# Averaging operators on $q$-deformed Witt and $q$-deformed $W(2,2)$ algebras

Ismail Laraiedh  Departement of Mathematics, Faculty of Sciences,
Sfax University, Box 1171, 3000 Sfax, Tunisia
<EMAIL_ADDRESS>Departement of Mathematics,
College of Sciences and Humanities Al Quwaiiyah,
Shaqra University, Kingdom of Saudi Arabia
<EMAIL_ADDRESS>Sergei Silvestrov  Division of Mathematics and
Physics,
School of Education, Culture and Communication,
Mälardalen University, Box 883, 72123 Västerås, Sweden
<EMAIL_ADDRESS>
###### Abstract

The aim of this paper is to give some constructions results of averaging
operators on Hom-Lie algebras. The homogeneous averaging operators on
$q$-deformed Witt and $q$-deformed $W(2,2)$ Hom-algebras are classified. As
applications, the induced Hom-Leibniz algebra structures are obtained and
their multiplicativity conditions are also given.

##  1 Introduction

The investigations of various quantum deformations or $q$-deformations of Lie
algebras began a period of rapid expansion in 1980’s stimulated by
introduction of quantum groups motivated by applications to the quantum Yang-
Baxter equation, quantum inverse scattering methods and constructions of the
quantum deformations of universal enveloping algebras of semi-simple Lie
algebras. Various $q$-deformed Lie algebras have appeared in physical contexts
such as string theory, vertex models in conformal field theory, quantum
mechanics and quantum field theory in the context of deformations of infinite-
dimensional algebras, primarily the Heisenberg algebras, oscillator algebras
and Witt and Virasoro algebras. In [5, 29, 30, 31, 32, 33, 34, 35, 40, 42, 52,
53, 54], it was in particular discovered that in these $q$-deformations of
Witt and Visaroro algebras and some related algebras, some interesting
$q$-deformations of Jacobi identities, extending Jacobi identity for Lie
algebras, are satisfied. This has been one of the initial motivations for the
development of general quasi-deformations and discretizations of Lie algebras
of vector fields using more general $\sigma$-derivations (twisted derivations)
in [39].

Hom-Lie algebras and more general quasi-Hom-Lie algebras were introduced first
by Hartwig, Larsson and Silvestrov [39], where the general quasi-deformations
and discretizations of Lie algebras of vector fields using more general
$\sigma$-derivations (twisted derivations) and a general method for
construction of deformations of Witt and Virasoro type algebras based on
twisted derivations have been developed, initially motivated by the
$q$-deformed Jacobi identities observed for the $q$-deformed algebras in
physics, along with $q$-deformed versions of homological algebra and discrete
modifications of differential calculi. Hom-Lie algebras, Hom-Lie
superalgebras, Hom-Lie color algebras and more general quasi-Lie algebras and
color quasi-Lie algebras where introduced first in [48, 49, 75]. Quasi-Lie
algebras and color quasi-Lie algebras encompass within the same algebraic
framework the quasi-deformations and discretizations of Lie algebras of vector
fields by $\sigma$-derivations obeying twisted Leibniz rule, and the well-
known generalizations of Lie algebras such as color Lie algebras, the natural
generalizations of Lie algebras and Lie superalgebras. In quasi-Lie algebras,
the skew-symmetry and the Jacobi identity are twisted by deforming twisting
linear maps, with the Jacobi identity in quasi-Lie and quasi-Hom-Lie algebras
in general containing six twisted triple bracket terms. In Hom-Lie algebras,
the bilinear product satisfies the non-twisted skew-symmetry property as in
Lie algebras, and the Hom-Lie algebras Jacobi identity has three terms twisted
by a single linear map, reducing to the Lie algebras Jacobi identity when the
twisting linear map is the identity map. Hom-Lie admissible algebras have been
considered first in [58], where in particular the Hom-associative algebras
have been introduced and shown to be Hom-Lie admissible, that is leading to
Hom-Lie algebras using commutator map as new product, and in this sense
constituting a natural generalization of associative algebras as Lie
admissible algebras. Since the pioneering works [39, 47, 49, 48, 50, 58], Hom-
algebra structures expanded into a popular area with increasing number of
publications in various directions. Hom-algebra structures of a given type
include their classical counterparts and open broad possibilities for
deformations, Hom-algebra extensions of cohomological structures and
representations, formal deformations of Hom-associative and Hom-Lie algebras,
Hom-Lie admissible Hom-coalgebras, Hom-coalgebras, Hom-Hopf algebras [6, 26,
36, 47, 51, 59, 61, 60, 71, 72, 80, 78]. Hom-Lie algebras, Hom-Lie
superalgebras and color Hom-Lie algebras and their $n$-ary generalizations
have been further investigated in various aspects for example in [1, 2, 3, 6,
7, 8, 9, 10, 11, 13, 14, 15, 12, 16, 17, 18, 19, 20, 21, 22, 23, 25, 24, 26,
28, 38, 44, 45, 46, 56, 58, 60, 59, 61, 57, 62, 64, 68, 69, 75, 71, 72, 73,
74, 76, 77, 79, 78, 80, 81, 82, 85, 86, 87, 88].

In the 1930s, the notion of averaging operator was explicitly defined by
Kolmogoroff and Kampé de Fériet [41, 63]. Then G. Birkhoff [27] continued its
study and showed that a positive bounded projection in the Banach algebra
$C(X)$, the algebra of scalar valued continuous functions on a compact
Hausdorff space $X$, onto a fixed range space is an idempotent averaging
operator. In 1954, S. T. C. Moy [65] made the connection between averaging
operators and conditional expectation. Furthermore, she studied the
relationship between integration theory and averaging operators in turbulence
theory and probability. Then her results were extended by G. C. Rota [70].
During the same period, the idempotent averaging operators on $C_{\infty}(X)$,
the algebra of all real valued continuous functions on a locally compact
Hausdorff space $X$ that vanish at the infinity, were characterized by J. L.
Kelley [43].

In this century, while averaging operators continued to find many applications
in its traditional areas of analysis and applied areas [37], their algebraic
study has been deepened and generalized. J. L. Loday [55] defined the
diassociative algebra as the enveloping algebra of the Leibniz algebra by
analogy with the associative algebra as the enveloping algebra of the Lie
algebra. More precisely, an averaging operator on an algebra $A$ over a field
$\mathbb{K}$ is a linear map $P:A\rightarrow A$ satisfying the averaging
relation:

$P(x)P(y)=P(P(x)y)=P(xP(y)).$

M. Aguiar in [4] showed that a diassociative algebra can be derived from an
averaging associative algebra by defining two new operations $x\dashv
y:=xP(y)$ and $x\vdash y:=P(x)y$. An analogue process gives a Leibniz algebra
from an averaging Lie algebra by defining a new operation
$\\{x,y\\}:=[P(x),y]$ and derives a (left) permutative algebra from an
averaging commutative associative algebra. In general, an averaging operator
was defined on any binary operad and this kind of process was systematically
studied in [66] by relating the averaging actions to a special construction of
binary operads called duplicators [84, 67].

The purpose of this paper is to give some constructions results of averaging
operators on Hom-Lie algebras and to classify the homogeneous averaging
operators on $q$-deformed Witt and $q$-deformed $W(2,2)$ algebras. Then the
induced Leibniz algebra structures are obtained. Section 2 contains some
necessary important basic notions, notations and examples on
$\mathbb{Z}$-graded Hom-Lie algebras which will be used in next sections and
we study the multiplicativity conditions of $q$-deformed Witt and $q$-deformed
$W(2,2)$ Hom algebras. Next, we present some useful methods for constructions
of averaging operator on Hom-Lie algebras. In section 3, we classify the
homogeneous averaging operators on the $q$-deformed Witt Hom-algebra
$\mathcal{V}^{q}$ and we give the induced Hom-Leibniz algebras from the
averaging operators on the $q$-deformed Witt Hom-algebra $\mathcal{V}^{q}$. In
section 4, we classify the homogeneous averaging operators on the $q$-deformed
$W(2,2)$ Hom-algebra $\mathcal{W}^{q}$. Also, we give the induced Hom-Leibniz
algebras from the averaging operators on the $q$-deformed $W(2,2)$ Hom-algebra
$\mathcal{W}^{q}$.

##  2 Constructions of averaging operators on Hom-Lie algebras

In this section, firstly, we review some important basic notions, notations
and examples on $\mathbb{Z}$-graded Hom-Lie algebras which will be used in
next sections. Then, we present some useful methods for constructions of
averaging operator on Hom Lie algebras.

In this article, all linear spaces are over a field $\mathbb{K}$ of
characteristic zero. A linear operator $T:A\mapsto A$ on a $\mathbb{Z}$-graded
linear space $A=\bigoplus_{j\in\mathbb{Z}}V_{j},$ is said to respect the
grading of the linear space $A$ if for any $i\in\mathbb{Z}$ there exists
$j\in\mathbb{Z}$ such that $T(A_{i})\subseteq A_{j}$. The linear operator
respecting grading is said to be homogeneous of degree $\deg T\in\mathbb{Z}$
if $T(A_{i})\subseteq A_{i+\deg T}$ for all $i\in\mathbb{Z}$, and $T$ is said
to be even if $\deg T=0$, that is $T(A_{i})\subseteq A_{i}$ for all for all
$i\in\mathbb{Z}$.

###  2.1 Hom-algebras, Hom-Lie algebras and multiplicativity

Hom-algebras in general are triples $(A,[\cdot,\cdot],\alpha)$ consisting of a
linear space $A$, bilinear product $[\cdot,\cdot]:A\times A\mapsto A$ and a
linear map (linear space homomorphism) $\alpha:A\mapsto A$. If, moreover, the
linear map $\alpha:A\rightarrow A$ is an algebra endomorphism, meaning that it
satisfies for all $x,y\in A$ the multiplicativity property

$\displaystyle\alpha([x,y])=[\alpha(x),\alpha(y)],$ (2.1)

then the Hom-Lie algebra is called multiplicative. Within specific classes of
Hom-algebras, it is important to characterize multiplicative and non-
multiplicative Hom-algebras belonging to the class.

A Hom-algebra, $(A,[\cdot,\cdot],\alpha)$ is said to be $\mathbb{Z}$-graded if
the linear space $A$ is $\mathbb{Z}$-graded,

$A=\bigoplus_{j\in\mathbb{Z}}A_{j},$

the bilinear product $[\cdot,\cdot]$ is $\mathbb{Z}$-graded, that is, for all
$m,n\in\mathbb{Z}$,

$[A_{m},A_{n}]\subseteq A_{m+n},$

and the linear operator $\alpha$ respects the $\mathbb{Z}$-grading of the
linear space $A$, that is, for any $i\in\mathbb{Z}$ there exists
$j\in\mathbb{Z}$ such that $\alpha(A_{i})\subseteq A_{j}$.

In any $\mathbb{Z}$-graded Hom-algebra
$(A=\bigoplus_{j\in\mathbb{Z}}A_{j},[\cdot,\cdot],\alpha)$, and the following
inclusions hold:

$\displaystyle[\alpha(A_{m}),\alpha(A_{n})]\subseteq[A_{m+k},A_{n+k}]\subseteq
A_{m+n+2k},$
$\displaystyle\alpha([A_{m},A_{n}])\subseteq\alpha(A_{m+n})\subseteq
A_{m+n+k},$
$\displaystyle[\alpha(A_{m}),\alpha(A_{n})]\cap\alpha([A_{m},A_{n}])\subseteq
A_{m+n+2k}\cap A_{m+n+k}=\left\\{\begin{array}[]{ll}A_{m+n},&\ \text{if}\
k=0\\\ 0,&\ \text{if}\ k\neq 0\end{array},\right.$
$\displaystyle\begin{array}[]{l}\ker([\cdot,\cdot])\cap((\ker(\alpha)\times
A)\cup(A\times\ker(\alpha)))\\\ \quad\subseteq
M_{A,[\cdot,\cdot],\alpha}=\\{(x,y)\in A\times
A\mid[\alpha(x),\alpha(y)]=\alpha([x,y])\\}\\\ \end{array}$

These inclusions directly yield the following handy conditions for checking
whether $\mathbb{Z}$-graded Hom-algebras are multiplicative or non-
multiplicative, based on an interaction between the bilinear product
$[\cdot,\cdot]$, the twisting map $\alpha$, the $\mathbb{Z}$-grading of $A$
and elements of its homogeneous subspaces $A_{j},j\in\mathbb{Z}$ in the
$\mathbb{Z}$-grading direct decomposition.

######  Theorem 2.1.

Let $(A=\displaystyle{\bigoplus_{n\in\mathbb{Z}}}A_{n},[\cdot,\cdot],\alpha)$
be a $\mathbb{Z}$-graded Hom-algebra where $\alpha$ is a linear operator
homogeneous of degree $\deg\alpha=k\in\mathbb{Z}$.

  1. (i)

The Hom-algebra $(A,[\cdot,\cdot],\alpha)$ is not multiplicative, if and only
if

$\exists\ m,n\in\mathbb{Z},x_{m}\in A_{m},x_{n}\in
A_{n}:[\alpha(x_{m}),\alpha(x_{n})]\neq\alpha([x_{m},x_{n}]).$

The Hom-algebra $(A,[\cdot,\cdot],\alpha)$ is multiplicative if and only if

$\forall\ m,n\in\mathbb{Z},x_{m}\in A_{m},x_{n}\in
A_{n}:[\alpha(x_{m}),\alpha(x_{n})]=\alpha([x_{m},x_{n}]).$

  2. (ii)

$(A,[\cdot,\cdot],\alpha)$ is not multiplicative, if and only if the strict
inclusion takes place

$\exists\ m,n\in\mathbb{Z}:\left\\{(x_{m},x_{n})\in A_{m}\times
A_{n}\mid[\alpha(x_{m}),\alpha(x_{n})]=\alpha([x_{m},x_{n}])\right\\}\subsetneq
A_{m}\times A_{n},$

or equivalently if and and only if

$\begin{array}[]{l}\exists\ m,n\in\mathbb{Z}:\\\ A_{m}\times
A_{n}\setminus\\{(x_{m},x_{n})\in A_{m}\times
A_{n}\mid[\alpha(x_{m}),\alpha(x_{n})]=\alpha([x_{m},x_{n}])\\}\neq\emptyset\end{array}$

  3. (iii)

If $(A,[\cdot,\cdot],\alpha)$ is multiplicative, then one of the following
alternatives holds:

    1. (a)

linear operator $\alpha$ is even, that is homogeneous of degree $k=0$;

    2. (b)

linear operator $\alpha$ is homogeneous of degree $k\neq 0$ and

$\forall\ m,n\in\mathbb{Z}:\
[\alpha(A_{m}),\alpha(A_{n})]\times\alpha([A_{m},A_{n}])=\\{0\\}\times\\{0\\}=\\{(0,0)\\},$

by linearity of $\alpha$ and bilinearity of $[\cdot,\cdot]$ equivalent to
$[\alpha(\cdot),\alpha(\cdot)]=\alpha([\cdot,\cdot])=0.$

  4. (iv)

If $k\neq 0$, then $(A,[\cdot,\cdot],\alpha)$ is not multiplicative if and
only if

$\exists\
m,n\in\mathbb{Z}:[\alpha(A_{m}),\alpha(A_{n})]\times\alpha([A_{m},A_{n}])\neq\\{0\\}\times\\{0\\}=\\{(0,0)\\}.$

If $k\neq 0$, then $(A,[\cdot,\cdot],\alpha)$ is multiplicative if and only if

$\forall\
m,n\in\mathbb{Z}:[\alpha(A_{m}),\alpha(A_{n})]\times\alpha([A_{m},A_{n}])=\\{0\\}\times\\{0\\}=\\{(0,0)\\},$

that is $\forall\
m,n\in\mathbb{Z}:[\alpha(A_{m}),\alpha(A_{n})]=\alpha([A_{m},A_{n}])=\\{0\\},$
which is the same as

$[A,A]\subseteq\ker(\alpha),\quad\alpha(A)\subseteq\ker([\cdot,\cdot])=\\{(x,y)\in
A\times A\mid[x,y]=0\\}.$

or equivalently, for elements of the homogeneous subspaces, if and only if

$\forall\ m,n\in\mathbb{Z},x_{m}\in A_{m},x_{n}\in
A_{n}:[\alpha(x_{m}),\alpha(x_{n})]=\alpha([x_{m},x_{n}])=0.$

If $\dim A_{m}=1$ for all $m\in\mathbb{Z}$ and $\\{x_{m}\in
A_{m},m\in\mathbb{Z}\\}$ is a homogeneous basis of the $\mathbb{Z}$-graded
linear space $A=\bigoplus\limits_{m\in\mathbb{Z}}A_{m}$, then for all
$m,n\in\mathbb{Z}$,

