as

$\displaystyle[\eta,\theta]^{(r)}(\varepsilon^{m+n-1}_{r})=\sum_{i=0}^{t_{n}}\sum_{j=0}^{t_{m}}b_{m-n+1,r}(n,i)\lambda_{i}-(-1)^{(m-1)(n-1)}(b_{m-n+1,r}(m,j)\beta_{j}$

provided $\lambda_{i},\beta_{j}$ are all paths of length 1.

###### Proof.

Write $\eta=\sum_{i=0}^{t_{n}}\eta^{(i)}$ where
$\eta^{(i)}=\begin{pmatrix}0&\cdots&0&(\lambda_{i})^{(i)}&0&\cdots&0\end{pmatrix}$
and $\theta=\sum_{j=0}^{t_{m}}\theta^{(j)}$ where
$\theta^{(j)}=\begin{pmatrix}0&\cdots&0&(\beta_{j})^{(j)}&0&\cdots&0\end{pmatrix}.$
Using the definition of Gerstenhaber bracket of 3.5, we get for $0\leq r\leq
t_{m+n-1}$,

$\displaystyle[\eta,\theta]^{(r)}(\varepsilon^{m+n-1}_{r})=[\sum_{i=0}^{t_{n}}\eta^{(i)},\sum_{j=0}^{t_{m}}\theta^{(j)}]^{r}(\varepsilon^{m+n-1}_{r})=\sum_{i=0}^{t_{n}}\sum_{j=0}^{t_{m}}[\eta^{(i)},\theta^{(j)}]^{r}(\varepsilon^{m+n-1}_{r})$
$\displaystyle=\sum_{i=0}^{t_{n}}\sum_{j=0}^{t_{m}}(\eta^{(i)}\psi_{\theta^{(j)}}-(-1)^{(m-1)(n-1)}\theta^{(j)}\psi_{\eta^{(i)}})(\varepsilon^{m+n-1}_{r})$

Since $\lambda_{i}=f^{1}_{w_{i}}$ for all $i$, and $\beta_{j}=f^{1}_{p_{j}}$
for all $j$, then the homotopy lifting maps can be defined as
$\psi_{\theta^{(j)}}(\varepsilon^{m+n-1}_{r})=b_{m-n+1,r}(n,i)\varepsilon^{n}_{i}$
for some $i$ and
$\psi_{\eta^{(i)}}(\varepsilon^{m+n-1}_{r})=b_{m-n+1,r}(m,j)\varepsilon^{m}_{j}$
for some $j$. Applying this, we get

$\displaystyle\sum_{i=0}^{t_{n}}\sum_{j=0}^{t_{m}}\eta^{(i)}(b_{m-n+1,r}(n,i)\varepsilon^{n}_{i})-(-1)^{(m-1)(n-1)}\theta^{(j)}(b_{m-n+1,r}(m,j)\varepsilon^{m}_{j})$
$\displaystyle=\sum_{i=0}^{t_{n}}\sum_{j=0}^{t_{m}}b_{m-n+1,r}(n,i)\lambda_{i}-(-1)^{(m-1)(n-1)}(b_{m-n+1,r}(m,j)\beta_{j}$

∎

##  5\. Application

In this section, we give an application of our results in Theorems 4.10 and
4.19 to specify solutions to the Maurer-Cartan equation. The Maurer-Cartan
equation is useful in the theory of deformation of algebras.

It is known that Hochschild cohomology is a differential graded Lie algebra
i.e.

$\bar{d}([f,g])=[\bar{d}(f),g]+(-1)^{(m-1)}[f,\bar{d}(g)],$

where $\bar{d}(f)=(-1)^{(m-1)}f\delta_{m+1}$. Since the resolution
$(\mathbb{K},d)$ embeds into the bar reduced resolution $(\mathcal{B},\delta)$
via $\mathbb{K}\xrightarrow{\iota}\mathcal{B}$, with $\iota d=\delta\iota$,
there are no sign changes, hence we take
$\bar{d}(\eta)=(-1)^{(m-1)}d^{*}_{m+1}\eta=(-1)^{(m-1)}\eta d_{m+1}.$ An
Hochschild 2-cocycle $\eta$ is then said to satisfy the Maurer-Cartan equation
if

(5.1) $\bar{d}(\eta)+\frac{1}{2}[\eta,\eta]=0$

Applying the definition of the bracket to a 2-cocycle, we obtain the following
version of the Maurer-Cartan equation

(5.2)
$-d^{*}_{3}(\eta)=-\frac{1}{2}(\eta\psi_{\eta}+\eta\psi_{\eta})=-\eta\psi_{\eta}$

######  Theorem 5.3.

Let $k$ be a field and $\Lambda=kQ/I$ be a Koszul algebra. Suppose
$\eta:\mathbb{K}_{2}\rightarrow\Lambda$ is a cocycle such that
$\eta=\begin{pmatrix}0&\cdots&0&(\lambda)^{(i)}&0&\cdots&0\end{pmatrix}$,
$i=0,1,2,\ldots,t_{2}$. If $\lambda\in kQ_{2}$, i.e. a linear combination of
paths of length 2, then $\eta$ satisfies the Maurer-Cartan equation.

###### Proof.

It is enough to check that for the case where $\lambda=f^{1}_{u}f^{1}_{v},$
the result is true. The left hand side of Equation 5.2 is given by

$\displaystyle d^{*}_{3}(\eta)(\varepsilon^{3}_{r})=\eta
d_{3}(\varepsilon^{3}_{r})=\eta\Big{[}\sum_{j=0}^{t_{2}}<\varepsilon^{2}_{j}>_{3,r}\Big{]}=\eta\sum_{j=0}^{t_{2}}\Big{[}\sum_{p=0}^{t_{1}}c_{pj}(3,r,1)f_{p}^{1}\varepsilon^{2}_{j}-\sum_{q=0}^{t_{1}}c_{jq}(3,r,2)\varepsilon^{2}_{j}f_{q}^{1}\Big{]}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{pi}(3,r,1)f_{p}^{1}\eta(\varepsilon^{2}_{i})-\sum_{q=0}^{t_{1}}c_{iq}(3,r,2)\eta(\varepsilon^{2}_{i})f_{q}^{1}$
$\displaystyle=\sum_{p=0}^{t_{1}}c_{pi}(3,r,1)f_{p}^{1}f^{1}_{u}f^{1}_{v}-\sum_{q=0}^{t_{1}}c_{iq}(3,r,2)f^{1}_{u}f^{1}_{v}f_{q}^{1}$

which is in $kQ_{3}$. On the other hand, the result of Theorem 4.20 states
that $[\eta,\eta](\varepsilon^{3}_{r})\in kQ_{3}$ since
$\lambda=f^{1}_{u}f^{1}_{v},$ a path of length 2. In particular, the right
hand side of Equation (5.2) becomes

$\displaystyle\frac{1}{2}[\eta,\eta](\varepsilon^{3}_{r})=\eta\psi_{\eta}(\varepsilon^{3}_{r})=\eta(b_{3,r}(2,i)f^{1}_{u}\varepsilon^{2}_{i}+b_{3,r}(2,j)\varepsilon^{2}_{j}f^{1}_{v})$
$\displaystyle=b_{3,r}(2,i)f^{1}_{u}\eta(\varepsilon^{2}_{i})=b_{3,r}(2,i)f^{1}_{u}f^{1}_{u}f^{1}_{v}.$

This expression could have yielded $b_{3,r}(2,i)f^{1}_{u}f^{1}_{v}f^{1}_{v}$
if we take $\psi_{\eta}(\varepsilon^{3}_{r})$ to be
$b_{3,r}(2,j)f^{1}_{u}\varepsilon^{2}_{j}+b_{3,r}(2,i)\varepsilon^{2}_{i}f^{1}_{v}$
but it doesn’t affect the result. The solution to this equation is obtained by
identifying possible scalars for which

$\displaystyle\sum_{p=0}^{t_{1}}c_{pi}(3,r,1)f_{p}^{1}f^{1}_{u}f^{1}_{v}-\sum_{q=0}^{t_{1}}c_{iq}(3,r,2)f^{1}_{u}f^{1}_{v}f_{q}^{1}=b_{3,r}(2,i)f^{1}_{u}f^{1}_{u}f^{1}_{v}.$

If $c_{pi}(3,r,1)=0$ for all $p\neq u$, $c_{iq}(3,r,2)=0$ for all $q$ and
$c_{pi}(3,r,1)=b_{3,r}(2,i)$ when $p=u$, the Maurer-Cartan equation (5.1)
holds.

######  Remark 5.4.

A 2-cocycle defined with $\lambda=f^{1}_{u}$ cannot satisfy the Maurer-Cartan
equation. This is because while the left hand side of Equation 5.2 yields a
linear combination of paths of length 2, the right hand side yields a linear
combination of paths of length 1.

∎

##  6\. Short Example

Let $k$ be a field of characteristics different from 2. Consider the quiver
algebra $A=kQ/I$ (also examined in [2, Example 5]) defined using the following
finite quiver:

${1}$$\scriptstyle{x}$$\scriptstyle{y}$

with one vertex and two arrows $x,y.$ We denote by $e_{1}$ the idempotent
associated with the only vertex. Let $I$, an ideal of the path algebra $kQ$ be
defined by

$I=\langle x^{2},xy+yx\rangle.$

Since $\\{x^{2},xy+yx\\}$ is a quadratic Grobner basis for the ideal generated
by relations under the length lexicographich order with $x>y>1$, the algebra
is Koszul.

