# Deformations of phase spaces of Lie algebras via $\mathfrak{s}$-matrices

Jiefeng Liu  School of Mathematics and Statistics, Northeast Normal
University, Changchun 130024, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we first use the graded Lie algebra for relative Rota-Baxter
operators on Lie algebras to give the graded Lie algebra whose Maurer-Cartan
elements characterize $\mathfrak{s}$-matrices on pre-Lie algebras. Then we
give the cohomology of $\mathfrak{s}$-matrices and study their deformations.
As an application, we study the deformations of phase spaces of Lie algebras
via $\mathfrak{s}$-matrices.

###### Key words and phrases:

pre-Lie algebra, $\mathfrak{s}$-matrix, deformation, para-Kähler Lie algebra,
phase space

00footnotetext: MSC: 17A30, 17B38

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Pre-Lie algebras, representations and cohomologies
    2. 2.2 Relative Rota-Baxter operators on Lie algebras and their Maurer-Cartan characterizations
  3. 3 Maurer-Cartan characterizations of $\mathfrak{s}$-matrices on pre-Lie algebras
  4. 4 Cohomology and deformations of $\mathfrak{s}$-matrices on pre-Lie algebras
  5. 5 Deformations of phase spaces of Lie algebras via $\mathfrak{s}$-matrices

##  1\. Introduction

Pre-Lie algebras (or left-symmetric algebras) are a class of nonassociative
algebras coming from the study of convex homogeneous cones, affine manifolds
and affine structures on Lie groups, deformation of associative algebras and
then appeared in many fields in mathematics and mathematical physics, such as
complex and symplectic structures on Lie groups and Lie algebras ([8, 18,
23]), Poisson brackets and infinite dimensional Lie algebras ([9]), vertex
algebras ([11]), $F$-manifold algebras ([20, 28]), homotopy algebra structures
([10, 21]) and operads ([17]). See the survey articles [6, 14] and the
references therein for more details.

A para-Kähler manifold is a symplectic manifold with a pair of transversal
Lagranian foliations. In particular, a para-Kähler Lie algebra $\mathfrak{g}$
is the Lie algebra of a Lie group $G$ with a $G$-invariant para-Kähler
structure ([23]), which can be seen as a symplectic Lie algebra with a
decomposition into a direct sum of the underlying vector spaces of two
Lagrangian subalgebras. The notion of phase space of a Lie algebra was
introduced by Kupershmidt in [25] and was generalized in [3]. Kupershmidt
pointed out the pre-Lie algebras are the underlying structures of phase space
of a Lie algebra and they form a natural category from the point of view of
classical and quantum mechanics ([26]). Furthermore, a phase space of a Lie
algebra is a para-Kähler Lie algebra. In [5], Bai showed that every para-
Kähler Lie algebra is isomorphic to a phase space of a Lie algebra. See [1, 2,
7, 13, 15, 16] for more details about para-Kähler Lie algebras and their
applications.

Motivated by the study of para-Kähler Lie algebras and phase spaces of Lie
algebras in terms of pre-Lie algebras, the notion of a pre-Lie bialgebra (also
called a left-symmetric bialgebra) was introduced in [5]. The notion of
coboundary pre-Lie bialgebra leads to an analogue of the classical Yang-Baxter
equation, called an “S-equation”. A symmetric solution of this equation,
called an “$\mathfrak{s}$-matrix”, gives a pre-Lie bialgebra and phase space
of a Lie algebra naturally. In particular, a Hessian structure on a pre-Lie
algebra, which corresponds to an affine Lie group $G$ with a $G$-invariant
Hessian metric ([29]), gives a non-degenerate symmetric $\mathfrak{s}$-matrix.

The concept of Rota-Baxter operators on associative algebras was introduced by
G. Baxter ([12]) and G.-C. Rota ([30]) in the 1960s. It also plays an
important role in the Connes-Kreimer’s algebraic approach ([19]) to the
renormalization in perturbative quantum field theory. In [27], Kupershmidt
introduced the notion of a relative Rota-Baxter operator (also called an
${\mathcal{O}}$-operator) on a Lie algebra in order to better understand the
relationship between the classical Yang-Baxter equation and the related
integrable systems. In addition, the defining relationship of a relative Rota-
Baxter operator was also called the Schouten curvature in [24]. See the book
[22] for more details on Rota-Baxter operators.

Recently, deformation theory of relative Rota-Baxter operators on Lie algebras
has been developed in [31]. They construct a graded Lie algebra whose Maurer-
Cartan elements are relative Rota-Baxter operators and then give a cohomology
for a relative Rota-Baxter operator. Furthermore, they use the above
cohomology to study deformations of relative Rota-Baxter operators. It is
important that an $\mathfrak{s}$-matrix on a pre-Lie algebra can be seen as a
relative Rota-Baxter operator on its sub-adjacent Lie algebra. The aim of this
paper is to develop cohomology and deformation of $\mathfrak{s}$-matrices on
pre-Lie algebras. In particular, we construct an explicit graded Lie algebra
whose Maurer-Cartan elements are $\mathfrak{s}$-matrices on pre-Lie algebras.
As an application, we study the deformations of phase spaces of Lie algebras
via $\mathfrak{s}$-matrices.

The paper is organized as following. In Section 2, we first recall the
representations and cohomology of pre-Lie algebras. Then we recall the graded
Lie algebra whose Maurer-Cartan elements are relative Rota-Baxter operators on
Lie algebras and the cohomology for a relative Rota-Baxter operator. In
Section 3, we use the graded Lie algebra for relative Rota-Baxter operators on
Lie algebras to construct a graded Lie algebra whose Maurer-Cartan elements
characterize $\mathfrak{s}$-matrices on pre-Lie algebras. In Section 4, we use
the graded Lie algebra for $\mathfrak{s}$-matrices to establish the cohomology
of $\mathfrak{s}$-matrices and study their deformations. In Section 5, we give
the notion of deformations of phase spaces of Lie algebras and apply the
deformations of $\mathfrak{s}$-matrices into the deformations of phase spaces.