$\displaystyle\alpha(x_{m})=\alpha_{m+k,m}x_{m+k},\quad\text{for some unique}\
\alpha_{m+k,m}\in\mathbb{K}$
$\displaystyle\alpha(x_{m+n})=\alpha_{m+n+k,m+n}x_{n+m+k},\quad\text{for some
unique}\ \alpha_{m+n+k,m+n}\in\mathbb{K}$
$\displaystyle\left[x_{m},x_{n}\right]=c^{m+n}_{m,n}x_{m+n},\quad\text{for
some unique}\ c^{m+n}_{m,n}\in\mathbb{K}$
$\displaystyle\left[\alpha(x_{m}),\alpha(x_{n})\right]=\alpha_{m+k,m}\alpha_{n+k,n}c^{m+n+2k}_{m+k,n+k}x_{m+n+2k},$
$\displaystyle\alpha([x_{m},x_{n}])=c^{m+n}_{m,n}\alpha(x_{m+n})=c^{m+n}_{m,n}\alpha_{m+n+k,m+n}x_{m+n+k}.$

######  Corollary 2.2.

Let
$\mathcal{A}=(A=\displaystyle{\bigoplus_{n\in\mathbb{Z}}}A_{n},[\cdot,\cdot],\alpha)$
be a $\mathbb{Z}$-graded Hom-algebra where $\alpha$ is a homogeneous linear
operator of degree $\deg\alpha=k\in\mathbb{Z}$. If $\dim A_{m}=1$ for all
$m\in\mathbb{Z}$ and $\\{x_{m}\in A_{m},m\in\mathbb{Z}\\}$ is a homogeneous
basis of the $\mathbb{Z}$-graded linear space
$A=\oplus_{m\in\mathbb{Z}}A_{m}$, then for all $m,n\in\mathbb{Z}$,

  1. (i)

If $\deg\alpha=k\neq 0$, then $\mathcal{A}$ is multiplicative if and only if
for all $m,n\in\mathbb{Z}$,

$\alpha_{m+k,m}\alpha_{n+k,n}c^{m+n+2k}_{m+k,n+k}=c^{m+n}_{m,n}\alpha_{m+n+k,m+n}=0,$

which is equivalent to
$\left\\{\begin{array}[]{l}\alpha_{m+n+k,m+n}=0,\quad\text{if}\quad
c^{m+n}_{m,n}\neq 0\\\ \alpha_{m+k,m}=0\ \text{or}\
\alpha_{n+k,n}=0,\quad\text{if}\quad c^{m+n+2k}_{m+k,n+k}\neq
0\end{array}\right.$

  2. (ii)

If $\deg\alpha=k=0$, then $\mathcal{A}$ is multiplicative if and only if, for
all $m,n\in\mathbb{Z}$,

$\alpha_{m,m}\alpha_{n,n}c^{m+n}_{m,n}=c^{m+n}_{m,n}\alpha_{m+n,m+n},$

that is if and only if, for all $m,n\in\mathbb{Z}$,

$c^{m+n}_{m,n}(\alpha_{m,m}\alpha_{n,n}-\alpha_{m+n,m+n})=0,$

or equivalently if and only if, for all $m,n\in\mathbb{Z}$,

$\alpha_{m,m}\alpha_{n,n}=\alpha_{m+n,m+n},\quad\text{if}\quad
c^{m+n}_{m,n}\neq 0.$

######  Definition 2.3 ([39, 58]).

Hom-Lie algebras are Hom-algebras $(A,[\cdot,\cdot],\alpha)$ consisting of a
linear space $A$ over a field $\mathbb{K}$, a bilinear map $[\cdot,\cdot]$:
$A\times A\rightarrow A$ and a linear map $\alpha:A\rightarrow A$, satisfying
for all $x,y,z\in A$,

$\displaystyle[x,y]$ $\displaystyle=-[y,x]$ (Skew-symmetry identity) (2.2)
$\displaystyle[\alpha(x),[y,z]]$
$\displaystyle+[\alpha(y),[z,x]]+[\alpha(z),[x,y]]=0.$ (Hom-Jacobi identity)
(2.3)

######  Definition 2.4 ([48, 58]).

Hom-Leibniz algebras are Hom-algebras $(A,[\cdot,\cdot],\alpha)$ consisting of
a linear space $A$ over a field $\mathbb{K}$, a bilinear map $[\cdot,\cdot]$:
$A\times A\rightarrow A$ and a linear map $\alpha:A\rightarrow A$ satisfying
for all $x,y,z\in A$,

$[\alpha(x),[y,z]]=[[x,y],\alpha(z)]+[\alpha(y),[x,z]].\quad\quad\text{(Hom-
Leibniz identity)}$ (2.4)

When, moreover, the linear map $\alpha:A\rightarrow A$ satisfies
multiplicativity (2.1), that is when $\alpha$ is an algebra endomorphism, the
Hom-Leibniz algebra $(A,[\cdot,\cdot],\alpha)$ is called multiplicative.

######  Remark 2.5.

Skewsymmetric Hom-algebras are Hom-algebras satisfying the skewsymmetry axiom
(2.2), and hence the Hom-Lie algebras form a special subclass of skewsymmetric
Hom-algebras where moreover the Hom-Jacobi identity (2.3) holds. In
skewsymmetric algebras however there is no requirement of any relations
between the linear operation $\alpha$ and bilinear operation $[\cdot,\cdot]$.
In this sence, the skewsymmetric Hom-algebras can be seen and studied just as
arbitrary pairs of skewsymmetric algebras and linear operators on them.
However, this is not the case in Hom-Lie or Hom-Leibniz algebras where the
linear and bilinear operations are dependent via Hom-Jacobi and Hom-Leibniz
identities in nontrivial ways.

######  Remark 2.6.

Every skewsymmetric Hom-Leibniz algebra is a Hom-Lie algebra, every Hom-Lie
algebra is a skewsymmetric Leibniz algebra, but not every Hom-Leibniz algebra
is skewsymmetric, and thus Hom-Lie algebras as a class of Hom-algebras
coincides with the intersection of the class of Hom-Leibniz algebras and the
class of skewsymmetric algebras, which is moroever properly included in each
of classes.

######  Example 2.7.

For $q\in\mathbb{K}\setminus\\{0\\}$ and $n\in\mathbb{Z}$, the $q$-numbers
$\\{n\\}$ defined by

$\\{n\\}=\left\\{\begin{array}[]{ll}\frac{1-q^{n}}{1-q},&\text{for}\ q\neq
1\\\ n,&\text{for}\ q=1\end{array}\right.$

have the following properties

$\begin{array}[]{c}\\{m+1\\}=1+q\\{m\\}=\\{m\\}+q^{m},\
\\{m+n\\}=\\{m\\}+q^{m}\\{n\\},\ q^{m}\\{-m\\}=-\\{m\\},\\\ \\{m\\}=0\
\Leftrightarrow\ q^{m}=1.\end{array}$ (2.5)

The linear space $\mathcal{V}^{q}$ with a basis $\\{L_{n}|n\in\mathbb{Z}\\}$
equipped with the bilinear operation $[\cdot,\cdot]$ and a linear map $\alpha$
on $\mathcal{V}^{q}$ on the basis, for all $m,n\in\mathbb{Z}$, by

$\displaystyle\left[L_{m},L_{n}\right]=(\\{m\\}-\\{n\\})L_{m+n},\quad$ (2.6)
$\displaystyle\alpha(L_{n})=(1+q^{n})L_{n}.$

Then, $(\mathcal{V}^{q},[\cdot,\cdot],\alpha)$ is a Hom-Lie algebra [39, 75,
76], called the $q$-deformed Witt Hom-Lie algebra or $q$-Witt Hom-Lie algebra.
There is a natural $\mathbb{Z}$-grading on $\mathcal{V}^{q}$,

$\mathcal{V}^{q}=\bigoplus_{n\in\mathbb{Z}}\mathcal{V}^{q}_{n},\
\mathcal{V}^{q}_{n}=\mathbb{K}L_{n},\ n\in\mathbb{Z}.$

######  Example 2.8.

If, in Example 2.7, the linear operator $\alpha$, homogeneous of degree $k$,
is defined for all $n\in\mathbb{Z}$, by
$\alpha=\alpha_{k}(L_{n})=(1+q^{n-k})L_{n+k},$ then
$(\mathcal{V}^{q},[\cdot,\cdot],\alpha_{k})$ are $\mathbb{Z}$-graded Hom-Lie
algebras for all $k\in\mathbb{Z}$.

######  Example 2.9.

For $q\neq 0$ and $n\in\mathbb{Z}$, let $[n]$ denote the $q$-number

$[n]=[n]_{q}=\left\\{\begin{array}[]{l}\frac{q^{n}-q^{-n}}{q-q^{-1}},\
\text{if}\ q\neq\pm 1\\\ n,\ \text{if}\ q=1\\\
(-1)^{n-1}n=(-1)^{n+1}n=-(-1)^{n}n,\ \text{if}\ q=-1.\end{array}\right.$

Note that these $q$-numbers are invariant under transformation replacing $q$
by $q^{-1}$, and satisfy for all $m,n\in\mathbb{Z}$,

$\left\\{\begin{array}[]{l}[-n]=-[n],\ q^{n}[m]-q^{m}[n]=[m-n],\
q^{-n}[m]+q^{m}[n]=[m+n],\ \text{for all}\ q\in\mathbb{K}\setminus\\{0\\}\\\
\left[n\right]=0\ \Leftrightarrow\ q^{2n}=1,\ \text{for all}\ q\neq\pm 1\\\
\left[n\right]=n=0\ \Rightarrow\ n=0,q^{2n}=1^{0}=1,\ \text{for}\ q=1\\\
\left[n\right]=(-1)^{n-1}n=0\ \Rightarrow\ n=0,q^{2n}=(-1)^{0}=1,\ \text{for}\
q=-1.\end{array}\right.$

Note that if $q=\pm 1$, then $q^{2n}=1$ for all $n\in\mathbb{Z}$, while
$\left[n\right]=\left\\{\begin{array}[]{l}n,\ \text{if}\ q=1\\\ (-1)^{n-1}n,\
\text{if}\ q=-1\\\ \end{array}\right.=0$ only for $n=0$.

Let $\mathcal{W}^{q}$ be a linear space with basis
$\\{L_{n},W_{n}|n\in\mathbb{Z}\\}$, and a bilinear operation on
$\mathcal{W}^{q}$ is defined on the basis, for all $m,n\in\mathbb{Z}$, by

$\displaystyle[L_{m},L_{n}]=[m-n]L_{m+n},\ \ {[L_{m},W_{n}]=[m-n]W_{m+n}},$
(2.7)

and with other brackets obtained by skew-symmetry or equal to 0. The linear
map $\alpha$ on $\mathcal{W}^{q}$ is defined, for all $n\in\mathbb{Z}$, by

$\alpha(L_{n})=(q^{n}+q^{-n})L_{n},\ \ \alpha(W_{n})=(q^{n}+q^{-n})W_{n}.$

It was proved in [83] that the triple ($\mathcal{W}^{q},[\cdot,\cdot],\alpha$)
forms a Hom-Lie algebra, which is called the $q$-deformed $W(2,2)$ Hom-Lie
algebra. By defining ${\rm deg}(L_{n})={\rm deg}(W_{n})=n$, we obtain that
$\mathcal{W}^{q}$ is $\mathbb{Z}$-graded Hom-Lie algebra, namely
$\mathcal{W}^{q}=\bigoplus_{n\in\mathbb{Z}}\mathcal{W}^{q}_{n}$ with
$\mathcal{W}^{q}_{n}={\rm span}_{\mathbb{K}}\\{L_{n},W_{n}\\}.$ Note that
$\mathcal{W}^{q}$ is not multiplicative since $\alpha$ is not a homomorphism
of Hom-Lie algebras.

######  Example 2.10.

In Example 2.9, if the homogeneous linear operator of degree $k$ is defined by
$\beta(L_{n})=(q^{n-k}+q^{k-n})L_{n+k}$,
$\beta(W_{n})=(q^{n-k}+q^{k-n})W_{n+k}$ for all $n\in\mathbb{Z},$ then
$(\mathcal{W}^{q},[\cdot,\cdot],\beta)$ is a $\mathbb{Z}$-graded Hom-Lie
algebra.

######  Proposition 2.11.

For any $k\in\mathbb{Z}$, the Hom-Lie algebra
$(\mathcal{V}^{q},[\cdot,\cdot],\alpha_{k})$ is multiplicative if and only if
$k=0$ and $q=-1$.

###### Proof.

If $k=\deg\alpha_{k}=\deg\alpha_{0}=0$, then by Corollary 2.2 (ii),

$\displaystyle(\mathcal{V}^{q},[\cdot,\cdot],\alpha)\quad\text{is
multiplicative}\ \Leftrightarrow$ $\displaystyle\forall\ m,n\in\mathbb{Z}:\
(\\{m\\}-\\{n\\})\big{(}(1+q^{m})(1+q^{n})-(1+q^{m+n})\big{)}=0\
\Leftrightarrow$ $\displaystyle\forall\ m,n\in\mathbb{Z}:\
(\\{m\\}-\\{n\\})(q^{n}+q^{m})=0\ \Leftrightarrow$ $\displaystyle\forall\
m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{ll}(q^{n}-q^{m})(q^{n}+q^{m})=0,&\text{for}\ q\neq
1\\\ (m-n)(q^{n}+q^{m})=(m-n)\cdot 2=0,&\text{for}\ q=1\end{array}\right.\
\Leftrightarrow$ $\displaystyle\forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{ll}q^{2(n-m)}=1,&\text{for}\ q\neq 1\\\
m-n=0,&\text{for}\ q=1\end{array}\right.\ \Leftrightarrow$ $\displaystyle
q\neq 1,\forall\ p\in\mathbb{Z}:\ q^{2p}=1\Leftrightarrow q\neq 1,q^{2}=1\
\text{(for $p=1$)}\Leftrightarrow\ q=-1.$

If $k=\deg\alpha_{k}\neq 0$, then by Corollary 2.2 (i),

$\displaystyle(\mathcal{V}^{q},[\cdot,\cdot],\beta)\quad\text{is
multiplicative}\ \Leftrightarrow$ $\displaystyle\forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{l}(\\{m\\}-\\{n\\})(1+q^{m+n-k})=0;\\\
(1+q^{m-k})(1+q^{n-k})(\\{m+k\\}-\\{n+k\\})=0,\end{array}\right.\Leftrightarrow$
(2.10) $\displaystyle\forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{l}\left\\{\begin{array}[]{l}(q^{n}-q^{m})(1+q^{m+n-k})=0,\
\text{if}\ q\neq 1\\\ (m-n)\cdot 2=0,\ \text{if}\ q=1\end{array}\right.;\\\
\left\\{\begin{array}[]{l}(1+q^{m-k})(1+q^{n-k})(q^{n+k}-q^{m+k})=0,\
\text{if}\ q\neq 1\\\ 4\cdot(m-n)=0,\text{if}\
q=1\end{array}\right.\end{array}\right.\Leftrightarrow$ (2.17) $\displaystyle
q\neq 1\ \text{and}\ \forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{lll}&q^{n}=q^{m}\ \text{or}\ q^{n+m-k}=-1;\\\
&q^{m-k}=-1\ \text{or}\ q^{n-k}=-1\ \text{or}\
q^{n}=q^{m}\end{array}\right.\Leftrightarrow$ (2.20) $\displaystyle q\neq 1\
\text{and}\ \forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{ll}&q^{m-n}=1\\\
&\left\\{\begin{array}[]{ll}&q^{n+m-k}=-1;\\\ &q^{m-k}=-1\ \text{or}\
q^{n-k}=-1\end{array}\right.\ \text{if}\ q^{m-n}\neq
1,\end{array}\right.\Leftrightarrow$ (2.25) $\displaystyle q\neq 1\
\text{and}\ \forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{ll}&q^{m-n}=1;\\\
&\left\\{\begin{array}[]{ll}&q^{n+m-k}=-1\\\ &q^{m-k}=-1,\end{array}\right.\
\text{if}\ q^{m-n}\neq 1,\ q^{n-k}\neq-1\\\
&\left\\{\begin{array}[]{ll}&q^{n+m-k}=-1;\\\ &\ q^{n-k}=-1\end{array}\right.\
\text{if}\ q^{m-n}\neq 1,\ q^{m-k}\neq-1,\\\
\end{array}\right.\Leftrightarrow$ (2.33) $\displaystyle q\neq 1\ \text{and}\
\forall\ m,n\in\mathbb{Z}:\ \left\\{\begin{array}[]{ll}&q^{m-n}=1;\\\
&q^{n}=1\ \text{for}\ q^{m}\neq 1,q^{k}\neq-1;\\\ &q^{m}=1\ \text{for}\
q^{n}\neq 1,q^{k}\neq-1,\end{array}\right.$ (2.37)

If $q\neq 1$ and $q^{k}=-1$, then (2.37) reduces to $q\neq 1\ \text{and}\
\forall\ m,n\in\mathbb{Z}:q^{m-n}=1$, which does not hold because
$q^{m-n}=-1\neq 1$ when $m-n=k$. If $q\neq 1$, $q^{k}\neq-1$ and $q^{k}=1$,
the (2.37) does not hold since for $m=2k+1$ and $n=k$,

$q^{m-n}=q^{k+1}=q\neq 1,\ q^{m}=q^{2k+1}=q\neq 1,\ q^{n}=q^{k+1}=q\neq 1,$

and if $q\neq 1$, $q^{k}\neq-1$ and $q^{k}\neq 1$, then (2.37) does not hold
since for $m=2k$ and $n=k$,

$q^{m-n}=q^{k}\neq 1,\ q^{m}=q^{2k}\neq 1,\ q^{n}=q^{k}=q\neq 1.$

Hence, if $k=\deg\alpha_{k}\neq 0$, then
$(\mathcal{V}^{q},[\cdot,\cdot],\alpha_{k})$ is not multiplicative for any
$q$. ∎

######  Proposition 2.12.

The $q$-deformed $W(2,2)$ Hom-Lie algebra
$(\mathcal{W}^{q},[\cdot,\cdot],\alpha)$ is multiplicative if and only if
$q^{2}=-1$ (which is equivalent to $q=\pm i$ if there exists $i\in\mathbb{K}$
such that $i^{2}=-1$, for example when $\mathbb{K}$ is algebraically closed
field, like $\mathbb{C}$).