In order to define a comultiplicative structure, we take $t_{0}=0,t_{n}=1$ for
all $n$,
$f_{0}^{0}=e_{1},f_{1}^{0}=0,f_{0}^{1}=x,f_{1}^{1}=y,f^{2}_{0}=x^{2},f^{2}_{1}=xy+yx,f_{0}^{3}=x^{3},f_{1}^{3}=x^{2}y+xyx+yx^{2},$
and in general $f^{n}_{0}=x^{n},f^{n}_{1}=\sum_{i+j=n-1}x^{i}yx^{j}$. We also
see that $f^{n}_{0}=f^{r}_{0}f^{n-r}_{0}$ and
$f^{n}_{1}=f^{r}_{0}f^{n-r}_{1}+f^{r}_{1}f^{n-r}_{0}$ so
$c_{00}(n,0,r)=c_{01}(n,1,r)=c_{10}(n,1,r)=1$ and all other $c_{pq}(n,i,r)=0$.
With the above stated, we can construct the resolution $\mathbb{K}$ for the
algebra $A$. A calculation shows that

$\displaystyle
d_{1}(\varepsilon^{1}_{0})=x\varepsilon^{0}_{0}-\varepsilon^{0}_{0}x,$
$\displaystyle
d_{1}(\varepsilon^{1}_{1})=y\varepsilon^{0}_{0}-\varepsilon^{0}_{0}y$
$\displaystyle
d_{2}(\varepsilon^{2}_{0})=x\varepsilon^{1}_{0}+\varepsilon^{1}_{0}x,$
$\displaystyle
d_{2}(\varepsilon^{2}_{1})=y\varepsilon^{1}_{0}+\varepsilon^{1}_{0}y+x\varepsilon^{1}_{1}+\varepsilon^{1}_{1}x.$

Consider the following maps $\chi,\theta:\mathbb{K}_{1}\xrightarrow{}A$
defined by $\chi=(xy\;\;0)$ and $\theta=(0\;\;y)$. Observe that
$\chi+\theta=(xy\;\;y)$ was originally given as an example in [2]. With the
following calculations

$\displaystyle\chi d_{2}(\varepsilon^{2}_{0})$
$\displaystyle=\chi(x\varepsilon^{1}_{0}+\varepsilon^{1}_{0}x)=x^{2}y+xyx=x^{2}y-x^{2}y=0$
$\displaystyle\chi d_{2}(\varepsilon^{2}_{1})$
$\displaystyle=\chi(y\varepsilon^{1}_{0}+\varepsilon^{1}_{0}y+x\varepsilon^{1}_{1}+\varepsilon^{1}_{1}x)=yxy+xy^{2}+0=(yx+xy)y=0$
$\displaystyle\theta d_{2}(\varepsilon^{2}_{0})$
$\displaystyle=\theta(x\varepsilon^{1}_{0}+\varepsilon^{1}_{0}x)=0$
$\displaystyle\theta d_{2}(\varepsilon^{2}_{1})$
$\displaystyle=\theta(y\varepsilon^{1}_{0}+\varepsilon^{1}_{0}y+x\varepsilon^{1}_{1}+\varepsilon^{1}_{1}x)=0+xy+yx=0,$

$\chi$ and $\theta$ are cocycles. The comultiplicative map
$\Delta:\mathbb{K}\xrightarrow{}\mathbb{K}\otimes_{A}\mathbb{K}$ on
$\varepsilon^{1}_{0}$,
$\varepsilon^{1}_{1},\varepsilon^{2}_{0},\varepsilon^{2}_{1}$ is given by

$\displaystyle\Delta(\varepsilon^{1}_{0})$
$\displaystyle=c_{00}(1,0,0)\varepsilon^{0}_{0}\otimes\varepsilon^{1}_{0}+c_{00}(1,0,1)\varepsilon^{1}_{0}\otimes\varepsilon^{0}_{0}=\varepsilon^{0}_{0}\otimes\varepsilon^{1}_{0}+\varepsilon^{1}_{0}\otimes\varepsilon^{0}_{0},$
$\displaystyle\Delta(\varepsilon^{1}_{1})$
$\displaystyle=\varepsilon^{0}_{0}\otimes\varepsilon^{1}_{1}+\varepsilon^{1}_{1}\otimes\varepsilon^{0}_{0},$
$\displaystyle\Delta(\varepsilon^{2}_{0})$
$\displaystyle=\varepsilon^{0}_{0}\otimes\varepsilon^{2}_{0}+\varepsilon^{1}_{0}\otimes\varepsilon^{1}_{0}+\varepsilon^{2}_{0}\otimes\varepsilon^{0}_{0},$
$\displaystyle\Delta(\varepsilon^{2}_{1})$
$\displaystyle=\varepsilon^{0}_{0}\otimes\varepsilon^{2}_{1}+\varepsilon^{1}_{0}\otimes\varepsilon^{1}_{1}+\varepsilon^{1}_{1}\otimes\varepsilon^{1}_{0}+\varepsilon^{2}_{1}\otimes\varepsilon^{0}_{0}.$

From Theorems 4.10 and 4.19, it can be verified by direct calculations that
the first, second and third degree of the homotopy lifting maps $\psi_{\chi}$
and $\psi_{\theta}$ associated with $\chi$ and $\eta$ respectively are the
following:

$\displaystyle\psi_{\chi_{0}}=0,\qquad$
$\displaystyle\psi_{\chi_{1}}(\varepsilon^{1}_{0})=x\varepsilon^{1}_{1}+\varepsilon^{1}_{0}y,\psi_{\chi_{1}}(\varepsilon^{1}_{1})=0$
$\displaystyle\psi_{\chi_{2}}(\varepsilon^{2}_{0})=x\varepsilon^{2}_{1},\psi_{\chi_{2}}(\varepsilon^{2}_{1})=\varepsilon^{2}_{1}y$
$\displaystyle\psi_{\theta_{0}}=0,\qquad$
$\displaystyle\psi_{\theta_{1}}(\varepsilon^{1}_{0})=0,\qquad\psi_{\theta_{1}}(\varepsilon^{1}_{1})=\varepsilon^{1}_{1}$
$\displaystyle\psi_{\theta_{2}}(\varepsilon^{2}_{0})=0,\psi_{\theta_{2}}(\varepsilon^{2}_{1})=\varepsilon^{2}_{1}.$

For the cocycle $\chi$ for instance, $b_{20}(2,1)=b_{2,1}(2,1)=1$,
$b_{10}(1,0)=b_{10}(1,1)=1$ and the Gerstenhaber bracket of $\chi$ and
$\theta$ is given by $[\chi,\theta]=-\chi.$

##  7\. Long Example

In this section, we give four examples of homotopy lifting maps that come from
a family of quiver algebras. While a member of this family was introduced in
[1] as a counterexample to the Snashall-Solberg finite generation conjecture,
the Hochschild cohomology modulo nilpotent cocycles of this family as a whole
was studied in [9]. See [1] for details on the Snashall-Solberg finite
generation conjecture. Two of these homotopy lifting maps come from degree 1
cocycles while the other two are homotopy lifting maps for degree 2 cocycles.
We adapted a version of the resolution $\mathbb{K}$ for this family and find
cocycles. We especially note how these homotopy lifting maps follow from their
generalized versions presented in Theorem 4.10 and Theorem 4.19 of Section 4.
We round off the section by making connections to derivation operators.

We begin with the following finite quiver:

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with two vertices and three arrows $a,b,c.$ We denote by $e_{1}$ and $e_{2}$
the idempotents associated with vertices 1 and 2. Let $kQ$ be the path algebra
associated with $Q$ and take for each $q\in k$, $I_{q}\subseteq kQ$ to be an
admissible ideal of $kQ$ generated as follows

$I_{q}=\langle a^{2},b^{2},ab-qba,ac\rangle.$

Let $\\{\Lambda_{q}\\}_{q\in k}=\\{kQ/I_{q}\\}_{q\in k}$ be a family of quiver
algebras generated by the quiver $Q$ and the ideal $I_{q}$. To define a set of
free basis for the resolution $\mathbb{K}$ we start by letting $kQ_{0}$ to be
the ideal of $kQ$ generated by the vertices of $Q$ with basis
$f_{0}^{0}=e_{1},f_{1}^{0}=e_{2}.$ Next, set $kQ_{1}$ to be the ideal
generated by paths with basis $f_{0}^{1}=a,f_{1}^{1}=b$ and $f_{2}^{1}=c.$ Set
$f^{2}_{j},$ $j=0,1,2,3$ to be the set of paths of length 2 that generates the
ideal $I$, that is $f_{0}^{2}=a^{2},f_{1}^{2}=ab-
qba,f_{2}^{2}=b^{2},f^{2}_{3}=ac,$ and define a comultiplicative equation on
the paths of length $n>2$ in the following way.

$\begin{cases}f^{n}_{0}=a^{n},\\\
f^{n}_{s}=f^{n-1}_{s-1}b+(-q)^{s}f^{n-1}_{s}a,&(0<s<n),\\\ f^{n}_{n}=b^{n},\\\
f^{n}_{n+1}=a^{(n-1)}c,\end{cases}$

The resolution $\mathbb{K}\rightarrow\Lambda_{q}$ has free basis elements
$\\{\varepsilon^{n}_{i}\\}_{i=0}^{t_{n}}$ such that for each $i$, we have
$\varepsilon^{n}_{i}=(0,\ldots,0,o(f^{n}_{i})\otimes_{k}t(f^{n}_{i}),0,\ldots,0)$.
The differentials on $\mathbb{K}_{n}$ are given explicitly for this family by

$\displaystyle d_{1}(\varepsilon^{1}_{2})$
$\displaystyle=c\varepsilon^{0}_{1}-\varepsilon^{0}_{0}c$ $\displaystyle
d_{n}(\varepsilon^{n}_{r})$
$\displaystyle=(1-\partial_{n,r})[a\varepsilon^{n-1}_{r})+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a]$
$\displaystyle+(1-\partial_{r,0})[(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(-1)^{n}\varepsilon^{n-1}_{r-1}b],\;\;\text{for}\;\;r\leq
n$ $\displaystyle d_{n}(\varepsilon^{n}_{n+1})$
$\displaystyle=a\varepsilon^{n-1}_{n}+(-1)^{n}\varepsilon^{n-1}_{0}c,\;\;\text{when}\;\;n\geq
2,$

where $\partial_{r,s}=1$ when $r=s$ and $0$ when $r\neq s$.

For any member $\Lambda_{q}$ of the above family of quiver algebras, we define
the bar resolution $(\mathcal{B},\delta)$ of $\Lambda_{q}$ by
$\mathcal{B}_{n}=(\Lambda_{q})^{\otimes_{\Lambda^{*}_{0}}(n+2)}$, the
$(n+2)$-fold tensor product of $\Lambda_{q}$ over $\Lambda^{*}_{0}$. Note here
that $\Lambda^{*}_{0}$ comes from the grading on $\Lambda_{q}$ for each $q$
and is not the same as the algebra $\Lambda_{q},q=0$. It is isomorphic to two
copies of $k$ generated by the two vertices of the quiver $Q$. Before we begin
to present the homotopy lifting maps, we give explicit description of the
comultiplicative map
$\Delta_{\mathbb{K}}:\mathbb{K}\rightarrow\mathbb{K}\otimes_{\Lambda_{q}}\mathbb{K}$
on the resolution $\mathbb{K}$. As was mentioned in the preliminaries, the
resolution $\mathbb{K}$ embeds into the reduced bar resolution $\mathcal{B}$
via the map $\iota:\mathbb{K}_{n}\rightarrow\mathcal{B}_{n}$ defined by
$\varepsilon^{n}_{r}\mapsto 1\otimes\widetilde{f^{n}_{r}}\otimes 1$, where
each $\widetilde{f^{n}_{r}}$ is viewed as a sum of tensor product of paths of
length 1 tensored over $\Lambda^{*}_{0}$ as given in Equation (2.7). For
example, $\widetilde{f^{2}_{0}}=f^{1}_{0}\otimes f^{1}_{0}=a\otimes a$,
$\widetilde{f^{2}_{1}}=f^{1}_{0}\otimes f^{1}_{1}-qf^{1}_{1}\otimes
f^{1}_{0}=a\otimes b-qb\otimes a.$ See [2, Proposition 2.1] for details about
this. Equation (2.8) gives the comultiplicative map
$\Delta:\mathcal{B}\rightarrow\mathcal{B}\otimes_{\Lambda_{q}}\mathcal{B}$ on
the bar resolution making the following diagram commutative.

${\mathbb{K}}$${\mathbb{K}\otimes_{\Lambda_{q}}\mathbb{K}}$${\mathcal{B}}$${\mathcal{B}\otimes_{\Lambda_{q}}\mathcal{B}.}$$\scriptstyle{\Delta_{\mathbb{K}}}$$\scriptstyle{\iota}$$\scriptstyle{\iota\otimes\iota}$$\scriptstyle{\Delta}$

It is clear that the following holds;

(7.1) $\begin{cases}\widetilde{f^{n}_{0}}=f^{1}_{0}\otimes
f^{1}_{0}\otimes\cdots\otimes f^{1}_{0},&{\rm n-times},\\\
\widetilde{f^{n}_{s}}=\widetilde{f^{n-1}_{s-1}}\otimes
f^{1}_{1}+(-q)^{s}\widetilde{f^{n-1}_{s}}\otimes f^{1}_{0},&{\rm(0<s<n)},\\\
\widetilde{f^{n}_{n}}=f^{1}_{1}\otimes f^{1}_{1}\otimes\cdots\otimes
f^{1}_{1},&{\rm n-times},\\\ \widetilde{f^{n}_{n+1}}=f^{1}_{0}\otimes
f^{1}_{0}\otimes\cdots\otimes f^{1}_{0}\otimes f^{1}_{2},&{\rm f^{1}_{0}\text{
appearing }(n-1)-times},\end{cases}$

In case $0<s<n$, it was also shown in [9] that
$f^{n}_{s}=\sum_{j=max\\{0,r+t-n\\}}^{min\\{t,s\\}}(-q)^{j(n-s+j-t)}f^{t}_{j}f^{n-t}_{s-j},$
hence,

(7.2)
$\widetilde{f^{n}_{s}}=\sum_{j=max\\{0,r+t-n\\}}^{min\\{t,s\\}}(-q)^{j(n-s+j-t)}\widetilde{f^{t}_{j}}\otimes\widetilde{f^{n-t}_{s-j}}.$

Calculations from [9] show that for this family, the comultiplicative map can
be expressed in the following way

$\Delta_{\mathbb{K}}(\varepsilon^{n}_{s})=\begin{cases}\displaystyle{\sum_{r=0}^{n}\varepsilon^{r}_{0}\otimes\varepsilon^{n-r}_{0},}&s=0\\\
\displaystyle{\sum_{w=0}^{n}\sum_{j=max\\{0,s+w-n\\}}^{min\\{w,s\\}}(-q)^{j(n-s+j-w)}\varepsilon^{w}_{j}\otimes\varepsilon^{n-w}_{s-j},}&0<s<n\\\
\displaystyle{\sum_{t=0}^{n}\varepsilon^{t}_{t}\otimes\varepsilon^{n-t}_{n-t},}&s=n\\\
\displaystyle{\varepsilon^{0}_{0}\otimes\varepsilon^{n}_{n+1}+\Big{[}\sum_{t=0}^{n}\varepsilon^{t}_{0}\otimes\varepsilon^{n-t}_{n-t+1}\Big{]}+\varepsilon^{n}_{n+1}\otimes\varepsilon^{0}_{0},}&s=n+1.\end{cases}$

###  7.1. Homotopy lifting maps from degree 2 cocycles

In this section, we will find Hochschild 2 cocycles. From these cocycles we
will select two cocycles and find homotopy lifting maps associated to them.
Our choice was arbitrary. However, these examples demonstrate that any
homotopy lifting map arises in the form described in Theorems 4.10 and 4.19.
Notation: As it was previously used in Section 4, we recall and use the
following standard notation for ease of computations. Since the set
$\\{\varepsilon^{n}_{r}\\}_{r=0}^{t_{n}},$ $t_{n}=n+1$ forms a basis for
$\mathbb{K}_{n}$, any module homomorphism
$\Theta:\mathbb{K}_{n}\rightarrow\Lambda_{q}$ taking $\varepsilon^{n}_{i}$ to
$\lambda_{i}$, $i=0,1,\ldots,n+1$ is denoted by

$\Theta=\begin{pmatrix}\lambda_{0}&\lambda_{1}&\cdots&\lambda_{t_{n}}\end{pmatrix}.$

Suppose that the $\Lambda_{q}^{e}$-module homomorphism
$\eta:\mathbb{K}_{2}\rightarrow\Lambda_{q}$ defined by
$\eta=\begin{pmatrix}\lambda_{0}&\lambda_{1}&\lambda_{2}&\lambda_{3}\end{pmatrix}$
is a cocycle, that is $d^{*}\eta=0,$ with $\lambda_{i}\in\Lambda_{q}$ for all
$i$. Since $d^{*}\eta:\mathbb{K}_{3}\rightarrow\Lambda_{q}$, we obtain using
$d^{*}\eta(\varepsilon^{3}_{r})=\eta d(\varepsilon^{3}_{r})$,

$\eta\cdot
d(\varepsilon^{3}_{i})=\eta\Big{(}\begin{cases}a\varepsilon^{2}_{0}-\varepsilon^{2}_{0}a&{\rm
if}\;i=0\\\
a\varepsilon^{2}_{1}+q\varepsilon^{2}_{1}a+q^{2}b\varepsilon^{2}_{0}-\varepsilon^{2}_{0}b&{\rm
if}\;i=1\\\
a\varepsilon^{2}_{2}-q^{2}\varepsilon^{2}_{2}a-qb\varepsilon^{2}_{1}-\varepsilon^{2}_{1}b&{\rm
if}\;i=2\\\ b\varepsilon^{2}_{2}-\varepsilon^{2}_{2}b&{\rm if}\;i=3\\\
a\varepsilon^{2}_{3}-\varepsilon^{2}_{0}c&{\rm if}\;i=4\end{cases}\Big{)}$

$\eta\cdot d$ may then be identified with the matrix

$\begin{pmatrix}a\lambda_{0}-\lambda_{0}a,&a\lambda_{1}+q\lambda_{1}a+q^{2}b\lambda_{0}-\lambda_{0}b,&a\lambda_{2}-q^{2}\lambda_{2}a-qb\lambda_{1}-\lambda_{1}b,&b\lambda_{2}-\lambda_{2}b,&a\lambda_{3}-\lambda_{0}c\end{pmatrix}$

which will be equated to $\begin{pmatrix}0&0&0&0&0\end{pmatrix}$ and solved.
We solve this system of equations with the following in mind. There is an
isomorphism of $\Lambda_{q}^{e}$-modules
$\operatorname{Hom}_{\Lambda^{e}}(\Lambda
o(f^{n}_{i})\otimes_{k}t(f^{n}_{i})\Lambda,\Lambda)\simeq
o(f^{n}_{i})\Lambda\;t(f^{n}_{i})$ ensuring that

$\displaystyle o(f^{2}_{i})\lambda_{i}t(f^{2}_{i})$
$\displaystyle=o(f^{2}_{i})\eta(\varepsilon^{2}_{i})t(f^{2}_{i})=o(f^{2}_{i})\eta(o(f^{2}_{i})\otimes_{k}t(f^{2}_{i}))t(f^{2}_{i})$
$\displaystyle=\phi(o(f^{2}_{i})^{2}\otimes_{k}t(f^{2}_{i})^{2})=\phi(o(f^{2}_{i})\otimes_{k}t(f^{2}_{i}))=\lambda_{i}.$

This means that for $i=0,1,2$ each $\lambda_{i}$ should satisfy
$e_{1}\lambda_{i}e_{1}=\lambda_{i}$ since the origin and terminal vertex of
$f^{2}_{0},f^{2}_{1},f^{2}_{2}$ is $e_{1}$ and
$e_{1}\lambda_{3}e_{2}=\lambda_{3}$. We obtain the following 9 solutions
presented in Table 1.

solutions | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---
$\lambda_{0}$ | a | ab | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\lambda_{1}$ | 0 | 0 | 0 | 0 | 0 | 0 | ab | 0 | 0
$\lambda_{2}$ | 0 | 0 | a | b | ab | $e_{1}$ | 0 | 0 | 0
$\lambda_{3}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | c | bc
Table 1. Possible values of $\eta(\varepsilon^{2}_{r})=\lambda_{r}$ for
different $r$.