###### Proof.

For all $n,m\in\mathbb{Z}$, we have

$\displaystyle[\alpha(L_{m}),\alpha(L_{n})]-\alpha([L_{m},L_{n}])=[(q^{m}+q^{-m})L_{m},(q^{n}+q^{-n})L_{n}]-\alpha([m-n]L_{m+n})$
$\displaystyle\quad=(q^{m}+q^{-m})(q^{n}+q^{-n})[m-n]L_{m+n}-[m-n](q^{m+n}+q^{-m-n})L_{m+n}$
$\displaystyle\quad=[m-n](q^{m-n}+q^{n-m})L_{m+n}=\left\\{\begin{array}[]{l}\frac{(q^{m-n}-q^{n-m})(q^{m-n}+q^{n-m})}{q-q^{-1}}L_{m+n},\
\text{if}\ q\neq\pm 1\\\ 2(m-n)L_{m+n},\ \text{if}\ q=1\\\ 2(n-m)L_{m+n}\
\text{if}\ q=-1\end{array}\right.$
$\displaystyle[\alpha(L_{m}),\alpha(W_{n})]-\alpha([L_{m},W_{n}])=[(q^{m}+q^{-m})L_{m},(q^{n}+q^{-n})W_{n}]-\alpha([m-n]W_{m+n})$
$\displaystyle\quad=(q^{m}+q^{-m})(q^{n}+q^{-n})[m-n]W_{m+n}-[m-n](q^{m+n}+q^{-m-n})W_{m+n}$
$\displaystyle\quad=[m-n](q^{m-n}+q^{n-m})W_{m+n}=\left\\{\begin{array}[]{l}\frac{(q^{m-n}-q^{n-m})(q^{m-n}+q^{n-m})}{q-q^{-1}}W_{m+n},\
\text{if}\ q\neq\pm 1\\\ 2(m-n)W_{m+n},\ \text{if}\ q=1\\\ 2(n-m)W_{m+n}\
\text{if}\ q=-1\end{array}\right.,$
$\displaystyle\alpha([W_{m},W_{n}])-[\alpha(W_{m}),\alpha(W_{n})]=0.$

So, by Theorem 2.1 (i),

$\displaystyle\text{$(\mathcal{W}^{q},[\cdot,\cdot],\alpha)$ is
multiplicative}\ \Leftrightarrow$ $\displaystyle\forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{l}(q^{m-n}-q^{n-m})(q^{m-n}+q^{n-m})=0,\ \text{if}\
q\neq\pm 1\\\ m-n=0,\ \text{if}\ q=\pm 1\end{array}\right.\Leftrightarrow$
$\displaystyle q\neq\pm 1\ \text{and}\ \forall\ m,n\in\mathbb{Z}:\
(q^{2(m-n)}-q^{2(n-m)})=0\ \Leftrightarrow$ $\displaystyle q\neq\pm 1\
\text{and}\ \forall\ m,n\in\mathbb{Z}:\ q^{4(m-n)}=1\Leftrightarrow q\neq\pm
1\ \text{and}\ \forall\ p\in\mathbb{Z}:\ q^{4p}=1\Leftrightarrow$
$\displaystyle q\neq\pm 1\ \text{and}\ \forall\ p\in\mathbb{Z}:\ q^{4}=1\
\Leftrightarrow\ q^{2}=-1.$ $\displaystyle\Leftrightarrow q=\pm i\ \text{if}\
\exists\ i\in\mathbb{K}:\ i^{2}=-1\ \text{(for example if $\mathbb{K}$ is
algebraically closed).}\qed$

######  Proposition 2.13.

The Hom-Lie algebra $(\mathcal{W}^{q},[\cdot,\cdot],\beta)$ is not
multiplicative for any $q\in\mathbb{K}\setminus\\{0\\}$.

###### Proof.

By Theorem 2.1 (iv),

$\displaystyle(\mathcal{W}^{q},[\cdot,\cdot],\beta)\ \text{is multiplicative}\
\Leftrightarrow\ \forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{llll}\beta([L_{m},L_{n}])=[\beta(L_{m}),\beta(L_{n})]=0\\\
\beta([L_{m},W_{n}])=[\beta(L_{m}),\beta(W_{n})]=0,\end{array}\right.\Leftrightarrow$
$\displaystyle\forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{l}(q^{m-k}+q^{k-m})(q^{n-k}+q^{k-n})[m-n]L_{m+n+2k}=0\\\
(q^{m+n-k}+q^{k-m-n})[m-n]L_{m+n+k}=0,\\\
(q^{m-k}+q^{k-m})(q^{n-k}+q^{k-n})[m-n]W_{m+n+2k}=0\\\
(q^{m+n-k}+q^{k-m-n})[m-n]W_{m+n+k}=0,\end{array}\right.\Leftrightarrow$
$\displaystyle q\neq\pm 1\ \text{and}\ \forall\
m,n\in\mathbb{Z}:\left\\{\begin{array}[]{lll}(q^{m-k}+q^{k-m})(q^{n-k}+q^{k-n})(q^{m-n}+q^{n-m})=0,\\\
(q^{m+n-k}+q^{k-m-n})(q^{m-n}+q^{n-m})=0,\end{array}\right.$

which does not hold since for $m=k+1,n=k$ it reduces to the impossible

$q\neq\pm 1\ \text{and}\ \left\\{\begin{array}[]{lll}2(q+q^{-1})^{2}=0,\\\
(q^{k+1}+q^{-(k+1)})(q+q^{-1})=0,\end{array}\right.$

Hence $(\mathcal{W}^{q},[\cdot,\cdot],\beta)$ is not multiplicative for any
$q\in\mathbb{K}\setminus\\{0\\}$. ∎

###  2.2 Averaging operators on Hom-algebras

######  Definition 2.14.

An averaging operator on a Hom-algebra $(A,[\cdot,\cdot],\alpha)$ over
$\mathbb{K}$ is a linear operator $P:A\rightarrow A$, satisfying for all
$x,y\in A,$

$\displaystyle\alpha\circ P=P\circ\alpha,$ (commutativity of $P$ with
$\alpha$) (2.38) $\displaystyle[P(x),P(y)]=P([P(x),y])=P([x,P(y)]),$
(averaging operator axiom) (2.39)

######  Remark 2.15.

In skewsymmetric Hom-algebras, and thus in the Hom-Lie algebras in particular,
the skewsymmetry of multiplication (2.2) implies that (2.39) is equivalent to

$[P(x),P(y)]=P([P(x),y]),\;\;\forall\ x,y\in A.$ (2.40)

######  Proposition 2.16.

If $P$ is an averaging operator on a Hom-algebra
$\mathcal{A}=(A,[\cdot,\cdot],\alpha)$, then

  1. (i)

$(P(A),[\cdot,\cdot],\alpha)$ is a Hom-subalgebra of the Hom-Lie algebra
$(A,[\cdot,\cdot],\alpha)$;

  2. (ii)

$[P(A),ker(P)]\subseteq ker(P)$ and $[ker(P),P(A)]\subseteq ker(P)$.

  3. (iii)

If $P$ is surjective, that is if $P(A)=A$, then $ker(P)$ is a two-sided Hom-
ideal in the Hom-algebra $\mathcal{A}$, meaning that $[A,ker(P)]\subseteq
ker(P)$, $[ker(P),A]\subseteq ker(P)$ and $\alpha(ker(P))\subseteq ker(P)$.

###### Proof.

Since $P$ is a linear operator, $P(A)$ is a linear subspace of $A$.

  1. (i)

By (2.38), $P$ and $\alpha$ commute, and hence $\alpha(P(x))=P(\alpha(x))\in
P(A).$ Since, for any $x,y\in P(A)$, there exist $x^{\prime},y^{\prime}\in A$
such that $x=P(x^{\prime}),\ y=P(y^{\prime}),$ the averaging operator axiom
(2.39) yields
$[x,y]=[P(x^{\prime}),P(y^{\prime})]=P([P(x^{\prime}),y^{\prime}])\in P(A)$.

  2. (ii)

Let $x=P(x^{\prime})$ for some $x^{\prime}\in A$. If $y^{\prime}\in ker(P)$,
then (2.39) yields
$P([x,y^{\prime}])=P([P(x^{\prime}),y^{\prime}])=[P(x^{\prime}),P(y^{\prime})]=0$,
and hence, $[P(A),ker(P)]\subseteq ker(P)$. Let $y=P(y^{\prime})$ for some
$y^{\prime}\in A$. If $x^{\prime}\in ker(P)$, then (2.39) yields
$P([x^{\prime},y])=P([x^{\prime},P(y^{\prime})])=[P(x^{\prime}),P(y^{\prime})]=0$,
and hence $[ker(P),P(A)]\subseteq ker(P)$.

  3. (iii)

The first two inclusions are a special case of (ii), and
$\alpha(ker(P))\subseteq ker(P)$ follows from commutativity of $\alpha$ and
$P$. ∎

The defining axioms of Hom-Leibniz algebras and Hom-Lie algebras are
multilinear in their arguments and are inherited by Hom-subalgebras.

######  Corollary 2.17.

Let $P$ be an averaging operator on a Hom-algebra $(A,[\cdot,\cdot],\alpha)$.
Then, if $(A,[\cdot,\cdot],\alpha)$ is a skewsymmetric Hom-algebra, or a Hom-
Leibniz algebra or a Hom-Lie algebra, then $(P(A),[\cdot,\cdot],\alpha)$ is
also a skewsymmetric Hom-algebra, or a Hom-Leibniz algebra or a Hom-Lie
algebra respectively.

######  Proposition 2.18.

If $\mathcal{A}=(A,[\cdot,\cdot],\alpha)$ is a Hom-Leibniz algebra and $P$ is
an averaging operator on $\mathcal{A}$, then with $\\{\cdot,\cdot\\}:A\times
A\rightarrow A$ defined for all $x,y\in A$ by $\\{x,y\\}=[P(x),y],$

  1. (i)

the triple $\mathcal{A}^{\prime}=(A,\\{\cdot,\cdot\\},\alpha)$ is a Hom-
Leibniz algebra;

  2. (ii)

If $\mathcal{A}=(A,[\cdot,\cdot],\alpha)$ is a Hom-Leibniz algebra, then
$\mathcal{A}^{\prime}=(A,\\{\cdot,\cdot\\},\alpha)$ is a Hom-Lie algebra if
and only if $[P(x),y]=-[P(y),x]$ for all $x,y\in A$;

  3. (iii)

If $\mathcal{A}=(A,[\cdot,\cdot],\alpha)$ is a Hom-Lie algebra, and the
averaging linear operator $P$ is surjective, that is $P(A)=A$, then
$\mathcal{A}^{\prime}=(A,\\{\cdot,\cdot\\},\alpha)$ is a Hom-Lie algebra.

###### Proof.

Let $x,y,z\in A$, Then (2.4) in $\mathcal{A}^{\prime}$ is proved as follows:

$\displaystyle\\{\alpha(x),\\{y,z\\}\\}-\\{\\{x,y\\},\alpha(z)\\}-\\{\alpha(y),\\{x,z\\}\\}$
$\displaystyle=[P(\alpha(x)),[P(y),z]]-[P([P(x),y],\alpha(z)]-[P(\alpha(y)),[P(x),P(z)]]$
$\displaystyle\begin{array}[b]{l}\text{\scriptsize($\alpha$, $P$ commute)}\\\
=[\alpha(P(x)),[P(y),z]]-[[P(x),P(y)],\alpha(z)]-[\alpha(P(y)),[P(x),z]]\end{array}\begin{array}[b]{l}\text{\scriptsize($\mathcal{A}$
is Hom-Leibniz algebra)}\\\ =0.\end{array}\qed$

######  Proposition 2.19.

Let $\\{P_{j}\\}_{1\leq j\leq n}$ be a finite set of averaging operators on a
Hom-Lie algebra $(A,[\cdot,\cdot],\alpha)$, and $\\{\lambda_{j}\\}_{1\leq
j\leq n}\subseteq\mathbb{K}$. Then

  1. (i)

The operator $S=\sum_{j=1}^{n}\lambda_{j}P_{j}$ is an averaging operator on
$A$ if

$\displaystyle\sum_{\begin{subarray}{c}j,k=1\\\ i\neq
j\end{subarray}}^{n}\lambda_{j}\lambda_{k}P_{j}([P_{k}(x),y])=\sum_{\begin{subarray}{c}j,k=1\\\
j\neq k\end{subarray}}^{n}\lambda_{j}\lambda_{k}[P_{j}(x),P_{k}(y)]$

  2. (ii)

If $P_{j}\circ P_{k}=P_{k}\circ P_{j}$ for $1\leq k,j\leq n$, then
$T={\displaystyle\prod_{j=1}^{n}}P_{j}=P_{1}\circ\dots\circ P_{n}$ is an
averaging operator.

  3. (iii)

If $P_{j}\circ P_{k}=P_{k}\circ P_{j}$ for $1\leq k,j\leq n$, then for any
polynomial $F\in\mathbb{K}[t_{1},\dots,t_{n}]$ with zero constant term
$F(0,\dots,0)=0$, the operator $F(P_{1},\dots,P_{n})$ is an averaging
operator.

  4. (iv)

If an averaging operator $P$ is invertible, then $P^{-1}$ is an averaging
operator.

###### Proof.

(i) The map $S$ is a linear operator on $A$ as a linear combination of the
linear operators $\\{P_{j}\\}_{1\leq j\leq n}$, and $\alpha\circ
S=S\circ\alpha$, since $\alpha\circ P_{j}=P_{j}\circ\alpha,\ 1\leq j\leq n$.
For all $x,y\in A$.

$\displaystyle S([S(x),y])=$
$\displaystyle(\sum_{j=1}^{n}\lambda_{j}P_{j})([(\sum_{k=1}^{n}\lambda_{k}P_{k})(x),y])=(\sum_{j=1}^{n}\lambda_{j}P_{j})([\sum_{k=1}^{n}\lambda_{k}P_{k}(x),y])$
$\displaystyle=$
$\displaystyle\sum_{j,k=1}^{n}\lambda_{j}\lambda_{k}P_{j}([P_{k}(x),y])$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\lambda_{j}^{2}P_{j}([P_{k}(x),y])+\sum_{\begin{subarray}{c}j,k=1\\\
j\neq k\end{subarray}}^{n}\lambda_{j}\lambda_{k}P_{j}([P_{k}(x),y])$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\lambda_{j}^{2}[P_{j}(x),P_{j}(y)]+\sum_{\begin{subarray}{c}j,k=1\\\
j\neq k\end{subarray}}^{n}\lambda_{j}\lambda_{k}([P_{j}(x),P_{k}(y)])$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}[\lambda_{j}P_{j}(x),\lambda_{j}P_{j}(y)]+\sum_{\begin{subarray}{c}j,k=1\\\
j\neq k\end{subarray}}^{n}([\lambda_{j}P_{j}(x),\lambda_{k}P_{k}(y)])$
$\displaystyle=$
$\displaystyle[\sum_{j=1}^{n}\lambda_{j}P_{j},\sum_{k=1}^{n}\lambda_{k}P_{k}]=[S(x),S(y)].$

(ii) The operator $T_{n}=P_{1}\circ\dots\circ P_{n}$ is linear as a
composition of the linear operators, and also $\alpha\circ
T_{n}=T_{n}\circ\alpha$ since $\alpha\circ P_{j}=P_{j}\circ\alpha$ for all
$1\leq j\leq n$. For $n=1$, $T_{1}={\displaystyle\prod_{j=1}^{n}}P_{j}=P_{1}$
is an averaging operator. Suppose that for $n=k$ the statement holds, that is
$T_{k}=P_{1}\circ\dots\circ P_{k}$ is an averaging operator. Then, for all
$x,y\in A$,

$\displaystyle T_{k+1}([T_{k+1}(x),y])=$ $\displaystyle T_{k}\circ
P_{k+1}([T_{k}(P_{k+1}(x)),y])$ ${}^{\text{($T_{k},P_{k+1}$ commute)}}=$
$\displaystyle T_{k}(P_{k+1}([P_{k+1}(T_{k}(x)),y]))$ ${}^{\text{($P_{k+1}$ is
an averaging operator)}}=$ $\displaystyle
T_{k}([P_{k+1}(T_{k}(x)),P_{k+1}(y)])$ ${}^{\text{($T_{k},P_{k+1}$
commute)}}=$ $\displaystyle T_{k}([T_{k}(P_{k+1}(x)),P_{k+1}(y)])$
${}^{\text{($T_{k}$ is an averaging operator)}}=$
$\displaystyle[T_{k}(P_{k+1}(x)),T_{k}(P_{k+1}(y))]$ $\displaystyle=$
$\displaystyle[T_{k+1}(x),T_{k+1}(y)].$

proving that $T_{n}$ is averaging operator for $n=k+1$, which completes the
proof by the principle of mathematical induction.
(iii) Since $F(P_{1},\dots,P_{n})$ is a linear combination of compositions of
averaging operators, it is also an averaging operator by (ii) and (i).
(iv) It is clear that if $P$ is invertible, then $P^{-1}$ is a linear map of
$A$ and $\alpha\circ P^{-1}=P^{-1}\circ\alpha$. Let $x,y\in A$. Since $P$ is
surjective, there exists $y^{\prime}\in A$ such that
$P(x^{\prime})=P^{-1}(x)$. Then,

$\displaystyle P(P^{-1}([P^{-1}(x),y]))$
$\displaystyle=[P^{-1}(x),y]=[P^{-1}(x),P(P^{-1}(y)]$
$\displaystyle=[P(x^{\prime}),P(P^{-1}(y)]=P([P(x^{\prime}),P^{-1}(y)])=P([P^{-1}(x),P^{-1}(y)]).$

Since $P$ is injective, we have $P^{-1}([P^{-1}(x),y])=[P^{-1}(x),P^{-1}(y)]$.
∎

######  Remark 2.20.

By Proposition 2.19, if $P$ and $Q$ are two averaging operators on a Hom-Lie
algebra $(A,[\cdot,\cdot],\alpha)$, then

  1. (i)

for any $\lambda\in\mathbb{K}$, $\lambda P$ is an averaging operator on $A$;

  2. (ii)

If $P([Q(x),y])+Q([P(x),y])=[Q(x),P(y)]+[P(x),Q(y)]$ holds for all $x,y\in A$,
then $P+Q$ is an averaging operator.