For the rest of this section, we are interested in the first and fifth
cocycles, that is

$\bar{\eta}=\begin{pmatrix}a&0&0&0\end{pmatrix}\text{ and
}\bar{\chi}=\begin{pmatrix}0&0&ab&0\end{pmatrix}.$

Again we are looking at the homotopy lifting maps corresponding to these two
cocycles because all of the other homotopy lifting maps corresponding to other
cocyles will have similar properties.
Homotopy lifting for the first and fifth maps $\bar{\eta}$ and $\bar{\chi}$
will be maps
$\psi_{\bar{\eta}},\psi_{\bar{\chi}}:\mathbb{K}\rightarrow\mathbb{K}[-1]$ such
that

$d\psi_{\bar{\eta}}+\psi_{\bar{\eta}}d=(\bar{\eta}\otimes
1_{\mathbb{K}}-1_{\mathbb{K}}\otimes\bar{\eta})\Delta_{\mathbb{K}},\;\text{
and }\;d\psi_{\bar{\chi}}+\psi_{\bar{\chi}}d=(\bar{\chi}\otimes
1_{\mathbb{K}}-1_{\mathbb{K}}\otimes\bar{\chi})\Delta_{\mathbb{K}}.$

We track the left hand side of the above equations using the following
diagram, which is not necessarily commutative.

${\mathbb{K}:=\cdots}$${\mathbb{K}_{4}}$${\mathbb{K}_{3}}$${\mathbb{K}_{2}}$${\mathbb{K}_{1}}$${\mathbb{K}_{0}}$${\mathbb{K}:=\cdots}$${\mathbb{K}_{3}}$${\mathbb{K}_{2}}$${\mathbb{K}_{1}}$${\mathbb{K}_{0}}$$\scriptstyle{\psi_{\bar{\eta}},\;\psi_{\bar{\chi}}}$$\scriptstyle{d_{5}}$$\scriptstyle{\psi_{\bar{\eta}_{4}},\;\psi_{\bar{\chi}_{4}}}$$\scriptstyle{d_{4}}$$\scriptstyle{\psi_{\bar{\eta}_{3}},\;\psi_{\bar{\chi}_{3}}}$$\scriptstyle{d_{3}}$$\scriptstyle{\psi_{\bar{\eta}_{2}},\;\psi_{\bar{\chi}_{2}}}$$\scriptstyle{d_{2}}$$\scriptstyle{\psi_{\bar{\eta}_{1}},\;\psi_{\bar{\chi}_{1}}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{4}}$$\scriptstyle{d_{3}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{1}}$

We now define
$\psi_{\bar{\eta}_{1}},\psi_{\bar{\eta}_{2}},\psi_{\bar{\eta}_{3}}$ and
$\psi_{\bar{\chi}_{1}},\psi_{\bar{\chi}_{2}},\psi_{\bar{\chi}_{3}}$ only, just
to point out that homotopy lifting maps can be defined as generalized in
Theorem (4.10) and (4.10). Calculations show that

(7.3)
$\psi_{\bar{\eta}_{1}}(\varepsilon^{1}_{i})=0,\;i=0,1,2,\;\psi_{\bar{\eta}_{2}}(\varepsilon^{2}_{i})=\begin{cases}\varepsilon^{1}_{0},&{\rm
if}\;i=0\\\ 0,&{\rm
if}\;i=1,2,3\end{cases},\quad\psi_{\bar{\eta}_{3}}(\varepsilon^{3}_{i})=\begin{cases}0,&{\rm
if}\;i=0\\\ \varepsilon^{2}_{1},&{\rm if}\;i=1\\\ 0,&{\rm if}\;i=2\\\ 0,&{\rm
if}\;i=3\\\ \varepsilon^{2}_{3},&{\rm if}\;i=4\end{cases}$

are the first, second and third degrees of the homotopy lifting map
$\psi_{\bar{\eta}}$ while

(7.4)
$\psi_{\bar{\chi}_{1}}(\varepsilon^{1}_{i})=0,\;i=0,1,2,\;\psi_{\bar{\chi}_{2}}(\varepsilon^{2}_{i})=\begin{cases}0&{\rm
if}\;i=0\\\ 0,&{\rm if}\;i=1\\\ a\varepsilon^{1}_{1}+\varepsilon^{1}_{0}b&{\rm
if}\;i=2\\\ 0&{\rm
if}\;i=3\end{cases},\quad\psi_{\bar{\chi}_{3}}(\varepsilon^{3}_{i})=\begin{cases}0,&{\rm
if}\;i=0\\\ 0,&{\rm if}\;i=1\\\ -a\varepsilon^{2}_{1},&{\rm if}\;i=2\\\
\varepsilon^{2}_{1}b,&{\rm if}\;i=3\\\ 0,&{\rm if}\;i=4\end{cases}$

are the first, second and third degrees of the homotopy lifting map
$\psi_{\bar{\chi}}$. The proof of this claim is by direct computations. We
illustrate with just one of the numerous computations i.e we will show that
for the cocycle $\bar{\eta}$ and $\bar{\chi}$ with $q=1$,

$\displaystyle(d_{2}\psi_{\bar{\eta}_{3}}+\psi_{\bar{\eta}_{2}}d_{3})(\varepsilon^{3}_{i})$
$\displaystyle=(\bar{\eta}\otimes
1-1\otimes\bar{\eta})\Delta_{\mathbb{K}}(\varepsilon^{3}_{i}),\quad
i=0,1,2,3,4$
$\displaystyle(d_{2}\psi_{\bar{\chi}_{3}}+\psi_{\bar{\chi}_{2}}d_{3})(\varepsilon^{3}_{i})$
$\displaystyle=(\bar{\chi}\otimes
1-1\otimes\bar{\chi})\Delta_{\mathbb{K}}(\varepsilon^{3}_{i}),\quad
i=0,1,2,3,4.$

The case of $\bar{\eta}$: We get
$(d_{2}\psi_{\bar{\eta}_{3}}+\psi_{\bar{\eta}_{2}}d_{3})(\varepsilon^{3}_{i})$
equal to

$\begin{cases}d_{2}(0)+\psi_{\bar{\eta}_{2}}(a\varepsilon^{2}_{0}-\varepsilon^{2}_{0}a)&{\rm
if}\;i=0\\\
d_{2}(\varepsilon^{2}_{1})+\psi_{\bar{\eta}_{2}}(a\varepsilon^{2}_{1}+q\varepsilon^{2}_{1}a+q^{2}b\varepsilon^{2}_{0}-\varepsilon^{2}_{0}b)&{\rm
if}\;i=1\\\
d_{2}(0)+\psi_{\bar{\eta}_{2}}(a\varepsilon^{2}_{2}+q^{2}\varepsilon^{2}_{2}a-qb\varepsilon^{2}_{1}-\varepsilon^{2}_{1}b)&{\rm
if}\;i=2\\\
d_{2}(0)+\psi_{\bar{\eta}_{2}}(b\varepsilon^{2}_{2}-\varepsilon^{2}_{2}b)&{\rm
if}\;i=3\\\
d_{2}(\varepsilon^{2}_{3})+\psi_{\bar{\eta}_{2}}(a\varepsilon^{2}_{3}-\varepsilon^{2}_{0}c)&{\rm
if}\;i=4\end{cases}$

which is equal to

$\begin{cases}0+a\varepsilon^{1}_{0}-\varepsilon^{1}_{0}a\\\
a\varepsilon^{1}_{1}-q\varepsilon^{1}_{1}a-qb\varepsilon^{1}_{0}+\varepsilon^{1}_{0}b+q^{2}b\varepsilon^{1}_{0}-\varepsilon^{1}_{0}b\\\
0+0\\\ 0+0\\\
a\varepsilon^{1}_{2}+\varepsilon^{1}_{0}c-\varepsilon^{1}_{0}c\end{cases}=\begin{cases}a\varepsilon^{1}_{0}-\varepsilon^{1}_{0}a&{\rm
if}\;i=0\\\ a\varepsilon^{1}_{1}-\varepsilon^{1}_{1}a&{\rm if}\;i=1\\\ 0&{\rm
if}\;i=2\\\ 0&{\rm if}\;i=3\\\ a\varepsilon^{1}_{2}&{\rm if}\;i=4\end{cases}.$

On the other hand, and using Koszul signs in the expansion of
$(1\otimes\bar{\eta})(\varepsilon^{n}_{r}\otimes\varepsilon^{m}_{s})$ to
obtain
$(-1)^{|\bar{\eta}|n}\varepsilon^{n}_{r}\bar{\eta}(\varepsilon^{m}_{s})$,
$(\bar{\eta}\otimes
1-1\otimes\bar{\eta})\Delta_{\mathbb{K}}(\varepsilon^{3}_{i})$ is equal to

$\begin{cases}(\bar{\eta}\otimes
1-1\otimes\bar{\eta})\big{[}\varepsilon^{0}_{0}\otimes\varepsilon^{3}_{0}+\varepsilon^{1}_{0}\otimes\varepsilon^{2}_{0}+\varepsilon^{2}_{0}\otimes\varepsilon^{1}_{0}+\varepsilon^{3}_{0}\otimes\varepsilon^{0}_{0}\big{]}\\\
(\bar{\eta}\otimes
1-1\otimes\bar{\eta})\big{[}\varepsilon^{0}_{0}\otimes\varepsilon^{3}_{1}+\varepsilon^{1}_{0}\otimes\varepsilon^{2}_{1}+q^{2}\varepsilon^{1}_{1}\otimes\varepsilon^{2}_{0}+\varepsilon^{2}_{0}\otimes\varepsilon^{1}_{1}-q\varepsilon^{2}_{1}\otimes\varepsilon^{1}_{0}+\varepsilon^{3}_{1}\otimes\varepsilon^{0}_{0}\big{]}\\\
(\bar{\eta}\otimes
1-1\otimes\bar{\eta})\big{[}\varepsilon^{0}_{0}\otimes\varepsilon^{3}_{2}+\varepsilon^{1}_{0}\otimes\varepsilon^{2}_{2}+q\varepsilon^{1}_{1}\otimes\varepsilon^{2}_{1}-\varepsilon^{2}_{1}\otimes\varepsilon^{1}_{1}+\varepsilon^{2}_{2}\otimes\varepsilon^{1}_{0}+\varepsilon^{3}_{2}\otimes\varepsilon^{0}_{0}\big{]}\\\
(\bar{\eta}\otimes
1-1\otimes\bar{\eta})\big{[}\varepsilon^{0}_{0}\otimes\varepsilon^{3}_{3}+\varepsilon^{1}_{1}\otimes\varepsilon^{2}_{2}+\varepsilon^{2}_{2}\otimes\varepsilon^{1}_{1}+\varepsilon^{3}_{3}\otimes\varepsilon^{0}_{0}\big{]}\\\
(\bar{\eta}\otimes
1-1\otimes\bar{\eta})\big{[}\varepsilon^{0}_{0}\otimes\varepsilon^{3}_{4}+\varepsilon^{1}_{0}\otimes\varepsilon^{2}_{3}+\varepsilon^{2}_{0}\otimes\varepsilon^{1}_{2}+\varepsilon^{3}_{4}\otimes\varepsilon^{0}_{0}\big{]}\end{cases}$

which is the same as

$\begin{cases}a\varepsilon^{1}_{0}-\varepsilon^{1}_{0}a&{\rm if}\;i=0\\\
a\varepsilon^{1}_{1}-q^{2}\varepsilon^{1}_{1}a&{\rm if}\;i=1\\\ 0&{\rm
if}\;i=2\\\ 0&{\rm if}\;i=3\\\ a\varepsilon^{1}_{2}&{\rm if}\;i=4.\end{cases}$

So we see that
$(d_{2}\psi_{\bar{\eta}_{3}}+\psi_{\bar{\eta}_{2}}d_{3})(\varepsilon^{3}_{i})=(\bar{\eta}\otimes
1-1\otimes\bar{\eta})\Delta_{\mathbb{K}}(\varepsilon^{3}_{i}),\quad
i=0,1,2,3,4.$
The case of $\bar{\chi}$:

$\displaystyle(d_{2}\psi_{\chi_{3}}+\psi_{\bar{\chi}_{2}}d_{3})(\varepsilon^{3}_{i})=\begin{cases}d_{2}(0)+\psi_{\bar{\chi}_{2}}(a\varepsilon^{2}_{0}-\varepsilon^{2}_{0}a)\\\
d_{2}(0)+\psi_{\bar{\chi}_{2}}(a\varepsilon^{2}_{1}+q\varepsilon^{2}_{1}a+q^{2}b\varepsilon^{2}_{0}-\varepsilon^{2}_{0}b)\\\
d_{2}(-a\varepsilon^{2}_{1})+\psi_{\chi_{2}}(a\varepsilon^{2}_{2}+q^{2}\varepsilon^{2}_{2}a-qb\varepsilon^{2}_{1}-\varepsilon^{2}_{1}b)\\\
d_{2}(\varepsilon^{2}_{1}b)+\psi_{\bar{\chi}_{2}}(b\varepsilon^{2}_{2}-\varepsilon^{2}_{2}b)\\\
d_{2}(0)+\psi_{\bar{\chi}_{2}}(a\varepsilon^{2}_{3}-\varepsilon^{2}_{0}c)\end{cases}$
$\displaystyle=\begin{cases}0+0\\\ 0+0\\\
qa\varepsilon^{1}_{1}a+qab\varepsilon^{1}_{0}-a\varepsilon^{1}_{0}b+a\varepsilon^{1}_{0}b-q^{2}a\varepsilon^{1}_{1}a-q^{2}\varepsilon^{1}_{0}ba\\\
a\varepsilon^{1}_{1}b-q\varepsilon^{1}_{1}ab-
qb\varepsilon^{1}_{0}b+ba\varepsilon^{1}_{1}-b\varepsilon^{1}_{0}b-a\varepsilon^{1}_{1}b\\\
0+0\end{cases}=\begin{cases}0&{\rm if}\;i=0\\\ 0&{\rm if}\;i=1\\\
ab\varepsilon^{1}_{0}-\varepsilon^{1}_{0}ab&{\rm if}\;i=2\\\
ab\varepsilon^{1}_{1}-\varepsilon^{1}_{1}ab&{\rm if}\;i=3\\\ 0&{\rm
if}\;i=4.\end{cases}$

On the other hand, and using Koszul signs convention as done in the previous
example, we get

$\displaystyle(\bar{\chi}\otimes
1-1\otimes\bar{\chi})\Delta_{\mathbb{K}}(\varepsilon^{3}_{i})=(\bar{\chi}\otimes
1-1\otimes\bar{\chi})$
$\displaystyle\begin{cases}\varepsilon^{0}_{0}\otimes\varepsilon^{3}_{0}+\varepsilon^{1}_{0}\otimes\varepsilon^{2}_{0}+\varepsilon^{2}_{0}\otimes\varepsilon^{1}_{0}+\varepsilon^{3}_{0}\otimes\varepsilon^{0}_{0}\\\
\varepsilon^{0}_{0}\otimes\varepsilon^{3}_{1}+\varepsilon^{1}_{0}\otimes\varepsilon^{2}_{1}+q^{2}\varepsilon^{1}_{1}\otimes\varepsilon^{2}_{0}+\varepsilon^{2}_{0}\otimes\varepsilon^{1}_{1}-q\varepsilon^{2}_{1}\otimes\varepsilon^{1}_{0}+\varepsilon^{3}_{1}\otimes\varepsilon^{0}_{0}\\\
\varepsilon^{0}_{0}\otimes\varepsilon^{3}_{2}+\varepsilon^{1}_{0}\otimes\varepsilon^{2}_{2}+q\varepsilon^{1}_{1}\otimes\varepsilon^{2}_{1}-\varepsilon^{2}_{1}\otimes\varepsilon^{1}_{1}+\varepsilon^{2}_{2}\otimes\varepsilon^{1}_{0}+\varepsilon^{3}_{2}\otimes\varepsilon^{0}_{0}\\\
\varepsilon^{0}_{0}\otimes\varepsilon^{3}_{3}+\varepsilon^{1}_{1}\otimes\varepsilon^{2}_{2}+\varepsilon^{2}_{2}\otimes\varepsilon^{1}_{1}+\varepsilon^{3}_{3}\otimes\varepsilon^{0}_{0}\\\
\varepsilon^{0}_{0}\otimes\varepsilon^{3}_{4}+\varepsilon^{1}_{0}\otimes\varepsilon^{2}_{3}+\varepsilon^{2}_{0}\otimes\varepsilon^{1}_{2}+\varepsilon^{3}_{4}\otimes\varepsilon^{0}_{0}\end{cases}=\begin{cases}0&{\rm
if}\;i=0\\\ 0&{\rm if}\;i=1\\\
ab\varepsilon^{1}_{0}-\varepsilon^{1}_{0}ab&{\rm if}\;i=2\\\
ab\varepsilon^{1}_{1}-\varepsilon^{1}_{1}ab&{\rm if}\;i=3\\\ 0&{\rm
if}\;i=4\end{cases}.$

So we see again that
$(d_{2}\psi_{\bar{\chi}_{3}}+\psi_{\bar{\chi}_{2}}d_{3})(\varepsilon^{3}_{i})=(\bar{\chi}\otimes
1-1\otimes\bar{\chi})\Delta_{\mathbb{K}}(\varepsilon^{3}_{i}),\quad
i=0,1,2,3,4.$

###  7.2. Homotopy liftings from degree 1 cocycles

In this section, we will find Hochschild 1 cocycles and their associated
homotopy lifting maps. The process of solving equations to obtain cocycles is
the same as what was done in the previous case, so we just state the results.
Unlike the previous section, we give explicit maps $\psi_{\eta_{n}}$ and
$\psi_{\chi_{n}}$ for all $n$.

Suppose that the $\Lambda_{q}^{e}$-module homomorphism
$\eta:\mathbb{K}_{1}\rightarrow\Lambda_{q}$ defined by
$\eta=\begin{pmatrix}\lambda_{0}&\lambda_{1}&\lambda_{2}\end{pmatrix}$ is a
cocycle, that is $d^{*}\eta=0,$ with $\lambda_{i}\in\Lambda_{q}$ for all $i$.
Since $d^{*}\eta:\mathbb{K}_{2}\rightarrow\Lambda_{q}$, we solve the equation
$d^{*}\eta(\varepsilon^{2}_{r})=\eta d_{2}(\varepsilon^{2}_{r})=0$ and present
the solutions in Table 2. Let us consider the first and second cocycles in
Table 2, i.e.

$\eta=\begin{pmatrix}a&0&0\end{pmatrix}\text{ and
}\quad\chi=\begin{pmatrix}ab&0&0\end{pmatrix}.$

There are homotopy lifting maps
$\psi_{\eta},\psi_{\chi}:\mathbb{K}_{n}\rightarrow\mathbb{K}_{n}$ associated
to $\eta$ and $\chi$ respectively satisfying

(7.5) $\displaystyle(d\psi_{\eta}-\psi_{\eta}d)(\varepsilon^{n}_{r})$
$\displaystyle=(\eta\otimes
1-1\otimes\eta)\Delta_{\mathbb{K}}(\varepsilon^{n}_{r})\qquad\text{and}$ (7.6)
$\displaystyle(d\psi_{\chi}-\psi_{\chi}d)(\varepsilon^{n}_{r})$
$\displaystyle=(\chi\otimes
1-1\otimes\chi)\Delta_{\mathbb{K}}(\varepsilon^{n}_{r}).$

solutions | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
$\lambda_{0}$ | a | ab | 0 | 0 | 0 | 0
$\lambda_{1}$ | 0 | 0 | b | ab | 0 | 0
$\lambda_{2}$ | 0 | 0 | 0 | 0 | c | bc
Table 2. Possible values of $\eta(\varepsilon^{1}_{r})=\lambda_{r}$ for
different $r$.