  3. (iii)

If $P\circ Q$ = $Q\circ P$, then $P\circ Q$ is an averaging operator;

  4. (iv)

for any polynomial $F\in\mathbb{K}[t]$ with zero constant term $F(0)=0$, the
operator $F(P)$ is an averaging operator.

######  Remark 2.21.

It is known that $P$ is an averaging operator on a Hom-Lie algebra
$(A,[\cdot,\cdot],\alpha)$ if and only if $\lambda P$ is an averaging operator
on $A$ for $\lambda\in\mathbb{K}^{\ast}=\mathbb{K}\setminus\\{0\\}$. So the
set of averaging operators on any Hom-Lie algebra carries an action of
$\mathbb{K}^{\ast}$ by scalar multiplication.

Next we provide the necessary and sufficient conditions for an idempotent
linear operator to be an averaging operator.

######  Definition 2.22.

An idempotent operator on a Hom-Lie algebra $(A,[\cdot,\cdot],\alpha)$ over
$\mathbb{K}$ is a linear map $P:A\rightarrow A$ satisfying $\alpha\circ
P=P\circ\alpha$ and $P^{2}=P.$

######  Remark 2.23.

Recall that there is a bijection

$\\{\text{idempotent linear operators on }A\\}\leftrightarrow\\{\text{direct
sum decompositions}\ A=A_{0}\oplus A_{1}\\}$

where $A_{0}=im(P)$ and $A_{1}=ker(P)$. The linear map $P$ corresponding to
$A=A_{0}\oplus A_{1}$ is called the projection onto $A_{0}$ along $A_{1}$. If
$P$ is the projection onto $A_{0}$ along $A_{1}$, then $I-P$ is the projection
onto $A_{1}$ along $A_{0}$ since $(I-P)^{2}=I-2P+P^{2}=I-2P+P=I-P$ and
$im(I-P)=ker(P),\ Ker(I-P)=im(P)$.

######  Proposition 2.24.

Let $(A,[\cdot,\cdot],\alpha)$ be a Hom-Lie algebra and let $P:A\to A$ be an
idempotent linear map. Let $A=A_{0}\oplus A_{1}$ be the corresponding linear
decomposition. Then $P$ is an averaging operator if and only if

$[A_{0},A_{0}]\subseteq A_{0},\quad[A_{0},A_{1}]\subseteq A_{1}.$ (2.41)

###### Proof.

For any $x,y\in A$, denote $x=x_{0}+x_{1}$ and $y=y_{0}+y_{1}$ with
$x_{i},y_{i}\in A_{i},i=0,1$. Suppose $P$ is an averaging operator. Then from
$P(A)=A_{0}$ and $[P(x),P(y)]=P([P(x),y]),$ we obtain $[A_{0},A_{0}]\subseteq
A_{0}$. Then we have

$\displaystyle[P(x),P(y)]$ $\displaystyle=$ $\displaystyle[x_{0},y_{0}],$
$\displaystyle P([P(x),y])$ $\displaystyle=$ $\displaystyle
P([x_{0},y_{0}]+[x_{0},y_{1}])=P([x_{0},y_{0}])+P([x_{0},y_{1}])=[x_{0},y_{0}]+P([x_{0},y_{1}]).$

Thus from (2.40) we obtain $P([x_{0},y_{1}])=0$ for all $x_{i},y_{i}\in
A_{i},i=0,1$. Therefore (2.41) holds since $A_{1}=kerP$ by the definition of
$P$. Conversely, suppose (2.41) holds. Then we have

$P([P(x),y])=P([x_{0},y_{0}]+[x_{0},y_{1}])=P([x_{0},y_{0}])+P([x_{0},y_{1}])=[x_{0},y_{0}]=[P(x),P(y)].$

Thus $P$ is an averaging operator. ∎

######  Corollary 2.25.

An idempotent endomorphism $P:A\to A$ is an averaging operator.

###### Proof.

Let $A_{0}:=imP$ and $A_{1}:=kerP$. Then we have $A=A_{0}\oplus A_{1}$ and $P$
is the projection to $A_{0}$ along $A_{1}$. Since $A_{1}$ is an ideal of $A$,
then (2.41) holds. Hence $P$ is an averaging operator. ∎

##  3 On homogeneous averaging operators on q-deformed Witt Hom-algebra

In this section we classify the homogeneous averaging operators on the
$q$-deformed Witt algebra $\mathcal{V}^{q}$ and we give the induced Hom-
Leibniz algebras from the averaging operators on the $q$-deformed Witt algebra
$\mathcal{V}^{q}$ and its multiplicativity condition is studied.

######  Definition 3.1.

A homogeneous operator $F$ with degree $d\in\mathbb{Z}$ on the $q$-deformed
Witt Hom-algebra $\mathcal{V}^{q}$ is a linear operator on $\mathcal{V}^{q}$
satisfying $F(\mathcal{V}^{q}_{m})\subseteq\mathcal{V}^{q}_{m+d}$ for all
$m\in\mathbb{Z}$.

Therefore, a homogeneous averaging operator $P_{d}$ with degree $d$ on the
$q$-deformed Witt Hom-algebra $\mathcal{V}^{q}$ is an averaging operator on
$\mathcal{V}^{q}$ of the following form

$P_{d}(L_{m})=f(m+d)L_{m+d},\quad\forall\ m\in\mathbb{Z},$ (3.1)

where $f$ is a $\mathbb{K}$-valued function defined on $\mathbb{Z}.$

Let $P_{d}$ be a homogeneous averaging operator with degree $d$ on the
$q$-deformed Witt Hom-algebra $(\mathcal{V}^{q},[\cdot,\cdot],\alpha)$
satisfying (3.1). Then by (2.6) and (2.40),

$\displaystyle{}[P_{d}(L_{m}),P_{d}(L_{n})]$
$\displaystyle=[f(m+d)L_{m+d},f(n+d)L_{n+d}]$
$\displaystyle=f(m+d)f(n+d)(\\{m+d\\}-\\{n+d\\})L_{m+n+2d},$ $\displaystyle
P_{d}([P_{d}(L_{m}),L_{n}])$ $\displaystyle=P_{d}([f(m+d)L_{m+d},L_{n}])$
$\displaystyle=f(m+d)f(m+n+2d)(\\{m+d\\}-\\{n\\})L_{m+n+2d}.$

We see that the function $f$ satisfies for all $m,n\in\mathbb{Z}$,

$f(m+d)f(n+d)(\\{m+d\\}-\\{n+d\\})=f(m+d)f(m+n+2d)(\\{m+d\\}-\\{n\\}),$

or equivalently, after changing $m\rightarrow m-d$ and $n\rightarrow n-d$,

$f(m)\big{(}f(n)(\\{m\\}-\\{n\\})-f(m+n)(\\{m\\}-\\{n-d\\})\big{)}=0.$ (3.2)

######  Lemma 3.2.

If $P_{d}$ is a non-zero averaging operator on the $q$-deformed Witt Hom-
algebra $(\mathcal{V}^{q},[\cdot,\cdot],\alpha)$ with degree $d$, then
$\alpha\circ P_{d}=P_{d}\circ\alpha$ if and only if $q^{d}=1$.

###### Proof.

For all $m\in\mathbb{Z}$,

$\displaystyle\alpha\circ
P_{d}(L_{m})=\alpha(f(m+d)L_{m+d})=f(m+d)\alpha(L_{m+d})=f(m+d)(1+q^{m+d})L_{m+d},$
$\displaystyle
P_{d}\circ\alpha(L_{m})=P_{d}(\alpha(L_{m}))=P_{d}((1+q^{m})L_{m})=(1+q^{m})P_{d}(L_{m})=(1+q^{m})f(m+d)L_{m+d},$
$\displaystyle\alpha\circ P_{d}(L_{m})=P_{d}\circ\alpha(L_{m})\
\Leftrightarrow\forall\ m\in\mathbb{Z}:\ f(m+d)(q^{d}-1)=0,$

is equivalent to $q^{d}=1$ when $P_{d}\neq 0$, as in this case $f(m+d)\neq 0$
for some $m\in\mathbb{Z}$. ∎

### Case 1: $q=1$

When $q=1$, equation (3.2) becomes for all $m,n\in\mathbb{Z}$,

$f(m)\big{(}f(n)(m-n)-f(m+n)(m-n+d)\big{)}=0.$ (3.3)

Plugging $n=0$ in (3.3), we have

$f(m)\big{(}mf(0)-(m+d)f(m)\big{)}=0.$ (3.4)

#### Subcase 1: $d=0$

######  Proposition 3.3.

With the notations as above, the averaging operator $P_{0}$ with degree $d=0$
is given by

$f(m)=\mu f(0)+\nu\delta_{m,0},\;\;\forall\
m\in\mathbb{Z},~{}for~{}some~{}(\nu,\mu)\in\mathbb{K}\times\\{0,1\\},$

where for any $x,y\in\mathbb{K},$
$\delta_{x,y}=\left\\{\begin{array}[]{lllll}1\text{ if }x=y\\\ 0\text{ if
}x\neq y.\end{array}\right.$

###### Proof.

When $d=0$, (3.4) becomes $mf(m)(f(0)-f(m))=0$. Hence $f(m)=\mu
f(0)+\nu\delta_{m,0}$ for some $\nu\in\mathbb{K}$ and $\mu\in\\{0,1\\}$. ∎

#### Subcase 2: $d\in\mathbb{Z}^{\ast}$

######  Proposition 3.4.

With the notations as above, when the degree $d\in\mathbb{Z}^{\ast}$ and
$f(0)=0$, we have

$f(m)=\nu\delta_{m+d,0},\;\;\forall\
m\in\mathbb{Z},~{}where~{}\nu\in\mathbb{K},$

###### Proof.

When $d\neq 0$ and $f(0)=0$, then by equation (3.4), we have for all
$m\in\mathbb{Z}$, $(m+d)f^{2}(m)=0$. Thus the function $f$ satisfies for any
$m\in\mathbb{Z}$, $f(m)=\nu\delta_{m+d,0}$ for some $\nu\in\mathbb{K}$. ∎

######  Proposition 3.5.

With the notations as above, when the degree $d\in\mathbb{Z}^{\ast}$ and
$f(0)\neq 0$, we have

$f(m)=\mu\frac{m}{m+d}f(0)\delta_{m,\mathbb{Z}\setminus\\{-d\\}},\;\;\forall\
m\in\mathbb{Z},~{}where~{}\mu\in\\{0,1\\},$

where
$\delta_{m,\mathbb{Z}\setminus\\{-d\\}}=\left\\{\begin{array}[]{lllll}1\text{
if }m\neq-d\\\ 0\text{ if }m=-d.\end{array}\right.$

###### Proof.

When $d\neq 0$ and $f(0)\neq 0$, it follows from equation (3.4) for $m=-d$
that $f(-d)=0$. Moreover for $m\neq-d$ in equation (3.4), we have
$f(m)=\mu\frac{m}{m+d}f(0),\;\;\text{ for some }\mu\in\\{0,1\\}.$ Therefore
for all $m\in\mathbb{Z}$,
$f(m)=\mu\frac{m}{m+d}f(0)\delta_{m,\mathbb{Z}\setminus\\{-d\\}},\;\;where~{}\mu\in\\{0,1\\}.$
∎

### Case 2: $q\neq 1$ and $q^{d}=1$

######  Proposition 3.6.

With the notations as above, Suppose that the degree $d$ satisfying $q\neq 1$
and $q^{d}=1$, then we have

$f(m)=\mu f(0)+\nu\delta_{q^{m},1},\;\;\forall\
m\in\mathbb{Z},~{}where~{}(\nu,\mu)\in\mathbb{K}\times\\{0,1\\}.$

###### Proof.

When $q\neq 1$ and $q^{d}=1$, (3.2) becomes for all $m,n\in\mathbb{Z},$

$f(m)\big{(}f(n)(\\{m\\}-\\{n\\})-f(m+n)(\\{m\\}-\\{n\\})\big{)}=0.$ (3.5)

Taking $n=0$ in (3.5) yields $\\{m\\}f(m)(f(0)-f(m))=0.$ Hence $f(m)=\mu
f(0)+\nu\delta_{q^{m},1}$ for some $\nu\in\mathbb{K}$ and $\mu\in\\{0,1\\}$. ∎

######  Theorem 3.7.

The homogeneous averaging operator $P_{d}$ with degree $d$ on the $q$-deformed
Witt Hom-algebra $(\mathcal{V}^{q},[\cdot,\cdot],\alpha)$ must be one of the
following operators, given for all $m\in\mathbb{Z}$, by

$\displaystyle
P_{d}^{1}(L_{m})=\beta+\nu\delta_{m+2d,0}L_{m+d},\quad\quad\text{ for
}q=1\text{ and }d\in\mathbb{Z},$ $\displaystyle
P_{d}^{2}(L_{m})=\mu\frac{m+d}{m+2d}\gamma\delta_{m+d,\mathbb{Z}\setminus\\{-d\\}}L_{m+d},\quad\quad\text{
for }q=1\text{ and }d\in\mathbb{Z},$ $\displaystyle
P_{d}^{3}(L_{m})=\beta+\nu\delta_{q^{m},1}L_{m+d},\quad\quad\text{ for }q\neq
1\text{ and }q^{d}=1,$

where $\beta,\nu\in\mathbb{K}$, $\gamma\in\mathbb{K}^{\ast}$ and
$\mu\in\\{0,1\\}$.

###### Proof.

Directly by ombining Lemma 3.2 and Propositions 3.3, 3.4, 3.5 and 3.6. ∎

Now, using the construction given in Proposition 2.18, we have the following:

######  Theorem 3.8.

The homogeneous averaging operators obtained in Theorem 3.7 for the
$q$-deformed Witt Hom-algebra $(\mathcal{V}^{q},[\cdot,\cdot],\alpha)$, give
rise to the following Hom-Leibniz algebras on the underlying linear space
$\mathcal{V}^{q}$,

$\displaystyle\\{L_{m},L_{n}\\}^{1}=(\beta+\nu\delta_{m+d,0})(m-n)L_{m+n},\quad\quad\forall
m,n\in\mathbb{Z},\text{ where }q=1\text{ and }d\in\mathbb{Z},$
$\displaystyle\\{L_{m},L_{n}\\}^{2}=(\mu\frac{m}{m+d}\gamma\delta_{m,\mathbb{Z}\setminus\\{-d\\}})(m-n)L_{m+n},\quad\quad\forall
m,n\in\mathbb{Z},\text{ where }q=1\text{ and }d\in\mathbb{Z},$
$\displaystyle\\{L_{m},L_{n}\\}^{3}=(\beta+\nu\delta_{q^{m},1})(\\{m\\}-\\{n\\})L_{m+n},\quad\quad\forall
m,n\in\mathbb{Z},\text{ where }q\neq 1\text{ and }q^{d}=1,$

where $\beta,\nu\in\mathbb{K}$, $\gamma\in\mathbb{K}^{\ast}$ and
$\mu\in\\{0,1\\}$.