We will prove that for each $n$ and $r$,

(7.7)
$\psi_{\eta_{n}}(\varepsilon^{n}_{r})=\begin{cases}(n-r)\varepsilon^{n}_{r}&{\rm
when}\;r=0,1,2,\ldots,n\\\ (n-1)\varepsilon^{n}_{r}&{\rm
when}\;r=n+1,\end{cases}$

is a homotopy lifting map associated to the cocycle $\eta$ and whenever $q=1$,

(7.8)
$\psi_{\chi_{n}}(\varepsilon^{n}_{r})=\begin{cases}(\frac{1+(-1)^{r}}{2})(-1)^{n+1}a\varepsilon^{n}_{r+1}+(n-r)\varepsilon^{n}_{r}b&\;\;r=0,1,2,\ldots,n-1,\\\
0&\;\;r=n,\\\
(n-1)(b\varepsilon^{n}_{r}+\varepsilon^{n}_{1}c)&\;\;r=n+1,\end{cases}$

is a homotopy lifting map associated to the cocycle $\chi.$ We present proofs
of these claims as follows.
The case for $\eta$. For $r=0,1,\ldots,n$, we get for the left hand side of
Equation (7.5),

$\displaystyle(d\psi_{\eta_{n}}-\psi_{\eta_{n-1}}d)(\varepsilon^{n}_{r})$
$\displaystyle=d\big{\\{}(n-r)\varepsilon^{n}_{r}\big{\\}}-\psi_{\eta_{n-1}}\big{\\{}\bar{\partial}_{n,r}[a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a]+\bar{\partial}_{r,0}[(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(-1)^{n}\varepsilon^{n-1}_{r-1}b]\big{\\}}$
$\displaystyle=(n-r)\big{\\{}\bar{\partial}_{n,r}[a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a]+\bar{\partial}_{r,0}[(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(-1)^{n}\varepsilon^{n-1}_{r-1}b]\big{\\}}$
$\displaystyle-\bar{\partial}_{n,r}[(n-r-1)a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}(n-r-1)\varepsilon^{n-1}_{r}a]$
$\displaystyle+\bar{\partial}_{r,0}[(-q)^{n-r}(n-r)b\varepsilon^{n-1}_{r-1}+(-1)^{n}(n-r)\varepsilon^{n-1}_{r-1}b]$
$\displaystyle=\bar{\partial}_{n,r}\big{[}(n-r)-(n-r-1)a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}((n-r)-(n-r-1))\varepsilon^{n-1}_{r}a\big{]}$
$\displaystyle+\bar{\partial}_{r,0}\big{[}(-q)^{n-r}((n-r)-(n-r))b\varepsilon^{n-1}_{r-1}+(-1)^{n}((n-r)-(n-r))\varepsilon^{n-1}_{r-1}b\big{]}$
$\displaystyle=\bar{\partial}_{n,r}\big{[}a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a\big{]}$
$\displaystyle=a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a,\;\text{\;if\;\;}r\neq
n.$

If $r=n$, the expression
$(d\psi_{\eta_{n}}-\psi_{\eta_{n-1}}d)(\varepsilon^{n}_{r})$ equals $0$. For
the case $r=n+1$, we obtain

$\displaystyle(d\psi_{\eta_{n}}-\psi_{\eta_{n-1}}d)(\varepsilon^{n}_{n+1})=d((n-1)\varepsilon^{n}_{n+1})-\psi_{\eta_{n-1}}(a\varepsilon^{n-1}_{n}+(-1)^{n}\varepsilon^{n-1}_{0}c)$
$\displaystyle=(n-1)(a\varepsilon^{n-1}_{n}+(-1)^{n}\varepsilon^{n-1}_{0}c)-(n-2)a\varepsilon^{n-1}_{n}-(-1)^{n}(n-1)\varepsilon^{n-1}_{0}c)$
$\displaystyle=a\varepsilon^{n-1}_{n}.$

On the right hand side of Equation (7.5), we obtain after using the Koszul
signs conventions that $(\eta\otimes
1-1\otimes\eta)\Delta_{\mathbb{K}}(\varepsilon^{n}_{r})$ is equal to

$\displaystyle(\eta\otimes
1-1\otimes\eta)\Big{(}\begin{cases}\displaystyle{\sum_{t=0}^{n}\varepsilon^{t}_{0}\otimes\varepsilon^{n-t}_{0}},&{\rm
if}\;r=0\\\
\displaystyle{\sum_{w=0}^{n}\sum_{j=max\\{0,s+w-n\\}}^{min\\{w,s\\}}(-q)^{j(n-s+j-w)}\varepsilon^{w}_{j}\otimes\varepsilon^{n-w}_{s-j}},&{\rm
if}\;0<r<n\\\
\displaystyle{\sum_{t=0}^{n}\varepsilon^{t}_{t}\otimes\varepsilon^{n-t}_{n-t}},&{\rm
if}\;r=n\\\
\displaystyle{\varepsilon^{0}_{0}\otimes\varepsilon^{n}_{n+1}+\Big{[}\sum_{t=0}^{n}\varepsilon^{t}_{0}\otimes\varepsilon^{n-t}_{n-t+1}\Big{]}+\varepsilon^{n}_{n+1}\otimes\varepsilon^{0}_{0}},&{\rm
if}\;r=n+1\end{cases}\Big{)}.$

In the case $r=0$, substitute $1$ for the index $t$ when applying $\eta\otimes
1$ and substitute $n-1$ for the index $t$ when applying $1\otimes\eta$.
Similarly for the case $0<r<n$, substitute $1,0,r$ respectively for the
indices $w,j,s$ when applying $\eta\otimes 1$ and substitute $n-1,r,r$
respectively for the indices $w,j,s$ when applying $1\otimes\eta$. When $r=n$,
everything is zero since $\eta(\varepsilon^{1}_{i})=0$ if $i\neq 0$ and
finally when $r=n+1$ take substitute $1$ for the index $t$. What we then have
is equal to the following

$\displaystyle\begin{cases}\displaystyle{\eta(\varepsilon^{1}_{0})\varepsilon^{n-1}_{0}-(-1)^{n-1}\varepsilon^{n-1}_{0}\eta(\varepsilon^{1}_{0})}\\\
\displaystyle{\eta(\varepsilon^{1}_{0})\varepsilon^{n-1}_{r}-(-1)^{n-1}(-q)^{r}\varepsilon^{n-1}_{r}\eta(\varepsilon^{n}_{0})}\\\
\displaystyle{\eta(\varepsilon^{1}_{1})\varepsilon^{n-1}_{0}-(-1)^{n-1}\varepsilon^{n-1}_{n-1}\eta(\varepsilon^{1}_{1})}\\\
\displaystyle{\eta(\varepsilon^{1}_{0})\varepsilon^{n-1}_{n}}\\\
\end{cases}=\begin{cases}a\varepsilon^{n-1}_{0}+(-1)^{n}\varepsilon^{n-1}_{0}a&{\rm
if}\;r=0\\\ a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a&{\rm
if}\;0<r<n\\\ 0&{\rm if}\;r=n\\\ a\varepsilon^{n-1}_{n}&{\rm if}\;r=n+1\\\
\end{cases}$
$\displaystyle=(d\psi_{\eta_{n}}-\psi_{\eta_{n-1}}d)(\varepsilon^{n}_{r}).$

Thus we have shown that for $r=0,1,\ldots,n+1$, Equation (7.5) holds.

The case for $\chi$. When $r=0$, the left hand side of Equation (7.6) is

$\displaystyle(d\psi_{\chi_{n}}-\psi_{\chi_{n-1}}d)(\varepsilon^{n}_{0})$
$\displaystyle=d((-1)^{n+1}a\varepsilon^{n}_{1}+n\varepsilon^{n}_{0}b)-\psi_{\chi}(a\varepsilon^{n-1}_{0}+(-1)^{n}\varepsilon^{n-1}_{0}a)$
$\displaystyle=(-1)^{n+1}a[a\varepsilon^{n-1}_{1}+(-1)^{n-1}q\varepsilon^{n-1}_{1}a+(-q)^{n-1}b\varepsilon^{n-1}_{0}+(-1)^{n}\varepsilon^{n-1}_{0}b]$
$\displaystyle+n[a\varepsilon^{n-1}_{0}+(-1)^{n}\varepsilon^{n-1}_{0}a]b-a[(-1)^{n}a\varepsilon^{n-1}_{1}+(n-1)\varepsilon^{n-1}_{0}b]$
$\displaystyle-(-1)^{n}[(-1)^{n}a\varepsilon^{n-1}_{1}+(n-1)\varepsilon^{n-1}_{0}b]a.$

Whenever $q=1,ab=ba$, so we obtain
$ab\varepsilon^{n-1}_{0}+(-1)^{n}\varepsilon^{n-1}_{0}ab$ which is equal to
the right hand side of (7.6). Therefore $(\chi\otimes
1-1\otimes\chi)\Delta_{\mathbb{K}}(\varepsilon^{n}_{0})$ because

$\displaystyle(\chi\otimes
1-1\otimes\chi)\sum_{t=0}^{n}\varepsilon^{t}_{0}\otimes\varepsilon^{n-t}_{0}=\sum_{t=0}^{n}\chi(\varepsilon^{t}_{0})\varepsilon^{n-t}_{0}-\sum_{t=0}^{n}(-1)^{t}\varepsilon^{t}_{0}\chi(\varepsilon^{n-t}_{0}).$

When $t=1$ in the first sum and $t=n-1$ in the second sum, the last expression
is equal to
$\displaystyle{\chi(\varepsilon^{1}_{0})\varepsilon^{n-1}_{0}-(-1)^{n-1}\varepsilon^{n-1}_{0}\eta(\varepsilon^{1}_{0})=ab\varepsilon^{n-1}_{0}+(-1)^{n}\varepsilon^{n-1}_{0}ab}.$
When $r$ is even and $0<r<n\;$, we obtain the following for the left hand side
of (7.6)

$\displaystyle(d\psi_{\chi_{n}}-\psi_{\chi_{n-1}}d)(\varepsilon^{n}_{r})$
$\displaystyle=d((-1)^{n+1}a\varepsilon^{n}_{r+1}+(n-r)\varepsilon^{n}_{r}b)$
$\displaystyle-\psi_{\chi}(a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a+(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(-1)^{n}\varepsilon^{n-1}_{r-1}b)$
$\displaystyle=(-1)^{n+1}a[a\varepsilon^{n-1}_{r+1}+(-1)^{n-r-1}q^{r+1}\varepsilon^{n-1}_{r+1}a+(-q)^{n-r-1}b\varepsilon^{n-1}_{r}+(-1)^{n}\varepsilon^{n-1}_{r}b]$
$\displaystyle+(n-r)[a\varepsilon^{n-r}_{r}+(-1)^{n-1}q^{r}\varepsilon^{n-1}_{r}a+(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(-1)^{n}\varepsilon^{n-1}_{r-1}b]b$
$\displaystyle-a[(-1)^{n}a\varepsilon^{n-1}_{r+1}+(n-r-1)\varepsilon^{n-1}_{r}b]-(-1)^{n-r}q^{r}[(-1)^{n}a\varepsilon^{n-1}_{r+1}+(n-r-1)\varepsilon^{n-1}_{r}b]a$
$\displaystyle-(-q)^{n-r}b[(n-r)\varepsilon^{n-1}_{r-1}b]-(-1)^{n}[(n-r)\varepsilon^{n-1}_{r-1}b]b$
$\displaystyle=[(-1)^{2n-r}q^{r+1}-(-1)^{2n-r}q^{r}]a\varepsilon^{n-1}_{r+1}a+[(-1)^{n+1}(-q)^{n-r-1}]ab\varepsilon^{n-1}_{r}$
$\displaystyle+[(-1)^{2n+1}+(n-r)-(n-r-1)]a\varepsilon^{n-1}_{r}b+[(-1)^{n-r}q^{r}(n-r)]\varepsilon^{n-1}_{r}ab$
$\displaystyle+[-(-1)^{n-r}q^{r}(n-r-1)]\varepsilon^{n-1}_{r}ba+[(-q)^{n-r}(n-r)-(-q)^{n-r}(n-r)]b\varepsilon^{n-1}_{r-1}b$

which with $q=1\;(ab=ba)$, we obtain
$ab\varepsilon^{n-1}_{r}+(-1)^{n-r}\varepsilon^{n-1}_{r}ab$ and all other
terms vanish. When $r$ is odd and $o\leq r\leq n$, we obtain the following for
the left hand side of (7.6)

$\displaystyle(d\psi_{\chi_{n}}-\psi_{\chi_{n-1}}d)(\varepsilon^{n}_{r})$
$\displaystyle=d((n-r)\varepsilon^{n}_{r}b)-\psi_{\chi}(a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a+(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(-1)^{n}\varepsilon^{n-1}_{r-1}b)$
$\displaystyle=(n-r)[a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a+(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(-1)^{n}\varepsilon^{n-1}_{r-1}b]b$
$\displaystyle-a[(n-r-1)\varepsilon^{n-1}_{r}b]-(-1)^{n-r}q^{r}[(n-r-1)\varepsilon^{n-1}_{r}b]a$
$\displaystyle-(-q)^{n-r}b[(-1)^{n}a\varepsilon^{n-1}_{r}+(n-r)\varepsilon^{n-1}_{r-1}b]-(-1)^{n}[(-1)^{n}a\varepsilon^{n-1}_{r}+(n-r)\varepsilon^{n-1}_{r-1}b]b$
$\displaystyle=[-(-1)^{n}(-q)^{n-r}]ba\varepsilon^{n-1}_{r}+[(n-r)-(n-r-1)-(-1)^{2n}]a\varepsilon^{n-1}_{r}b+[(-1)^{n-r}q^{r}(n-r)]\varepsilon^{n-1}_{r}ab$
$\displaystyle+[-(-1)^{n-r}q^{r}(n-r-1)]\varepsilon^{n-1}_{r}ba+[(-q)^{n-r}(n-r)-(-q)^{n-r}(n-r)]b\varepsilon^{n-1}_{r-1}b$

which with $q=1\;$, we get the result obtain previously:
$ab\varepsilon^{n-1}_{r}+(-1)^{n-r}\varepsilon^{n-1}_{r}ab.$ On the other
hand, the right hand side of (7.6) when $0<r<n$ becomes

$\displaystyle(\chi\otimes
1-1\otimes\chi)\Big{[}\sum_{w=0}^{n}\sum_{j=max\\{0,r+w-n\\}}^{min\\{w,r\\}}(-q)^{j(n-r+j-w)}\varepsilon^{w}_{j}\otimes\varepsilon^{n-w}_{r-j}\Big{]}.$

To get a non-zero term, substitute $w=1,j=0$ then apply $\chi\otimes 1$, and
substitute $w=n-1,j=r$ and apply $1\otimes\chi$:

$\displaystyle(-q)^{0(n-r+1)}\chi(\varepsilon^{1}_{0})\varepsilon^{n-1}_{r}-(-1)^{n-1}(-q)^{r(n-r+r-n+1)}\varepsilon^{n-1}_{r}\chi(\varepsilon^{1}_{0})=ab\varepsilon^{n-1}_{r}+(-1)^{n-r}\varepsilon^{n-1}_{r}ab$

When $r=n$, the left hand side of (7.6) becomes

$\displaystyle(d\psi_{\chi}-\psi_{\chi}d)(\varepsilon^{n}_{n})$
$\displaystyle=d(0)-\psi_{\chi}(b\varepsilon^{n-1}_{n-1}+(-1)^{n}\varepsilon^{n-1}_{n-1}b)=0-b\cdot
0+(-1)^{n+1}0\cdot b=0,$

while the right hand side $(\chi\otimes
1-1\otimes\chi)\Delta_{\mathbb{K}}(\varepsilon^{n}_{n})$ becomes

$\displaystyle(\chi\otimes
1-1\otimes\chi)\sum_{t=0}^{n}\varepsilon^{t}_{1}\otimes\varepsilon^{n-t}_{n-t}=\chi(\varepsilon^{1}_{1})\varepsilon^{n-1}_{1}-(-1)^{n-1}\varepsilon^{n-1}_{1}\chi(\varepsilon^{1}_{1})=0$

and they are equal. It is also true whenever $r=n+1$:

$\displaystyle(d\psi_{\chi}-\psi_{\chi}d)(\varepsilon^{n}_{n+1})=d((n-1)\varepsilon^{n}_{1}c+b\varepsilon^{n}_{n+1})-\psi_{\chi}(a\varepsilon^{n-1}_{n}+(-1)^{n}\varepsilon^{n-1}_{0}c)$
$\displaystyle=[(n-1)-(n-2)]a\varepsilon^{n-1}_{1}c+(n-1)[(-q)^{n-1}+(-1)^{n}]b\varepsilon^{n-1}_{0}c$
$\displaystyle+(n-1)[(-1)^{n}+(-1)^{n-1}]\varepsilon^{n-1}_{0}bc+(n-1)ba\varepsilon^{n-1}_{n}-(n-2)ab\varepsilon^{n-1}_{n}$
$\displaystyle=(n-1-n+2)ab\varepsilon^{n-1}_{n}=ab\varepsilon^{n-1}_{n}$

is equal to

$\displaystyle(\chi\otimes
1-1\otimes\chi)\Delta_{\mathbb{K}}(\varepsilon^{n}_{n+1})=(\chi\otimes
1-1\otimes\chi)\sum_{t=0}^{n}\varepsilon^{t}_{0}\otimes\varepsilon^{n-t}_{n-t+1}+\varepsilon^{n}_{n+1}\otimes\varepsilon^{0}_{0}$
$\displaystyle=\chi\otimes
1(\varepsilon^{1}_{0}\otimes\varepsilon^{n-1}_{n})-1\otimes\chi((-1)^{n-1}\varepsilon^{n-1}_{0}\otimes\varepsilon^{1}_{2})+(\chi\otimes
1-1\otimes\chi)(\varepsilon^{n}_{n+1}\otimes\varepsilon^{0}_{0})$
$\displaystyle=ab\varepsilon^{n-1}_{n}.$

We have therefore shown that for $r=0,1,\ldots,n+1$, Equation (3.1) holds.

######  Remark 7.9.

Table 3 shows some bracket computations based on these examples. Recall that
$\eta=\begin{pmatrix}a&0&0\end{pmatrix}$ and
$\chi=\begin{pmatrix}ab&0&0\end{pmatrix}$ are the degree 1 cocycles with
homtopy lifting maps given in (7.7) and (7.8) respectively. Also take
$\overline{\eta}=\begin{pmatrix}a&0&0&0\end{pmatrix}$ and
$\overline{\chi}=\begin{pmatrix}0&0&ab&0\end{pmatrix}$ to be the degree 2
cocycles whose homtopy lifting maps were given in (7.3) and (7.4)
respectively. Take $\theta=\begin{pmatrix}ab&0&0&0\end{pmatrix}$ to be the
second degree 2 cocycle appearing in Table 1. The following bracket structure
can be verified by direct computations.

$[\cdot\;,\cdot]$ | $\eta$ | $\chi$ | $\overline{\eta}$ | $\overline{\chi}$
---|---|---|---|---
$\eta$ | 0 | 0 | $-\overline{\eta}$ | $\overline{\chi}$
$\chi$ | 0 | 0 | $-\theta$ | 0
$\overline{\eta}$ | $\overline{\eta}$ | $\theta$ | 0 | 0
$\overline{\chi}$ |  -$\overline{\chi}$ | 0 | 0 | 0
Table 3. Some bracket computations

For instance
$[\bar{\eta},\eta](\varepsilon^{2}_{0})=\bar{\eta}\psi_{\eta}(\varepsilon^{2}_{0})-\eta\psi_{\bar{\eta}}(\varepsilon^{2}_{0})=\bar{\eta}(2\varepsilon^{2}_{0})-\eta(\varepsilon^{1}_{0})=2a-a=a=\bar{\eta}(\varepsilon^{2}_{0})$
and $[\bar{\eta},\eta](\varepsilon^{2}_{r})=0$ for all $r\neq 0$.

###  7.3. Connections to derivation operators

To compute the Gerstenhaber bracket of a $1$-cocycle and any $n$-cocycle, M.
Súarez-Álvarez in [10] introduced the idea of derivation operators ( called
$\delta$-operators in the general settings). We give the definition of a
derivation operator in Lemma 7.10. Since an Hochschild $1$-cocycle can be
viewed as a derivation, it was shown that there are derivation operators
associated to each $1$-cocycle. In particular if
$\delta^{e}:A^{e}\xrightarrow{}A^{e}$ is a derivation, that is
$\delta^{e}(ab)=\delta^{e}(a)b+a\delta^{e}(b)$ for all $a,b\in A^{e}$ and
$\mathbb{P}\xrightarrow{}A$ is a projective bimodule resolution of $A$, the
derivation operators are chain maps lifting $\delta^{e}$ from $\mathbb{P}_{n}$
to $\mathbb{P}_{n}$. Derivation operators are not module homomorphism in
general.