###### Proof.

By Proposition 2.18, the Hom-Leibniz algebra induced by
$P_{d}^{i},~{}i=1,2,3,$ is given for $m,n\in\mathbb{Z}$ by

$\displaystyle\\{L_{m},L_{n}\\}^{1}=[P_{d}^{1}(L_{m}),L_{n}]=[(\beta+\nu\delta_{m+2d,0}L_{m+d},L_{n}]=(\beta+\nu\delta_{m+2d,0})(m+d-n)L_{m+n+d}$
$\displaystyle=(\beta+\nu\delta_{m+d,0})(m-n)L_{m+n},$
$\displaystyle\\{L_{m},L_{n}\\}^{2}=[P_{d}^{2}(L_{m}),L_{n}]=[\mu\frac{m+d}{m+2d}\gamma\delta_{m+d,\mathbb{Z}\setminus\\{-d\\}}L_{m+d},L_{n}]$
$\displaystyle=(\mu\frac{m+d}{m+2d}\gamma\delta_{m+d,\mathbb{Z}\setminus\\{-d\\}})(m+d-n)L_{m+n+d}=(\mu\frac{m}{m+d}\gamma\delta_{m,\mathbb{Z}\setminus\\{-d\\}})(m-n)L_{m+n},$
$\displaystyle\\{L_{m},L_{n}\\}^{3}=[P_{d}^{3}(L_{m}),L_{n}]=[(\beta+\nu\delta_{q^{m},1})L_{m+d},L_{n}]=(\beta+\nu\delta_{q^{m},1})(\\{m\\}-\\{n\\})L_{m+n+d}$
$\displaystyle=(\beta+\nu\delta_{q^{m},1})(\\{m\\}-\\{n-d\\})L_{m+n}\begin{array}[b]{l}\text{\scriptsize($\\{n-d\\}=\\{n\\}+q^{n}\\{-d\\}$)}\\\
=(\beta+\nu\delta_{q^{m},1})(\\{m\\}-(\\{n\\}+q^{n}\\{-d\\})L_{m+n}\end{array}$
${}^{\text{($\\{-d\\}=0$)}}=(\beta+\nu\delta_{q^{m},1})(\\{m\\}-\\{n\\})L_{m+n}.\qed$

######  Proposition 3.9.

The non-trivial Hom-Leibniz algebra
$(\mathcal{V}^{q},~{}\\{\cdot,\cdot\\}^{i},\alpha)$ induced by $P_{d}^{i}$ is
multiplicative if and only if $i=3$ and $q=-1$.

###### Proof.

By Corollary 2.2 item (ii):

  1. $(\mathcal{V}^{q},\\{\cdot,\cdot\\}^{1},\alpha)\ \text{is multiplicative}\ \Leftrightarrow\forall\ m,n\in\mathbb{Z}:\ 2(\beta+\nu\delta_{m+d,0})(m-n)=0\ \Leftrightarrow$
$\forall\ m\in\mathbb{Z}:\
(\beta+\nu\delta_{m+d,0})=0\Leftrightarrow\beta=\nu=0\Leftrightarrow\\{\cdot,\cdot\\}^{1}=0.$

  2. $\text{$(\mathcal{V}^{q},\\{\cdot,\cdot\\}^{2},\alpha)$ is multiplicative}\Leftrightarrow\forall\ m,n\in\mathbb{Z}:\ 2\mu\frac{m}{m+d}\gamma\delta_{m,\mathbb{Z}\setminus\\{-d\\}})(m-n)=0\ \Leftrightarrow$ $\forall\ m\in\mathbb{Z}:\ \mu\frac{m}{m+d}\gamma\delta_{m,\mathbb{Z}\setminus\\{-d\\}})=0\stackrel{{\scriptstyle\gamma\neq 0}}{{\Leftrightarrow}}\mu=0\Leftrightarrow\\{\cdot,\cdot\\}^{2}=0.$

  3. $\text{$(\mathcal{V}^{q},\\{\cdot,\cdot\\}^{3},\alpha)$ is multiplicative}\Leftrightarrow$
$\forall\ m,n\in\mathbb{Z}:\
(\beta+\nu\delta_{q^{m},1})(\\{m\\}-\\{n\\})\big{(}(1+q^{m+n})-(1+q^{m})(1+q^{n})\big{)}=0\
\Leftrightarrow$
$\forall\ m\in\mathbb{Z}:\
(\beta+\nu\delta_{q^{m},1})(q^{n}-q^{m})(q^{n}+q^{m})=0\Leftrightarrow$
$\forall\ m\in\mathbb{Z}:\
(\beta+\nu\delta_{q^{m},1})(q^{2n}-q^{2m})=0\Leftrightarrow$
$\forall\ m,n\in\mathbb{Z}:\ \left\\{\begin{array}[]{ll}(q^{2(n-m)})=1,\text{
or }\\\ (\beta+\nu\delta_{q^{m},1})=0,\end{array}\right.\Leftrightarrow$
$\left\\{\begin{array}[]{ll}q^{2p}=1,~{}\forall p\in\mathbb{Z},\text{ or }\\\
\forall\ m\in\mathbb{Z}:\
\beta+\nu\delta_{q^{m},1}=0,\end{array}\right.\Leftrightarrow\left\\{\begin{array}[]{ll}q=-1,\text{
or }\hskip 28.45274pt(\text{since }q\neq 1)\\\ \beta=\nu=0.\end{array}\right.$
∎

##  4 On homogeneous averaging operators on q-deformed W(2,2) Hom-algebra

In this section we classify the homogeneous averaging operators on the
$q$-deformed $W(2,2)$ Hom-algebra $\mathcal{W}^{q}$. Also, we give the induced
Hom-Leibniz algebras from the averaging operators on the $q$-deformed $W(2,2)$
Hom-algebra and its multiplicativity condition is studied.

######  Definition 4.1.

A homogeneous operator $F$ with degree $d\in\mathbb{Z}$ on the $q$-deformed
$W(2,2)$ Hom-algebra $\mathcal{W}^{q}$ is a linear operator on
$\mathcal{W}^{q}$ satisfying
$F(\mathcal{W}^{q}_{m})\subseteq\mathcal{W}^{q}_{m+d}$ for all
$m\in\mathbb{Z}$.

Hence a homogeneous averaging operator $P_{d}$ with degree $d$ on the
$q$-deformed $W(2,2)$ Hom-algebra $\mathcal{W}^{q}$ is an averaging operator
on $\mathcal{W}^{q}$ with the following form:

$\displaystyle P_{d}(L_{m})=f_{1}(m+d)L_{m+d}+g_{1}(m+d)W_{m+d},$ (4.1)
$\displaystyle P_{d}(W_{m})=f_{2}(m+d)L_{m+d}+g_{2}(m+d)W_{m+d},$ (4.2)

where $f_{i}$ and $g_{i}$ are $\mathbb{K}$-valued functions defined on
$\mathbb{Z}$.

Let $P_{d}$ be a homogeneous averaging operator of degree $d$ on the
$q$-deformed $W(2,2)$ Hom-algebra $\mathcal{W}^{q}$ satisfying equations (4.1)
and (4.2). Then, by equations (2.7) and (2.40),

$\displaystyle[P_{d}(L_{m}),P_{d}(L_{n})]=[f_{1}(m+d)L_{m+d}+g_{1}(m+d)W_{m+d},f_{1}(n+d)L_{n+d}+g_{1}(n+d)W_{n+d}]$
$\displaystyle\quad=f_{1}(m+d)f_{1}(n+d)[m-n]L_{m+n+2d}+f_{1}(m+d)g_{1}(n+d)[m-n]W_{n+m+2d}$
$\displaystyle\hskip 227.62204pt-g_{1}(m+d)f_{1}(n+d)[n-m]W_{m+n+2d},$
$\displaystyle
P_{d}([P_{d}(L_{m}),L_{n}])=P_{d}([f_{1}(m+d)L_{m+d}+g_{1}(m+d)W_{m+d},L_{n}])$
$\displaystyle\quad=f_{1}(m+d)[m+d-n]P_{d}(L_{m+n+d})-g_{1}(m+d)[n-m-d]P_{d}(W_{m+n+d})$
$\displaystyle\quad=f_{1}(m+d)f_{1}(m+n+2d)[m+d-n]L_{m+n+2d}$
$\displaystyle\quad\quad+f_{1}(m+d)g_{1}(m+n+2d)[m+d-n]W_{m+n+2d}$
$\displaystyle\quad\quad-g_{1}(m+d)f_{2}(m+n+2d)[n-m-d]L_{m+n+2d}$
$\displaystyle\quad\quad-g_{1}(m+d)g_{2}(m+n+2d)[n-m-d]W_{m+n+2d},$

we see that the functions $f_{i}$ and $g_{i}$ satisfy, for all
$m,n\in\mathbb{Z}$, the following equations:

$\displaystyle
f_{1}(m)f_{1}(n)[m-n]-f_{1}(m)f_{1}(m+n)[m+d-n]+g_{1}(m)f_{2}(m+n)[n-m-d]=0,$
(4.3)
$\displaystyle\begin{array}[]{l}f_{1}(m)g_{1}(n)[m-n]-f_{1}(n)g_{1}(m)[n-m]-f_{1}(m)g_{1}(m+n)[m+d-n]\\\
\hskip 227.62204pt+g_{1}(m)g_{2}(m+n)[n-m-d]=0.\end{array}$ (4.6)

and from

$\displaystyle[P_{d}(L_{m}),P_{d}(W_{n})]=[f_{1}(m+d)L_{m+d}+g_{1}(m+d)W_{m+d},f_{2}(n+d)L_{n+d}+g_{2}(n+d)W_{n+d}]$
$\displaystyle\quad=f_{1}(m+d)f_{2}(n+d)[m-n]L_{m+n+2d}+f_{1}(m+d)g_{2}(n+d)[m-n]W_{m+n+2d}$
$\displaystyle\hskip 227.62204pt-g_{1}(m+d)f_{2}(n+d)[n-m]W_{m+n+2d},$
$\displaystyle
P_{d}([P_{d}(L_{m}),W_{n}])=P_{d}([f_{1}(m+d)L_{m+d}+g_{1}(m+d)W_{m+d},W_{n}])$
$\displaystyle\quad=f_{1}(m+d)f_{2}(m+n+2d)[m+d-n]L_{m+n+2d}$
$\displaystyle\hskip 170.71652pt+f_{1}(m+d)g_{2}(m+n+2d)[m+d-n]W_{m+n+2d},$

we see that the functions $f_{i}$ and $g_{i}$ satisfy, for all
$m,n\in\mathbb{Z}$, the following equations:

$\displaystyle f_{1}(m)f_{2}(n)[m-n]-f_{1}(m)f_{2}(m+n)[m+d-n]=0,$ (4.7)
$\displaystyle
f_{1}(m)g_{2}(n)[m-n]-g_{1}(m)f_{2}(n)[n-m]-f_{1}(m)g_{2}(m+n)[m+d-n]=0.$
(4.8)

In the same way, from

$\displaystyle[P_{d}(W_{m}),P_{d}(W_{n})]=[f_{2}(m+d)L_{m+d}+g_{2}(m+d)W_{m+d},f_{2}(n+d)L_{n+d}+g_{2}(n+d)W_{n+d}]$
$\displaystyle\quad=[f_{2}(m+d)f_{2}(n+d)[m-n]L_{n+m+2d}+f_{2}(m+d)g_{2}(n+d)[m-n]W_{m+n+2d}$
$\displaystyle\hskip 227.62204pt-g_{2}(m+d)f_{2}(n+d)[n-m]W_{m+n+2d},$
$\displaystyle
P_{d}([P_{d}(W_{m}),W_{n}])=P_{d}([f_{2}(m+d)L_{m+d}+g_{2}(m+d)W_{m+d},W_{n}]$
$\displaystyle\quad=f_{2}(m+d)f_{2}(m+n+2d)[m+d-n]L_{m+n+2d}+f_{2}(m+d)g_{2}(m+n+2d)W_{m+n+2d},$
$\displaystyle
P_{d}([W_{m},P_{d}(W_{m}])=P_{d}([W_{m},f_{2}(n+d)L_{n+d}+g_{2}(n+d)W_{n+d}])$
$\displaystyle\quad\quad-f_{2}(n+d)f_{2}(m+n+2d)[m-n-d]L_{m+n+2d}$
$\displaystyle\quad\quad-f_{2}(n+d)g_{2}(m+n+2d)[m-n-d]W_{m+n+2d},$

we see that the functions $f_{i}$ and $g_{i}$ satisfy for all
$m,n\in\mathbb{Z}$, the following equations:

$\displaystyle f_{2}(m)f_{2}(n)[m-n]-f_{2}(m)f_{2}(m+n)[m+d-n]=0,$ (4.9)
$\displaystyle
f_{2}(m)g_{2}(n)[m-n]-f_{2}(n)g_{2}(m)[n-m]-f_{2}(m)g_{2}(m+n)[m+d-n]=0.$
(4.10)

Similarly from

$\displaystyle[P_{d}(W_{m}),P_{d}(L_{n})]=[f_{2}(m+d)L_{m+d}+g_{2}(m+d)W_{m+d},f_{1}(n+d)L_{n+d}+g_{1}(n+d)W_{n+d}]$
$\displaystyle\quad=f_{2}(m+d)f_{1}(n+d)[m-n]L_{m+n+2d}$
$\displaystyle\quad\quad+f_{2}(m+d)g_{1}(n+d)[m-n]W_{m+n+2d}-g_{2}(m+d)f_{1}(n+d)[m-n]W_{m+n+2d},$
$\displaystyle
P_{d}([P_{d}(W_{m}),L_{n}])=P_{d}([f_{2}(m+d)L_{m+d}+g_{2}(m+d)W_{m+d},L_{n}])$
$\displaystyle\quad=f_{2}(m+d)[m+d-n](f_{1}(m+n+2d)L_{m+n+2d}+g_{1}(m+n+2d)L_{m+n+2d})$
$\displaystyle\quad\quad
g_{2}(m+d)[n-d-m](f_{2}(m+n+2d)L_{m+n+2d}+g_{2}(m+n+2d)W_{m+n+2d}),$

we see that the functions $f_{i}$ and $g_{i}$ satisfy, for all
$m,n\in\mathbb{Z}$,

$\displaystyle
f_{2}(m)f_{1}(n)[m-n]-f_{2}(m)f_{1}(m+n)[m+d-n]+g_{2}(m)f_{2}(m+n)[n-m-d]=0,$
(4.11)
$\displaystyle\begin{array}[]{l}f_{2}(m)g_{1}(n)[m-n]-g_{2}(m)f_{1}(n)[n-m]-f_{2}(m)g_{1}(m+n)[m+d-n]\\\
\hskip 227.62204pt+g_{2}(m)g_{2}(m+n)[n-m-d]=0.\end{array}$ (4.14)

######  Lemma 4.2.

If $P_{d}$ be a non zero averaging operator on $q$-deformed $W(2,2)$ Hom-
algebra $\mathcal{W}^{q}$ with degree $d$. Then $\alpha\circ
P_{d}=P_{d}\circ\alpha$ if and only if $q^{d}=1$.

###### Proof.

For all $m\in\mathbb{Z},$ we have

$\displaystyle\alpha\circ P_{d}(L_{m})$
$\displaystyle=\alpha(P_{d}(L_{m}))=\alpha(f_{1}(m+d)L_{m+d}+g_{1}(m+d)W_{m+d})$
$\displaystyle=f_{1}(m+d)(q^{m+d}+q^{-(m+d)})L_{m+d}+g_{1}(m+d)(q^{m+d}+q^{-(m+d)})W_{m+d},$
$\displaystyle P_{d}\circ\alpha(L_{m})$
$\displaystyle=P_{d}(\alpha(L_{m}))=P_{d}((q^{m}+q^{-m})L_{m})$
$\displaystyle=f_{1}(m+d)(q^{m}+q^{-m})L_{m}+g_{1}(m+d)(q^{m}+q^{-m})W_{m}.$

Then $\forall m\in\mathbb{Z},~{}\alpha\circ
P_{d}(L_{m})=P_{d}\circ\alpha(L_{m})$ if and only if
$q^{m}+q^{-m}=q^{m+d}+q^{-m-d}.$ Similarly, $\forall
m\in\mathbb{Z},~{}\alpha\circ P_{d}(W_{m})=P_{d}\circ\alpha(W_{m})$ if and
only if $q^{m}+q^{-m}=q^{m+d}+q^{-m-d}.$ Thus,

  1. (i)

if $q^{d}=1$, it is clear that $\alpha\circ P_{d}=P_{d}\circ\alpha$,

  2. (ii)

if $q^{d}\neq 1$, we have

$\displaystyle\alpha\circ P_{d}=P_{d}\circ\alpha$
$\displaystyle\Longleftrightarrow\forall
m\in\mathbb{Z},~{}q^{m}+q^{-m}=q^{m+d}+q^{-m-d}\Longleftrightarrow\forall
m\in\mathbb{Z},~{}q^{m}(1-q^{d})=q^{-m-d}(1-q^{d})$
$\displaystyle\begin{array}[b]{l}\text{\scriptsize($q^{d}\neq 1$)}\\\
\Longleftrightarrow\forall
m\in\mathbb{Z},~{}q^{m}=q^{-m-d}\end{array}\Longleftrightarrow\forall
m\in\mathbb{Z},~{}q^{2m+d}=1,$

this implies for $m=0$, $q^{d}=1$ which impossible since $q^{d}\neq 1$. ∎

### Case 1: $q\neq-1,1$ and $q^{d}=1$

#### Subcase 1: $q^{2m}=1$

Take $n=0$ in (4.3)-(4.14). For $q^{d}=1$ and $q^{2m}=1$, the functions
$f_{1},~{}f_{2},~{}g_{1}$ and $g_{2}$ satisfy

$f_{1}(m)=\nu_{1},\ f_{2}(m)=\nu_{2},\ g_{1}(m)=\nu_{3},\ g_{2}(m)=\nu_{4},\
\nu_{i}\in\mathbb{K}.$

Then we have the following Proposition.

######  Proposition 4.3.

If $P_{d}$ is an averaging operator on $\mathcal{W}^{q}$ with degree $d$,
where $q^{d}=1,q^{2m}=1$, then

$\left\\{\begin{array}[]{ll}f_{1}(m)=\nu_{1},&f_{2}(m)=\nu_{2},\\\
g_{1}(m)=\nu_{3},&g_{2}(m)=\nu_{4},\end{array}\right.\quad\text{where
$\nu_{i}\in\mathbb{K}.$}$

#### Subcase 2: $q^{2m}\neq 1$.

Taking $n=0$ in (4.9) yields

$\begin{array}[t]{l}f_{2}(m)f_{2}(0)[m]=f_{2}^{2}(m)[m+d]=f_{2}^{2}(m)(q^{-d}[m]+q^{m}[d])\begin{array}[t]{l}=f_{2}^{2}(m)[m].\\\
\text{\small(since $q^{d}=1$)}\end{array}\end{array}$

This gives $f_{2}(m)(f_{2}(0)-f_{2}(m))=0$. Hence,

$f_{2}(m)=\mu_{1}f_{2}(0),\quad\mu_{1}\in\\{0,1\\}.$

Then, we have the following Proposition.

######  Proposition 4.4.

If $P_{d}$ is an averaging operators on $\mathcal{W}^{q}$ with degree $d$ such
that $q^{d}=1$, $q^{2m}\neq 1$ and $f_{2}(0)=0,$ then

  1. (i)

if $f_{1}(0)=0$, then $f_{1}(m)=0,\ f_{2}(m)=0,\ g_{1}(m)=\gamma,\
g_{2}(m)=0,$ where $\gamma\in\mathbb{K}$;

  2. (ii)

if $f_{1}(0)\neq 0$, then

    1. (a)

$f_{1}(m)=f_{1}(0),\ f_{2}(m)=0,\ g_{1}(m)=\gamma,\ g_{2}(m)=0$, where
$\gamma\in\mathbb{K};$

    2. (b)

$f_{1}(m)=f_{1}(0),\ f_{2}(m)=0,\ g_{1}(m)=0,\ g_{2}(m)=f_{1}(0)$, where
$\gamma\in\mathbb{K};$

    3. (c)

$f_{1}(m)=0,\ f_{2}(m)=0,\ g_{1}(m)=\gamma,\ g_{2}(m)=f_{1}(0)$, where
$\gamma\in\mathbb{K}.$

###### Proof.

Let $q^{d}=1$, $q^{2m}\neq 1$ and $f_{2}(0)=0$ as assumed. Taking $n=0$ in
(4.3) yields

$\displaystyle f_{1}(m)f_{1}(0)[m]=$ $\displaystyle
f_{1}^{2}(m)[m+d]-g_{1}(m)f_{2}(m)[-m-d]$ $\displaystyle=$ $\displaystyle
f_{1}^{2}(m)[m]+g_{1}(m)f_{2}(m)[m].\quad\text{(since $q^{d}=1$)}$

Since $q^{2m}\neq 0$, we get $f_{1}(m)f_{1}(0)=f_{1}^{2}(m)+g_{1}(m)f_{2}(m)$,
and since $f_{2}(m)=0$, we obtain $f_{1}(m)(f_{1}(0)-f_{1}(m))=0$. Thus,
$f_{1}(m)=\mu_{2}f_{1}(0)$, $\mu_{2}\in\\{0,1\\}$. Setting $n=0$ in (4.11)
yields

$\displaystyle f_{2}(m)g_{1}(0)[m]=$ $\displaystyle
g_{2}(m)f_{1}(0)[-m]+f_{2}(m)g_{1}(m)[m+d]-g_{2}^{2}(m)[-m-d]$
$\displaystyle=$ $\displaystyle-
g_{2}(m)f_{1}(0)[m]+f_{2}(m)g_{1}(m)[m]+g_{2}^{2}(m)[m].\quad\text{(since
$q^{d}=1$)}$

Since $q^{2m}\neq 1$, we have
$f_{2}(m)g_{1}(0)=-g_{2}(m)f_{1}(0)+f_{2}(m)g_{1}(m)+g_{2}^{2}(m)$, from which
together with $f_{2}(m)=0$, we get $g_{2}(m)=\mu_{3}f_{1}(0)$,
$\kappa_{3}\in\\{0,1\\}.$ Taking $n=0$ in (4.6) gives

$\displaystyle f_{1}(m)g_{1}(0)[m]=$ $\displaystyle
f_{1}(0)g_{1}(m)[-m]+f_{1}(m)g_{1}(m)[m+d]-g_{1}(m)g_{2}(m)[-m-d]$
$\displaystyle=$ $\displaystyle-
f_{1}(0)g_{1}(m)[m]+f_{1}(m)g_{1}(m)[m]+g_{1}(m)g_{2}(m)[m],\quad\text{(since
$q^{d}=1$)}$

from which with $q^{2m}\neq 1$ we get
$f_{1}(m)g_{1}(0)=-f_{1}(0)g_{1}(m)+f_{1}(m)g_{1}(m)+g_{1}(m)g_{2}(m)$, and
with $f_{1}(m)=\mu_{2}f_{1}(0)$ and $g_{2}(m)=\mu_{3}f_{1}(0)$, we obtain
$g_{1}(m)(\mu_{2}+\mu_{3}-1)f_{1}(0)=\mu_{2}f_{1}(0)g_{1}(0)$. Then, we have
the two cases:

  1. (i)

if $f_{1}(0)=0$, then $g_{1}(m)=\gamma$,

  2. (ii)

if $f_{1}(0)\neq 0$ we have $\mu_{2}g_{1}(0)=(\mu_{2}+\mu_{3}-1)g_{1}(m)$ then
for $\mu_{2}=1$ and $\mu_{3}=0$ gives $g_{1}(0)=0$. Then
$\left\\{\begin{array}[]{ll}g_{1}(m)=0&\ \text{if}\
(\mu_{2},\mu_{3})\in\\{(0,0),(1,1)\\},\\\ g_{1}(m)=\gamma&\ \text{if}\
(\mu_{2},\mu_{3})\in\\{(1,0),(0,1)\\}.\end{array}\right.$ ∎

######  Proposition 4.5.

If $P_{d}$ is the averaging operator on $\mathcal{W}^{q}$ with degree $d$
satisfying $q^{d}=1$, $q^{2m}\neq 1$ and $f_{2}(0)\neq 0$, then
$\left\\{\begin{array}[]{ll}f_{1}(m)=\gamma,&f_{2}(m)=f_{2}(0),\\\
g_{1}(m)=\frac{\gamma
f_{1}(0)-\gamma^{2}}{f_{2}(0)},&g_{2}(m)=f_{1}(0)-\gamma,\end{array}\right.$
where $\gamma\in\mathbb{K}$.

###### Proof.

Let $q^{d}=1,~{}~{}q^{2m}\neq 1$ and $f_{2}(0)\neq 0,$ as assumed. Taking
$n=0$ in (4.7) yields

$\displaystyle f_{1}(m)f_{2}(0)[m]=$ $\displaystyle f_{1}(m)f_{2}(m)[m+d]$
$\displaystyle=$ $\displaystyle f_{1}(m)f_{2}(m)[m].\quad\text{(since
$q^{d}=1$)}$

Since $q^{2m}\neq 1$ we have $f_{1}(m)f_{2}(0)=f_{1}(m)f_{2}(m)$. This
together with $f_{2}(m)=\mu_{1}f_{2}(0)$ gives $f_{1}(m)(\mu_{1}-1)=0$. Then
$f_{1}(m)=\left\\{\begin{array}[]{lllll}0&if&\mu=0,\\\
\gamma&if&\mu_{1}=1.\end{array}\right.$ Taking $n=0$ in (4.10) yields

$\displaystyle f_{2}(m)g_{2}(0)[m]=$ $\displaystyle
f_{2}(0)g_{2}(m)[-m]+f_{2}(m)g_{2}(m)[m+d]$ $\displaystyle=$ $\displaystyle-
f_{2}(0)g_{2}(m)[m]+f_{2}(m)g_{2}(m)[m].\quad\text{(since $q^{d}=1$})$

Since $q^{2m}\neq 1$, we have
$f_{2}(m)g_{2}(0)=-f_{2}(0)g_{2}(m)+f_{2}(m)g_{2}(m)$. This, with
$f_{2}(m)=\mu_{1}f_{2}(0)$, gives
$\mu_{1}f_{2}(0)g_{2}(0)=-f_{2}(0)g_{2}(m)+\mu_{1}f_{2}(0)g_{2}(m)$. Then

  1. (i)

if $\mu_{1}=0$ we have $g_{2}(m)=0$,

  2. (ii)

if $\mu_{1}=1$ we have $g_{2}(0)=0$.

Taking $n=0$ in (4.3) yields

$\displaystyle f_{1}(m)f_{1}(0)[m]=$ $\displaystyle
f_{1}^{2}(m)[m+d]-g_{1}(m)f_{2}(m)[-m-d]$ $\displaystyle=$ $\displaystyle
f_{1}^{2}(m)[m]+g_{1}(m)f_{2}(m)[m].\quad\text{(since $q^{d}=1$)}$

Since $q^{2m}\neq 1$, we have
$f_{1}(m)f_{1}(0)=f_{1}^{2}(m)+g_{1}(m)f_{2}(m)$. This, with
$f_{2}(m)=\mu_{1}f_{2}(0)$, $f_{1}(m)=\gamma$ and $\mu_{1}=1$, yields
$g_{1}(m)=\frac{\gamma f_{1}(0)-\gamma^{2}}{f_{2}(0)}$.

Taking $n=0$ in (4.8) yields

$\displaystyle f_{1}(m)g_{2}(0)[m]=$ $\displaystyle
f_{2}(0)g_{1}(m)[-m]+f_{1}(m)g_{2}(m)[m+d]$ $\displaystyle=$ $\displaystyle-
f_{2}(0)g_{1}(m)[m]+f_{1}(m)g_{2}(m)[m].\quad\text{(since $q^{d}=1$)}$

Since $q^{2m}\neq 0$ we have
$f_{1}(m)g_{2}(0)=-f_{2}(0)g_{1}(m)+f_{1}(m)g_{2}(m)$. This together with
$g_{2}(0)=0$ and $f_{1}(m)=g_{2}(m)=0$ for $\mu_{1}=0$ gives $g_{1}(m)=0$.

Taking $n=0$ in the equation (4.14), we have

$\displaystyle f_{2}(m)f_{1}(0)[m]=$ $\displaystyle
f_{2}(m)f_{1}(m)[m+d]-g_{2}(m)f_{2}(m)[-m-d]$ $\displaystyle=$ $\displaystyle
f_{2}(m)f_{1}(m)[m]+g_{2}(m)f_{2}(m)[m].\quad\text{(since $q^{d}\neq 1$)}$

Then $f_{2}(m)f_{1}(0)=f_{2}(m)f_{1}(m)+g_{2}(m)f_{2}(m)$. This, together with
$f_{1}(m)=\gamma$ for $\mu_{1}=1$, gives $g_{2}(m)=f_{1}(0)-\gamma$. ∎

######  Theorem 4.6.

Homogeneous averaging operators on the $q$-deformed $W(2,2)$ Hom-algebra
$\mathcal{W}^{q}$ with degree $d$ such that $q^{d}=1$ and $q\neq-1,1$ must be
one of the following operators, given for all $m\in\mathbb{Z}$, by

$\displaystyle\left\\{\begin{array}[]{lllll}P_{d}^{1}(L_{m})=\nu_{1}\delta_{q^{2m},1}L_{m+d}+(\nu_{3}\delta_{q^{2m},1}+\gamma)W_{m+d},\\\
P_{d}^{1}(W_{m})=\nu_{2}\delta_{q^{2m},1}L_{m+d}+\nu_{4}\delta_{q^{2m},1}W_{m+d},\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{lllll}P_{d}^{2}(L_{m})=(\nu_{1}\delta_{q^{2m},1}+\beta)L_{m+d}+(\nu_{3}\delta_{q^{2m},1}+\gamma)W_{m+d},\\\
P_{d}^{2}(W_{m})=\nu_{2}\delta_{q^{2m},1}L_{m+d}+\nu_{4}\delta_{q^{2m},1}W_{m+d},\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{lllll}P_{d}^{3}(L_{m})=(\nu_{1}\delta_{q^{2m},1}+\beta)L_{m+d}+\nu_{3}\delta_{q^{2m},1}W_{m+d},\\\
P_{d}^{3}(W_{m})=\nu_{2}\delta_{q^{2m},1}L_{m+d}+(\nu_{4}\delta_{q^{2m},1}+\beta)W_{m+d},\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{l}P_{d}^{4}(L_{m})=(\nu_{1}\delta_{q^{2m},1}+\gamma)L_{m+d}+\nu_{3}\delta_{q^{2m},1}W_{m+d},\\\
P_{d}^{4}(W_{m})=\nu_{2}\delta_{q^{2m},1}L_{m+d}+(\nu_{4}\delta_{q^{2m},1}+\beta)W_{m+d},\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{lllll}P_{d}^{5}(L_{m})=(\nu_{1}\delta_{q^{2m},1}+\gamma)L_{m+d}+(\nu_{3}\delta_{q^{2m},1}+\frac{\gamma\theta-\gamma^{2}}{\beta})W_{m+d},\\\
P_{d}^{5}(W_{m})=(\nu_{2}\delta_{q^{2m},1}+\beta)L_{m+d}+(\nu_{4}\delta_{q^{2m},1}+\theta-\gamma)W_{m+d},\end{array}\right.$

where $\gamma,\theta,\nu_{1},\nu_{2},\nu_{3},\nu_{4}\in\mathbb{K}$ and
$\beta\in\mathbb{K}^{\ast}$.

###### Proof.

Directly by ombining Lemma 4.2 and Propositions4.3-4.5. ∎

######  Theorem 4.7.

The homogeneous averaging operators on the $q$-deformed $W(2,2)$ Hom-algebra
$\mathcal{W}^{q}$ with of degree $d$ such that $q^{d}=1$ and $q\neq-1,1$
obtained in Theorem 4.6 provide the following Hom-Leibniz algebras on the
underlying linear space $\mathcal{W}^{q}:$

  1. (i)

$\\{L_{m},L_{n}\\}^{1}=\nu_{1}\delta_{q^{2m},1}[m-n]L_{m+n}+(\nu_{3}\delta_{q^{2m},1}+\gamma)[m-n]W_{m+n}\\\
\\{L_{m},W_{n}\\}^{1}=\nu_{1}\delta_{q^{2m},1}[m-n]W_{m+n}\\\
\\{W_{m},L_{n}\\}^{1}=\nu_{2}[m-n]\delta_{q^{2m},1}L_{m+n}+\nu_{4}\delta_{q^{2m},1}[m-n]W_{m+n}\\\
\\{W_{m},W_{n}\\}^{1}=\nu_{2}\delta_{q^{2m},1}[m-n]L_{m+n},$

  2. (ii)

$\\{L_{m},L_{n}\\}^{2}=(\nu_{1}\delta_{q^{2m},1}+\beta)[m-n]L_{m+n}+(\nu_{3}\delta_{q^{2m},1}+\gamma)[m-n]W_{m+n}\\\
\\{L_{m},W_{n}\\}^{2}=(\nu_{1}\delta_{q^{2m},1}+\beta)[m-n]W_{m+n}\\\
\\{W_{m},L_{n}\\}^{2}=\nu_{2}[m-n]\delta_{q^{2m},1}L_{m+n}+\nu_{4}\delta_{q^{2m},1}[m-n]W_{m+n}\\\
\\{W_{m},W_{n}\\}^{2}=\nu_{2}\delta_{q^{2m},1}[m-n]W_{m+n},$

  3. (iii)

$\\{L_{m},L_{n}\\}^{3}=(\nu_{1}\delta_{q^{2m},1}+\beta)[m-n]L_{m+n}+\nu_{3}\delta_{q^{2m},1}[m-n]W_{m+n}\\\
\\{L_{m},W_{n}\\}^{3}=(\nu_{1}\delta_{q^{2m},1}+\beta)[m-n]W_{m+n}\\\
\\{W_{m},L_{n}\\}^{3}=\nu_{2}[m-n]\delta_{q^{2m},1}L_{m+n}+(\nu_{4}\delta_{q^{2m},1}+\beta)[m-n]W_{m+n}\\\
\\{W_{m},W_{n}\\}^{3}=\nu_{2}\delta_{q^{2m},1}[m-n]W_{m+n},$

  4. (iv)

$\\{L_{m},L_{n}\\}^{4}=(\nu_{1}\delta_{q^{2m},1}+\gamma)[m-n]L_{m+n}+\nu_{3}\delta_{q^{2m},1}[m-n]W_{m+n}\\\
\\{L_{m},W_{n}\\}^{4}=(\nu_{1}\delta_{q^{2m},1}+\gamma)[m-n]W_{m+n}\\\
\\{W_{m},L_{n}\\}^{4}=\nu_{2}[m-n]\delta_{q^{2m},1}L_{m+n}+(\nu_{4}\delta_{q^{2m},1}+\beta)[m-n]W_{m+n}\\\
\\{W_{m},W_{n}\\}^{4}=\nu_{2}\delta_{q^{2m},1}[m-n]W_{m+n},$

  5. (v)

$\\{L_{m},L_{n}\\}^{5}=(\nu_{1}\delta_{q^{2m},1}+\gamma)[m-n]L_{m+n}+(\nu_{3}\delta_{q^{2m},1}+\frac{\gamma\theta-\gamma^{2}}{\beta}[m-n]W_{m+n}\\\
\\{L_{m},W_{n}\\}^{5}=(\nu_{1}\delta_{q^{2m},1}+\gamma)[m-n]W_{m+n}\\\
\\{W_{m},L_{n}\\}^{5}=(\nu_{2}\delta_{q^{2m},1}+\beta)[m-n]L_{m+n}+(\nu_{4}\delta_{q^{2m},1}+\theta-\gamma)[m-n]W_{m+n}\\\
\\{W_{m},W_{n}\\}^{5}=(\nu_{2}\delta_{q^{2m},1}+\beta)[m-n]W_{m+n}$,

where $\nu_{i},\gamma,\theta\in\mathbb{K}~{}and~{}\beta\in\mathbb{K}^{\ast}$.