In the build up to this work, we in fact realized Equations (7.7) and (7.8)
first as derivation operators before observing that they are homotopy lifting
maps. It was remarked in [5, Example 6.3.8], that for an $n$-cocycle $g$,
there is an associated homotopy lifting map $\psi_{g}$ which can also be
realized as a coderivation on the tensor algebra of $A$. If $g$ is a
$1$-cocycle, then $\psi_{g}$ may be extended to a derivation operator
$\tilde{g}$ in some sense. This describes how a derivation operator may be
obtained from a homotopy lifting map. Question 7.17 asks whether the converse
of this observation is true.

Our goal in this section is to draw connections between derivation operators
and homotopy lifting maps by showing that the homotopy lifting maps of
Equations (7.7) and (7.8) associated with
$\eta=\begin{pmatrix}a&0&0\end{pmatrix}$ and
$\chi=\begin{pmatrix}ab&0&0\end{pmatrix}$ respectively are indeed derivation
operators.

######  Lemma 7.10.

[5, Lemma 6.2.2] Let $\gamma:\Lambda\xrightarrow{}\Lambda$ be a derivation.
There is a $k$-linear chain map
$\tilde{\gamma}_{n}:\mathbb{K}_{n}\xrightarrow{}\mathbb{K}_{n}$ lifting
$\gamma$ with the property that for each $n$

(7.11)
$\tilde{\gamma}_{n}(a\varepsilon^{n}_{r}b)=\gamma(a)\varepsilon^{n}_{r}b+a\tilde{\gamma}_{n}(\varepsilon^{n}_{r})b+a\varepsilon^{n}_{r}\gamma(b)$

for all $a,b\in\Lambda$ and each basis element
$\varepsilon^{n}_{r}\in\mathbb{K}_{n}.$ Moreover $\tilde{\gamma}$ is unique up
to $\Lambda^{e}$-module chain homotopy.

The chain map $\tilde{\gamma}_{n}$ is called a derivation operator. The
following theorem due to M. Súarez-Álvarez provides a way for computing the
bracket $[\operatorname{HH}^{1}(\Lambda),\operatorname{HH}^{m}(\Lambda)]$ on
Hochschild cohomology. It was first given in [10, Lemma 1.8] and reformulated
by S. Witherspoon in [5, Theorem 6.2.5].

######  Theorem 7.12.

[10, 5] Let $\eta:\Lambda\xrightarrow{}\Lambda$ be a derivation. Let
$\mathbb{K}$ be a projective resolution of $\Lambda$ as a
$\Lambda^{e}$-module. Let
$\chi\in\operatorname{Hom}_{\Lambda^{e}}(\mathbb{K}_{n},\Lambda)$ be a cocycle
and let $\tilde{\eta}_{n}:\mathbb{K}_{n}\xrightarrow{}\mathbb{K}_{n}$ be
derivation operators satisfying Equation (7.11). The Gerstenhaber bracket of
$\eta$ and $\chi$ is represented by

$[\eta,\chi]=\eta\chi-\chi\tilde{\eta}_{n}$

as a cocycle on $\mathbb{K}_{n}$.

######  Proposition 7.13.

When regarded as $k$-linear chain maps, the homotopy lifting maps of Equations
(7.7) and (7.8) associated with $\eta=\begin{pmatrix}a&0&0\end{pmatrix}$ and
$\chi=\begin{pmatrix}ab&0&0\end{pmatrix}$ respectively are indeed derivation
operators.

###### Proof.

We make a slight change of notation by defining

(7.14)
$\tilde{\eta}_{n}(\varepsilon^{n}_{r}):=\psi_{\eta_{n}}(\varepsilon^{n}_{r})=\begin{cases}(n-r)\varepsilon^{n}_{r}&{\rm
when}\;r=0,1,2,\ldots,n\\\ (n-1)\varepsilon^{n}_{r}&{\rm
when}\;r=n+1.\end{cases}$

to distinguish derivation operators $\tilde{\eta}_{\bullet}$ from homotopy
lifting maps $\psi_{\eta_{\bullet}}$. Since $\varepsilon^{1}_{i}$ is generated
from $(o(f^{1}_{i})\otimes_{k}t(f^{1}_{i}))$ which is equal to $a\otimes_{k}a$
when $i=0$, $b\otimes_{k}b$ when $i=1$ and $c\otimes_{k}c$ when $i=2$, we take
$\eta=\begin{pmatrix}a&0&0\end{pmatrix}$ to mean $\eta(a)=a,\eta(b)=0$ and
$\eta(c)=0$ as a derivation. The next calculation shows that derivation
operators are chain maps i.e. $\tilde{\eta}_{n-1}d_{n}=d_{n}\tilde{\eta}_{n}$
for all $n$. Whenever $0<r\leq n$, the expression
$\tilde{\eta}_{n-1}d_{n}(\varepsilon^{n}_{r})$ is equal to

$\displaystyle\tilde{\eta}_{n-1}\Big{[}(1-\partial_{n,r})[a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a]+(1-\partial_{r,0})[(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(-1)^{n}\varepsilon^{n-1}_{r-1}b]\Big{]}$
$\displaystyle=(1-\partial_{n,r})[\eta(a)\varepsilon^{n-1}_{r})+a\tilde{\eta}_{n-1}(\varepsilon^{n-1}_{r})+(-1)^{n-r}q^{r}\tilde{\eta}_{n-1}(\varepsilon^{n-1}_{r})a+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}\eta(a)]$
$\displaystyle+(1-\partial_{r,0})[(-q)^{n-r}\eta(b)\varepsilon^{n-1}_{r-1}+(-q)^{n-r}b\tilde{\eta}_{n-1}(\varepsilon^{n-1}_{r-1})+(-1)^{n}\tilde{\eta}_{n-1}(\varepsilon^{n-1}_{r-1})b+(-1)^{n}\varepsilon^{n-1}_{r-1}\eta(b)]$
$\displaystyle=(1-\partial_{n,r})[(n-r)a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}(n-r)\varepsilon^{n-1}_{r}a]$
$\displaystyle+(1-\partial_{r,0})[(n-r)(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(n-r)(-1)^{n}\varepsilon^{n-1}_{r-1}b]$
$\displaystyle(n-r)\Big{[}(1-\partial_{n,r})[a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a]+(1-\partial_{r,0})[(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(-1)^{n}\varepsilon^{n-1}_{r-1}b]\Big{]}$

while the expression
$d_{n}\tilde{\eta}_{n}(\varepsilon^{n}_{r})=d_{n}((n-r)\varepsilon^{n}_{r})$
is equal to

$\displaystyle(n-r)\Big{[}(1-\partial_{n,r})[a\varepsilon^{n-1}_{r}+(-1)^{n-r}q^{r}\varepsilon^{n-1}_{r}a]+(1-\partial_{r,0})[(-q)^{n-r}b\varepsilon^{n-1}_{r-1}+(-1)^{n}\varepsilon^{n-1}_{r-1}b]\Big{]}.$

When $r=n+1$, we get $\tilde{\eta}_{n-1}d_{n}(\varepsilon^{n}_{n+1})$ equal to

$\displaystyle\tilde{\eta}_{n-1}[a\varepsilon^{n-1}_{n}+(-1)^{n}\varepsilon^{n-1}_{0}c]=\eta(a)\varepsilon^{n-1}_{n}+a\tilde{\eta}_{n-1}(\varepsilon^{n-1}_{n})+(-1)^{n}\tilde{\eta}_{n-1}(\varepsilon^{n-1}_{0})c+(-1)^{n}\varepsilon^{n-1}_{0}\eta(c)$
$\displaystyle=(n-1)a\varepsilon^{n-1}_{n}+(n-1)(-1)^{n}\varepsilon^{n-1}_{0}c=(n-1)[a\varepsilon^{n-1}_{n}+(-1)^{n}\varepsilon^{n-1}_{0}c].$

while
$d_{n}\tilde{\eta}_{n}(\varepsilon^{n}_{n+1})=d_{n}((n-1)\varepsilon^{n}_{n+1})$
is equal to $(n-1)[a\varepsilon^{n-1}_{n}+(-1)^{n}\varepsilon^{n-1}_{0}c].$

Similarly if we consider $\chi=\begin{pmatrix}ab&0&0\end{pmatrix}$ to mean
$\chi(a)=ab,\chi(b)=0$ and $\chi(c)=0$ as a derivation, the following is a
derivation operator associated with $\chi$.

(7.15)
$\tilde{\chi}_{n}(\varepsilon^{n}_{r}):=\psi_{\chi_{n}}(\varepsilon^{n}_{r})=\begin{cases}(\frac{1+(-1)^{r}}{2})(-1)^{n+1}a\varepsilon^{n}_{r+1}+(n-r)\varepsilon^{n}_{r}b&\;\;r=0,1,2,\ldots,n-1,\\\
0&\;\;r=n,\\\
(n-1)(b\varepsilon^{n}_{r}+\varepsilon^{n}_{1}c)&\;\;r=n+1,\end{cases}$

For instance whenever $0\leq r<n-1$ and $r$ is even, then
$\tilde{\chi}_{n}(\varepsilon^{n}_{r})=(n-r)\varepsilon^{n}_{r}b+(-1)^{n+1}a\varepsilon^{n}_{r+1}$
and $\tilde{\chi}_{n}(\varepsilon^{n}_{r-1})=(n-r+1)\varepsilon^{n}_{r-1}b.$
Therefore the expression $\tilde{\chi}_{n}d_{n+1}(\varepsilon^{m+1}_{r})$ is
equal to

$\displaystyle\tilde{\chi}_{n}{[}a\varepsilon^{n}_{r}+(-1)^{n+1-r}\varepsilon^{n}_{r}a+(-1)^{n+1-r}b\varepsilon^{n}_{r-1}+(-1)^{n+1}\varepsilon^{n}_{r-1}b{]}$