###### Proof.

We demonstrate a proof of (i). The others are proved analogously. For any
$m,n\in\mathbb{Z}$,

$\displaystyle\\{L_{m},L_{n}\\}^{1}=[P_{d}^{1}(L_{m}),L_{n}]=[\nu_{1}\delta_{q^{2m},1}L_{m+d}+(\nu_{3}\delta_{q^{2m},1}+\gamma)W_{m+d},L_{n}]$
$\displaystyle\quad=\nu_{1}[m+d-n]\delta_{q^{2m},1}L_{m+n+d}+(\nu_{3}\delta_{q^{2m},1}+\gamma)[m+d-n]W_{m+n+d}$
$\displaystyle\quad=\nu_{1}(m-n)\delta_{q^{2m},1}L_{m+n}+(\nu_{3}\delta_{q^{2m},1}+\gamma)[m-n]W_{m+n},$
$\displaystyle\\{L_{m},W_{n}\\}^{1}=[P_{d}^{1}(L_{m}),W_{n}]=[\nu_{1}\delta_{q^{2m},1}L_{m+d}+(\nu_{3}\delta_{q^{2m},1}+\gamma)W_{m+d},W_{n}]$
$\displaystyle\quad=\nu_{1}(m+d-n)\delta_{q^{2m},1}L_{m+n+d},=\nu_{1}[m-n]\delta_{q^{2m},1}L_{m+n},$
$\displaystyle\\{W_{m},L_{n}\\}^{1}=[P_{d}^{1}(W_{m}),L_{n}]=[\nu_{1}\delta_{q^{2m},1}L_{m+d}+\nu_{4}\delta_{q^{2m},1}W_{m+d},L_{n}]$
$\displaystyle\quad=\nu_{2}[m+d-n]\delta_{q^{2m},1}L_{m+n+d}+\nu_{4}\delta_{q^{2m},1}[m+d-n]W_{m+n+d}$
$\displaystyle\quad=\nu_{2}(m-n)\delta_{q^{2m},1}L_{m+n}+\nu_{4}\delta_{q^{2m},1}[m-n]W_{m+n},$
$\displaystyle\\{W_{m},W_{n}\\}^{1}=[P_{d}^{1}(W_{m}),W_{n}]$
$\displaystyle\quad=[\nu_{2}\delta_{q^{2m},1}L_{m+d}+\nu_{4}\delta_{q^{2m},1}W_{m+d},W_{n}]=\nu_{2}(m+d-n)\delta_{q^{2m},1}W_{m+n+d}$
$\displaystyle\hskip 204.85974pt=\nu_{2}(m-n)\delta_{q^{2m},1}W_{m+n}.\qed$

######  Proposition 4.8.

The Hom-Leibniz algebras $(\mathcal{W}^{q},~{}\\{\cdot,\cdot\\}^{i},\alpha)$
for $i\in\\{1,\cdots,5\\}$ given in Theorem 4.7 items (i)-(v) are respectively
multiplicatives if and only if

  1. (i)

$q^{2}=-1$ or $~{}\nu_{1}=\nu_{2}=\nu_{3}=\nu_{4}=\gamma=0;$

  2. (ii)

$q^{2}=-1$;

  3. (iii)

$q^{2}=-1$;

  4. (iv)

$q^{2}=-1$;

  5. (v)

$q^{2}=-1$.

###### Proof.

We prove (i), the others are proved analogously. For any $m,n\in\mathbb{Z}$,
we have

$\displaystyle\alpha(\\{L_{m},L_{n}\\}^{1})-\\{\alpha(L_{m}),\alpha(L_{n})\\}^{1}$
$\displaystyle\quad=\alpha\Big{(}\nu_{1}\delta_{q^{2m},1}[m-n]L_{m+n}+(\nu_{3}\delta_{q^{2m},1}+\gamma)[m-n]W_{m+n}\Big{)}$
$\displaystyle\quad\quad-\\{(q^{m}+q^{-m})L_{m},(q^{n}+q^{-n})L_{n}\\}^{1}$
$\displaystyle\quad=\nu_{1}\delta_{q^{2m},1}[m-n](q^{m+n}+q^{-m-n})L_{m+n}$
$\displaystyle\quad\quad+(\nu_{3}\delta_{q^{2m},1}+\gamma)[m-n](q^{m+n}+q^{-m-n})W_{m+n}$
$\displaystyle\quad\quad-(q^{m}+q^{-m})(q^{n}+q^{-n})\Big{(}(\nu_{1}\delta_{q^{2m},1}+\beta)[m-n]L_{m+n}$
$\displaystyle\quad\quad+(\nu_{3}\delta_{q^{2m},1}+\gamma)[m-n]W_{m+n}\Big{)}$
$\displaystyle\quad=\nu_{1}\delta_{q^{2m},1}[m-n](q^{m-n}+q^{n-m})L_{m+n}+(\nu_{3}\delta_{q^{2m},1}+\gamma)[m-n](q^{m-n}+q^{n-m})W_{m+n}$
$\displaystyle\quad=-(q^{m-n}-q^{n-m})(q^{m-n}+q^{n-m})\big{(}\frac{\nu_{1}\delta_{q^{2m},1}}{q-q^{-1}}L_{m+n}-\frac{\nu_{3}\delta_{q^{2m},1}+\gamma}{q-q^{-1}}W_{m+n}\big{)}$
$\displaystyle\quad=-(q^{2(m-n)}-q^{2(n-m)})\big{(}\frac{\nu_{1}\delta_{q^{2m},1}}{q-q^{-1}}L_{m+n}+\frac{\nu_{3}\delta_{q^{2m},1}+\gamma}{q-q^{-1}}W_{m+n}\big{)},$
$\displaystyle\alpha(\\{L_{m},W_{n}\\}^{1})-\\{\alpha(L_{m}),\alpha(W_{n})\\}^{1}$
$\displaystyle\quad=\alpha\Big{(}(\nu_{1}\delta_{q^{2m},1})[m-n]W_{m+n}-\\{(q^{m}+q^{-m})L_{m},(q^{n}+q^{-n})W_{n}\\}^{2}$
$\displaystyle\quad=\nu_{1}\delta_{q^{2m},1}[m-n](q^{m+n}+q^{-m-n})W_{m+n}-\nu_{1}\delta_{q^{2m},1}[m-n](q^{m}+q^{-m})(q^{n}+q^{-n})W_{m+n}$
$\displaystyle\quad=-(q^{m-n}-q^{n-m})(q^{m-n}+q^{n-m})\big{(}\frac{\nu_{1}\delta_{q^{2m},1}}{q-q^{-1}}W_{m+n}\big{)}$
$\displaystyle\quad=-(q^{2(m-n)}-q^{2(n-m)})\big{(}\frac{\nu_{1}\delta_{q^{2m},1}}{q-q^{-1}}W_{m+n}\big{)},$
$\displaystyle\alpha(\\{W_{m},L_{n}\\}^{1})-\\{\alpha(W_{m}),\alpha(L_{n})\\}^{1}$
$\displaystyle\quad=\alpha\Big{(}\nu_{2}\delta_{q^{2m},1}[m-n]L_{m+n}+\nu_{4}\delta_{q^{2m},1}[m-n]W_{m+n}\Big{)}$
$\displaystyle\quad\quad-\\{(q^{m}+q^{-m})W_{m},(q^{n}+q^{-n})L_{n}\\}^{3}$
$\displaystyle\quad=\nu_{2}\delta_{q^{2m},1}[m-n](q^{m+n}+q^{-m-n})L_{m+n}$
$\displaystyle\quad\quad+\nu_{4}\delta_{q^{2m},1}[m-n](q^{m+n}+q^{-m-n})W_{m+n}$
$\displaystyle\quad\quad-(q^{m}+q^{-m})(q^{n}+q^{-n})\Big{(}(\nu_{2}\delta_{q^{2m},1}+\beta)[m-n]L_{m+n}$
$\displaystyle\quad\quad+\nu_{4}\delta_{q^{2m},1}[m-n]W_{m+n}\Big{)}$
$\displaystyle\quad=\nu_{2}\delta_{q^{2m},1}[m-n](q^{m-n}+q^{n-m})L_{m+n}+\nu_{4}\delta_{q^{2m},1}[m-n](q^{m-n}+q^{n-m})W_{m+n}$
$\displaystyle\quad=-(q^{m-n}-q^{n-m})(q^{m-n}+q^{n-m})\big{(}\frac{\nu_{2}\delta_{q^{2m},1}}{q-q^{-1}}L_{m+n}-\frac{\nu_{4}\delta_{q^{2m},1}}{q-q^{-1}}W_{m+n}\big{)}$
$\displaystyle\quad=-(q^{2(m-n)}-q^{2(n-m)})\big{(}\frac{\nu_{2}\delta_{q^{2m},1}}{q-q^{-1}}L_{m+n}+\frac{\nu_{4}\delta_{q^{2m},1}}{q-q^{-1}}W_{m+n}\big{)},$
$\displaystyle\alpha(\\{W_{m},W_{n}\\}^{1})-\\{\alpha(W_{m}),\alpha(W_{n})\\}^{1}$
$\displaystyle\quad=\alpha\Big{(}(\nu_{2}\delta_{q^{2m},1})[m-n]W_{m+n}-\\{(q^{m}+q^{-m})L_{m},(q^{n}+q^{-n})W_{n}\\}^{4}$
$\displaystyle\quad=\nu_{2}\delta_{q^{2m},1}[m-n](q^{m+n}+q^{-m-n})L_{m+n}-\nu_{2}\delta_{q^{2m},1}[m-n](q^{m}+q^{-m})(q^{n}+q^{-n})L_{m+n}$
$\displaystyle\quad=-(q^{m-n}-q^{n-m})(q^{m-n}+q^{n-m})\big{(}\frac{\nu_{2}\delta_{q^{2m},1}}{q-q^{-1}}L_{m+n}\big{)}$
$\displaystyle\quad=-(q^{2(m-n)}-q^{2(n-m)})\big{(}\frac{\nu_{2}\delta_{q^{2m},1}}{q-q^{-1}}L_{m+n}\big{)}.$

So, by Theorem 2.1 (i),

$\displaystyle\text{$(\mathcal{W}^{q},\\{\cdot,\cdot\\}^{1},\alpha)$ is
multiplicative}\ \Leftrightarrow$ $\displaystyle\forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{llll}q^{2(m-n)}-q^{2(n-m)}=0,or\\\
\nu_{1}\delta_{q^{2m},1}=\nu_{3}\delta_{q^{2m},1}+\gamma=\\\
\nu_{1}\delta_{q^{2m},1}=\nu_{2}\delta_{q^{2m},1}=\nu_{4}\delta_{q^{2m},1}=0,\end{array}\right.,\
\Leftrightarrow$ $\displaystyle\forall\ m,n\in\mathbb{Z}:\
\left\\{\begin{array}[]{llll}q^{4(m-n)}=0,or\\\
\nu_{1}=\nu_{2}=\nu_{3}=\nu_{4}=\gamma=0,\end{array}\right.,\Leftrightarrow$
$\displaystyle\quad\quad\quad\quad\quad\quad\left\\{\begin{array}[]{llll}\forall
p\in\mathbb{Z},~{}q^{4p}=0,or\\\
\nu_{1}=\nu_{2}=\nu_{3}=\nu_{4}=\gamma=0,\end{array}\right.,\Leftrightarrow$
$\displaystyle\quad\quad\quad\quad\quad\quad\left\\{\begin{array}[]{llll}q^{4}=1,or\\\
\nu_{1}=\nu_{2}=\nu_{3}=\nu_{4}=\gamma=0,\end{array}\right.,\Leftrightarrow$
$\displaystyle\quad\quad\quad\quad\quad\quad\left\\{\begin{array}[]{llll}q^{2}=-1,or\\\
\nu_{1}=\nu_{2}=\nu_{3}=\nu_{4}=\gamma=0,\end{array}\right.\Leftrightarrow$
$\displaystyle q=\pm i\ \text{if}\ \exists\ i\in\mathbb{K}:\ i^{2}=-1\
\text{(for example if $\mathbb{K}$ is algebraically closed)},$ $\displaystyle
or~{}~{}\nu_{1}=\nu_{2}=\nu_{3}=\nu_{4}=\gamma=0.\qed$

### Case 2: $(q,d)\in\\{1\\}\times\mathbb{Z}\cup\\{-1\\}\times 2\mathbb{Z}$

######  Remark 4.9.

The equations (4.3)-(4.14) are equivalents for
$(q,d)\in\\{1\\}\times\mathbb{Z}$ and for $(q,d)\in\\{-1\\}\times
2\mathbb{Z}$.

#### Subcase 1: $m=0$ and $d=0$

Take $n=0$ in (4.3)-(4.14). For $d=0$ and $m=0$, the functions
$f_{1},~{}f_{2},~{}g_{1}$ and $g_{2}$ satisfy

$f_{1}(0)=\nu_{1},\ f_{2}(0)=\nu_{2},\ g_{1}(0)=\nu_{3},\ g_{2}(0)=\nu_{4},\
\nu_{i}\in\mathbb{K}.$

Then we have the following Proposition.

######  Proposition 4.10.

If $P_{0}$ is an averaging operator on $\mathcal{W}^{q}$ with degree $d=0$,
then

$\left\\{\begin{array}[]{ll}f_{1}(0)=\nu_{1},&f_{2}(0)=\nu_{2},\\\
g_{1}(0)=\nu_{3},&g_{2}(0)=\nu_{4},\end{array}\right.\quad\text{where
$\nu_{i}\in\mathbb{K}.$}$

#### Subcase 1: $m\neq 0$ and $d=0$

Taking $n=0$ in (4.9) we give $f_{2}(m)(f_{2}(0)-f_{2}(m))=0$. Hence,

$f_{2}(m)=\mu_{1}f_{2}(0),\quad\mu_{1}\in\\{0,1\\}.$

Then, we have the following Proposition.

######  Proposition 4.11.

If $P_{0}$ is an averaging operators on $\mathcal{W}^{q}$ with degree $d=0$
such that $m\neq 0$ and $f_{2}(0)=0,$ then

  1. (i)

if $f_{1}(0)=0$, then $f_{1}(m)=0,\ f_{2}(m)=0,\ g_{1}(m)=\gamma,\
g_{2}(m)=0,$ where $\gamma\in\mathbb{K}$;

  2. (ii)

if $f_{1}(0)\neq 0$, then

    1. (a)

$f_{1}(m)=f_{1}(0),\ f_{2}(m)=0,\ g_{1}(m)=\gamma,\ g_{2}(m)=0$, where
$\gamma\in\mathbb{K};$

    2. (b)

$f_{1}(m)=f_{1}(0),\ f_{2}(m)=0,\ g_{1}(m)=0,\ g_{2}(m)=f_{1}(0)$, where
$\gamma\in\mathbb{K};$

    3. (c)

$f_{1}(m)=0,\ f_{2}(m)=0,\ g_{1}(m)=\gamma,\ g_{2}(m)=f_{1}(0)$, where
$\gamma\in\mathbb{K}.$

###### Proof.

Let $m\neq 0$, $d=0$ and $f_{2}(0)=0$ as assumed. Taking $n=0$ in (4.3) we
obtain $f_{1}(m)(f_{1}(0)-f_{1}(m))=0$. Thus, $f_{1}(m)=\mu_{2}f_{1}(0)$,
$\mu_{2}\in\\{0,1\\}$. Setting $n=0$ in (4.11) yields

$\displaystyle mf_{2}(m)g_{1}(0)=$ $\displaystyle-
mg_{2}(m)f_{1}(0)+mf_{2}(m)g_{1}(m)+mg_{2}^{2}(m).$

Since $m\neq 0$, we have
$f_{2}(m)g_{1}(0)=g_{2}(m)f_{1}(0)-f_{2}(m)g_{1}(m)-g_{2}^{2}(m)$, from which
together with $f_{2}(m)=0$, we get

$g_{2}(m)=\mu_{3}f_{1}(0),\quad\mu_{3}\in\\{0,1\\}.$

Taking $n=0$ in (4.6) and from $m\neq 0$ we get
$f_{1}(m)g_{1}(0)=-f_{1}(0)g_{1}(m)+f_{1}(m)g_{1}(m)+g_{1}(m)g_{2}(m)$, and
with $f_{1}(m)=\mu_{2}f_{1}(0)$ and $g_{2}(m)=\mu_{3}f_{1}(0)$, we obtain
$g_{1}(m)(\mu_{2}+\mu_{3}-1)f_{1}(0)=\mu_{2}f_{1}(0)g_{1}(0)$. Then, we have
the two cases:

  1. (i)

if $f_{1}(0)=0$, then $g_{1}(m)=\gamma$,

  2. (ii)

if $f_{1}(0)\neq 0$ we have $\mu_{2}g_{1}(0)=(\mu_{2}+\mu_{3}-1)g_{1}(m)$.
Then,
$\left\\{\begin{array}[]{ll}g_{1}(m)=0&\ \text{if}\
(\mu_{2},\mu_{3})\in\\{(0,0),(1,1)\\},\\\ g_{1}(m)=\gamma&\ \text{if}\
(\mu_{2},\mu_{3})\in\\{(1,0),(0,1)\\}.\end{array}\right.$ ∎

######  Proposition 4.12.

If $P_{d}$ is the averaging operator on $\mathcal{W}^{q}$ with degree $d=0$
such that $m\neq 0$ and $f_{2}(0)\neq 0$, then
$\left\\{\begin{array}[]{ll}f_{1}(m)=\gamma,&f_{2}(m)=f_{2}(0),\\\
g_{1}(m)=\frac{\gamma
f_{1}(0)-\gamma^{2}}{f_{2}(0)},&g_{2}(m)=f_{1}(0)-\gamma,\end{array}\right.$
where $\gamma\in\mathbb{K}$.

###### Proof.

Let $d=0,~{}~{}m\neq 0$ and $f_{2}(0)\neq 0,$ as assumed. Taking $n=0$ in
(4.7) yields $mf_{1}(m)f_{2}(0)=mf_{1}(m)f_{2}(m)$. Since $m\neq 0$ we have
$f_{1}(m)f_{2}(0)=f_{1}(m)f_{2}(m)$. This together with
$f_{2}(m)=\mu_{1}f_{2}(0)$ gives $f_{1}(m)(\mu_{1}-1)=0$. Then
$f_{1}(m)=\left\\{\begin{array}[]{lllll}0&if&\mu=0,\\\
\gamma&if&\mu_{1}=1.\end{array}\right.$ Taking $n=0$ in (4.10) yields
$mf_{2}(m)g_{2}(0)=-mf_{2}(0)g_{2}(m)+mf_{2}(m)g_{2}(m).$ Since $m\neq 0$, we
have $f_{2}(m)g_{2}(0)=-f_{2}(0)g_{2}(m)+f_{2}(m)g_{2}(m)$. This, with
$f_{2}(m)=\mu_{1}f_{2}(0)$, gives
$\mu_{1}f_{2}(0)g_{2}(0)=-f_{2}(0)g_{2}(m)+\mu_{1}f_{2}(0)g_{2}(m)$. Then

  1. (i)

if $\mu_{1}=0$ we have $g_{2}(m)=0$,

  2. (ii)

if $\mu_{1}=1$ we have $g_{2}(0)=0$.

Taking $n=0$ in (4.3) yields
$f_{1}(m)f_{1}(0)=mf_{1}^{2}(m)+mg_{1}(m)f_{2}(m).$ Since $m\neq 0$, we have
$f_{1}(m)f_{1}(0)=f_{1}^{2}(m)+g_{1}(m)f_{2}(m)$. This, with
$f_{2}(m)=\mu_{1}f_{2}(0)$, $f_{1}(m)=\gamma$ and $\mu_{1}=1$, yields
$g_{1}(m)=\frac{\gamma f_{1}(0)-\gamma^{2}}{f_{2}(0)}$.

Taking $n=0$ in (4.8) yields
$mf_{1}(m)g_{2}(0)=-mf_{2}(0)g_{1}(m)+mf_{1}(m)g_{2}(m).$ Since $m\neq 0$ we
have $f_{1}(m)g_{2}(0)=-f_{2}(0)g_{1}(m)+f_{1}(m)g_{2}(m)$. This together with
$g_{2}(0)=0$ and $f_{1}(m)=g_{2}(m)=0$ for $\mu_{1}=0$ gives $g_{1}(m)=0$.

Taking $n=0$ in the equation (4.14) yields
$mf_{2}(m)f_{1}(0)=mf_{2}(m)f_{1}(m)+mg_{2}(m)f_{2}(m).$ Then
$f_{2}(m)f_{1}(0)=f_{2}(m)f_{1}(m)+g_{2}(m)f_{2}(m)$. This, together with
$f_{1}(m)=\gamma$ for $\mu_{1}=1$, gives $g_{2}(m)=f_{1}(0)-\gamma$. ∎

######  Theorem 4.13.

The Homogeneous averaging operators on the $q$-deformed $W(2,2)$ Hom-algebra
$\mathcal{W}^{q}$ with degree $d=0$. must be one of the following operators,
given for all $m\in\mathbb{Z}$, by

$\displaystyle\left\\{\begin{array}[]{lllll}P_{0}^{1}(L_{m})=\nu_{1}\delta_{m,0}L_{m}+(\nu_{3}\delta_{m,0}+\gamma)W_{m},\\\
P_{0}^{1}(W_{m})=\nu_{2}\delta_{m,0}L_{m}+\nu_{4}\delta_{m,0}W_{m},\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{lllll}P_{0}^{2}(L_{m})=(\nu_{1}\delta_{m,0}+\beta)L_{m}+(\nu_{3}\delta_{m,0}+\gamma)W_{m},\\\
P_{0}^{2}(W_{m})=\nu_{2}\delta_{m,0}L_{m}+\nu_{4}\delta_{m,0}W_{m},\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{lllll}P_{0}^{3}(L_{m})=(\nu_{1}\delta_{m,0}+\beta)L_{m}+\nu_{3}\delta_{m,0}W_{m},\\\
P_{0}^{3}(W_{m})=\nu_{2}\delta_{m,0}L_{m}+(\nu_{4}\delta_{m,0}+\beta)W_{m},\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{l}P_{0}^{4}(L_{m})=(\nu_{1}\delta_{m,0}+\gamma)L_{m}+\nu_{3}\delta_{m,0}W_{m}\\\
P_{0}^{4}(W_{m})=\nu_{2}\delta_{m,0}L_{m}+(\nu_{4}\delta_{m,0}+\beta)W_{m},\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{lllll}P_{0}^{5}(L_{m})=(\nu_{1}\delta_{m,0}+\gamma)L_{m}+(\nu_{3}\delta_{m,0}+\frac{\gamma\theta-\gamma^{2}}{\beta})W_{m},\\\
P_{0}^{5}(W_{m})=(\nu_{2}\delta_{m,0}+\beta)L_{m}+(\nu_{4}\delta_{m,0}+\theta-\gamma)W_{m}.\end{array}\right.$

where $\gamma,\theta,\nu_{1},\nu_{2},\nu_{3},\nu_{4}\in\mathbb{K}$ and
$\beta\in\mathbb{K}^{\ast}$.

###### Proof.

Directly by ombining Lemma 4.2 and Propositions 4.10-4.12. ∎

######  Theorem 4.14.

The homogeneous averaging operators on the $q$-deformed $W(2,2)$ Hom-algebra
$\mathcal{W}^{q}$ with of degree $d=0$ obtained in Theorem 4.6 provide the
following Hom-Leibniz algebras on the underlying linear space
$\mathcal{W}^{q}:$

  1. (i)

$\\{L_{m},L_{n}\\}^{1}=\nu_{1}\delta_{m,0}[m-n]L_{m+n}+(\nu_{3}\delta_{m,0}+\gamma)[m-n]W_{m+n}\\\
\\{L_{m},W_{n}\\}^{1}=\nu_{1}\delta_{m,0}[m-n]W_{m+n}\\\
\\{W_{m},L_{n}\\}^{1}=\nu_{2}[m-n]\delta_{m,0}L_{m+n}+\nu_{4}\delta_{m,0}[m-n]W_{m+n}\\\
\\{W_{m},W_{n}\\}^{1}=\nu_{2}\delta_{m,0}[m-n]L_{m+n},$

  2. (ii)

$\\{L_{m},L_{n}\\}^{2}=(\nu_{1}\delta_{m,0}+\beta)[m-n]L_{m+n}+(\nu_{3}\delta_{m,0}+\gamma)[m-n]W_{m+n}\\\
\\{L_{m},W_{n}\\}^{2}=(\nu_{1}\delta_{m,0}+\beta)[m-n]W_{m+n}\\\
\\{W_{m},L_{n}\\}^{2}=\nu_{2}[m-n]\delta_{m,0}L_{m+n}+\nu_{4}\delta_{m,0}[m-n]W_{m+n}\\\
\\{W_{m},W_{n}\\}^{2}=\nu_{2}\delta_{m,0}[m-n]W_{m+n},$

  3. (iii)

$\\{L_{m},L_{n}\\}^{3}=(\nu_{1}\delta_{m,0}+\beta)[m-n]L_{m+n}+\nu_{3}\delta_{m,0}[m-n]W_{m+n}\\\
\\{L_{m},W_{n}\\}^{3}=(\nu_{1}\delta_{m,0}+\beta)[m-n]W_{m+n}\\\
\\{W_{m},L_{n}\\}^{3}=\nu_{2}[m-n]\delta_{m,0}L_{m+n}+(\nu_{4}\delta_{m,0}+\beta)[m-n]W_{m+n}\\\
\\{W_{m},W_{n}\\}^{3}=\nu_{2}\delta_{m,0}[m-n]W_{m+n},$

  4. (iv)

$\\{L_{m},L_{n}\\}^{4}=(\nu_{1}\delta_{m,0}+\gamma)[m-n]L_{m+n}+\nu_{3}\delta_{m,0}[m-n]W_{m+n}\\\
\\{L_{m},W_{n}\\}^{4}=(\nu_{1}\delta_{m,0}+\gamma)[m-n]W_{m+n}\\\
\\{W_{m},L_{n}\\}^{4}=\nu_{2}[m-n]\delta_{m,0}L_{m+n}+(\nu_{4}\delta_{m,0}+\beta)[m-n]W_{m+n}\\\
\\{W_{m},W_{n}\\}^{4}=\nu_{2}\delta_{m,0}[m-n]W_{m+n},$

  5. (v)

$\\{L_{m},L_{n}\\}^{5}=(\nu_{1}\delta_{m,0}+\gamma)[m-n]L_{m+n}+(\nu_{3}\delta_{m,0}+\frac{\gamma\theta-\gamma^{2}}{\beta}[m-n]W_{m+n}\\\
\\{L_{m},W_{n}\\}^{5}=(\nu_{1}\delta_{m,0}+\gamma)[m-n]W_{m+n}\\\
\\{W_{m},L_{n}\\}^{5}=(\nu_{2}\delta_{m,0}+\beta)[m-n]L_{m+n}+(\nu_{4}\delta_{m,0}+\theta-\gamma)[m-n]W_{m+n}\\\
\\{W_{m},W_{n}\\}^{5}=(\nu_{2}\delta_{m,0}+\beta)[m-n]W_{m+n}$,

where $\nu_{i},\gamma,\theta\in\mathbb{K}~{}and~{}\beta\in\mathbb{K}^{\ast}$.

###### Proof.

We demonstrate a proof of (i). The others are proved analogously. For any
$m,n\in\mathbb{Z}$,

$\displaystyle\\{L_{m},L_{n}\\}^{1}=[P_{0}^{1}(L_{m}),L_{n}]=[\nu_{1}\delta_{m,0}L_{m}+(\nu_{3}\delta_{m,0}+\gamma)W_{m},L_{n}]$
$\displaystyle\quad=\nu_{1}[m-n]\delta_{m,0}L_{m+n}+(\nu_{3}\delta_{m,0}+\gamma)[m-n]W_{m+n},$
$\displaystyle\\{L_{m},W_{n}\\}^{1}=[P_{d}^{1}(L_{m}),W_{n}]=[\nu_{1}\delta_{m,0}L_{m}+(\nu_{3}\delta_{m,0}+\gamma)W_{m},W_{n}]$
$\displaystyle\quad=\nu_{1}[m-n]\delta_{m,0}L_{m+n},$
$\displaystyle\\{W_{m},L_{n}\\}^{1}=[P_{0}^{1}(W_{m}),L_{n}]=[\nu_{1}\delta_{m,0}L_{m}+\nu_{4}\delta_{m,0}W_{m},L_{n}]$
$\displaystyle\quad=\nu_{2}(m-n)\delta_{m,0}L_{m+n}+\nu_{4}\delta_{m,0}[m-n]W_{m+n},$
$\displaystyle\\{W_{m},W_{n}\\}^{1}=[P_{0}^{1}(W_{m}),W_{n}]$
$\displaystyle\quad=[\nu_{2}\delta_{m,0}L_{m}+\nu_{4}\delta_{m,0}W_{m},W_{n}]=\nu_{2}(m-n)\delta_{m,0}W_{m+n}.\qed$

######  Proposition 4.15.

The Hom-Leibniz algebras $(\mathcal{W}^{q},~{}\\{\cdot,\cdot\\}^{i},\alpha)$
induced by $P_{0}^{i}$ for all $i\in\\{1,\dots,5\\}$ is multiplicative if and
only if $i=1$ and $\nu_{1}=\nu_{2}=\nu_{3}=\nu_{4}=\gamma=0.$

###### Proof.

For any $m,n\in\mathbb{Z}$, we have

$\displaystyle\alpha(\\{L_{m},L_{n}\\}^{1})-\\{\alpha(L_{m}),\alpha(L_{n})\\}^{1}$
$\displaystyle\quad=\alpha\Big{(}\nu_{1}\delta_{m,0}[m-n]L_{m+n}+(\nu_{3}\delta_{m,0}+\gamma)[m-n]W_{m+n}\Big{)}$
$\displaystyle\quad\quad-\\{2q^{m}L_{m},2q^{N}L_{n}\\}^{1}$
$\displaystyle\quad=-2q^{m+n}[m-n](\nu_{1}\delta_{m,0}L_{m+n}+(\nu_{3}\delta_{m,0}+\gamma)W_{m+n})$
$\displaystyle\alpha(\\{L_{m},W_{n}\\}^{1})-\\{\alpha(L_{m}),\alpha(W_{n})\\}^{1}$
$\displaystyle\quad=\alpha\Big{(}(\nu_{1}\delta_{m,0})[m-n]W_{m+n}-\\{2q^{m}L_{m},2q^{n}W_{n}\\}^{1}$
$\displaystyle\quad=2q^{m+n}\nu_{1}\delta_{m,0}[m-n]W_{m+n}-4q^{m+n}\nu_{1}\delta_{m,0}[m-n]W_{m+n}$
$\displaystyle\quad=-2q^{m+n}\nu_{1}\delta_{m,0}[m-n]W_{m+n}$
$\displaystyle\alpha(\\{W_{m},L_{n}\\}^{1})-\\{\alpha(W_{m}),\alpha(L_{n})\\}^{1}$
$\displaystyle\quad=2q^{m+n}[m-n](\nu_{2}\delta_{m,0}L_{m+n}+\nu_{4}\delta_{m,0}W_{m+n})-4q^{m+n}[m-n](\nu_{2}\delta_{m,0}L_{m+n}+\nu_{4}\delta_{m,0}W_{m+n})$
$\displaystyle\alpha(\\{W_{m},W_{n}\\}^{1})-\\{\alpha(W_{m}),\alpha(W_{n})\\}^{1}$
$\displaystyle\quad=-2q^{m+n}\nu_{2}\delta_{m,0}[m-n]L_{m+n}.$

So, by Theorem 2.1 (i),

$(\mathcal{W}^{q},\\{\cdot,\cdot\\}^{1},\alpha)$ is multiplicative
$\Leftrightarrow\forall
m\in\mathbb{Z},~{}\nu_{1}\delta_{m,0}=\nu_{3}\delta_{m,0}+\gamma=\nu_{2}\delta_{m,0}=\nu_{4}\delta_{m,0}=0\Leftrightarrow\nu_{1}=\nu_{2}=\nu_{3}=\nu_{4}=\gamma=0.$

Similarly, for all $i\in\\{2,\dots,5\\}$ we prove that the Hom-Leibniz
algebras $(\mathcal{W}^{q},\\{\cdot,\cdot\\}^{i},\alpha)$ are not
multiplicative. ∎

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