In this paper, all the vector spaces are over algebraically closed field
$\mathbb{K}$ of characteristic $0$ and finite dimensional.

Acknowledgement: This research is supported by the National Key Research and
Development Program of China (2021YFA1002000) and the Fundamental Research
Funds for the Central Universities (2412022QD033).

##  2\. Preliminaries

###  2.1. Pre-Lie algebras, representations and cohomologies

######  Definition 2.1.

A pre-Lie algebra is a pair $(\mathfrak{g},\cdot_{\mathfrak{g}})$, where
$\mathfrak{g}$ is a vector space and
$\cdot_{\mathfrak{g}}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$
is a bilinear multiplication satisfying that for all $x,y,z\in\mathfrak{g}$,
the associator
$(x,y,z)=(x\cdot_{\mathfrak{g}}y)\cdot_{\mathfrak{g}}z-x\cdot_{\mathfrak{g}}(y\cdot_{\mathfrak{g}}z)$
is symmetric in $x,y$, i.e.

$(x,y,z)=(y,x,z),\;\;{\rm or}\;\;{\rm
equivalently,}\;\;(x\cdot_{\mathfrak{g}}y)\cdot_{\mathfrak{g}}z-x\cdot_{\mathfrak{g}}(y\cdot_{\mathfrak{g}}z)=(y\cdot_{\mathfrak{g}}x)\cdot_{\mathfrak{g}}z-y\cdot_{\mathfrak{g}}(x\cdot_{\mathfrak{g}}z).$

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra. The commutator
$[x,y]_{\mathfrak{g}}=x\cdot_{\mathfrak{g}}y-y\cdot_{\mathfrak{g}}x$ defines a
Lie algebra structure on $\mathfrak{g}$, which is called the sub-adjacent Lie
algebra of $(\mathfrak{g},\cdot_{\mathfrak{g}})$ and denoted by
$\mathfrak{g}^{c}$. Furthermore,
$L:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g})$ with $x\rightarrow
L_{x}$, where $L_{x}y=x\cdot_{\mathfrak{g}}y$, for all $x,y\in\mathfrak{g}$,
gives a representation of the Lie algebra $\mathfrak{g}^{c}$ on
$\mathfrak{g}$.

######  Definition 2.2.

([5]) Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and $V$ a
vector space. A representation of $\mathfrak{g}$ on $V$ consists of a pair
$(\rho,\mu)$, where $\rho:\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ is a
representation of the Lie algebra $\mathfrak{g}^{c}$ on $V$ and
$\mu:\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ is a linear map satisfying

(1)
$\displaystyle\rho(x)\mu(y)u-\mu(y)\rho(x)u=\mu(x\cdot_{\mathfrak{g}}y)u-\mu(y)\mu(x)u,\quad\forall~{}x,y\in\mathfrak{g},~{}u\in
V.$

Usually, we denote a representation by $(V;\rho,\mu)$. It is obvious that
$(\mathbb{K};\rho=0,\mu=0)$ is a representation, which we call the trivial
representation. Let $R:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g})$ be
a linear map with $x\longrightarrow R_{x}$, where the linear map
$R_{x}:\mathfrak{g}\longrightarrow\mathfrak{g}$ is defined by
$R_{x}(y)=y\cdot_{\mathfrak{g}}x,$ for all $x,y\in\mathfrak{g}$. Then
$(\mathfrak{g};\rho=L,\mu=R)$ is also a representation, which we call the
regular representation. Define two linear maps
$L^{*},R^{*}:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g}^{*})$ with
$x\longrightarrow L^{*}_{x}$ and $x\longrightarrow R^{*}_{x}$ respectively
(for all $x\in\mathfrak{g}$) by

(2) $\langle L_{x}^{*}(\alpha),y\rangle=-\langle\alpha,x\cdot
y\rangle,\;\;\langle R_{x}^{*}(\alpha),y\rangle=-\langle\alpha,y\cdot
x\rangle,\;\;\forall x,y\in\mathfrak{g},\alpha\in\mathfrak{g}^{*}.$

Then $(\mathfrak{g}^{*};\rho={\rm ad}^{*}=L^{*}-R^{*},\mu=-R^{*})$ is a
representation of $(\mathfrak{g},\cdot_{\mathfrak{g}})$. In fact, it is the
dual representation of the regular representation $(\mathfrak{g};L,R)$.

The cohomology complex for a pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ with a representation $(V;\rho,\mu)$ is
given as follows ([14]). The set of $n$-cochains is given by
$\mathrm{Hom}(\wedge^{n-1}\mathfrak{g}\otimes\mathfrak{g},V),\ n\geq 1.$ For
all $\phi\in\mathrm{Hom}(\wedge^{n-1}\mathfrak{g}\otimes\mathfrak{g},V)$, the
coboundary operator
$\mathrm{d}:\mathrm{Hom}(\wedge^{n-1}\mathfrak{g}\otimes\mathfrak{g},V)\longrightarrow\mathrm{Hom}(\wedge^{n}\mathfrak{g}\otimes\mathfrak{g},V)$
is given by

(3) $\displaystyle\mathrm{d}\phi(x_{1},\cdots,x_{n+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}\rho(x_{i})\phi(x_{1},\cdots,\hat{x_{i}},\cdots,x_{n+1})$
$\displaystyle+\sum_{i=1}^{n}(-1)^{i+1}\mu(x_{n+1})\phi(x_{1},\cdots,\hat{x_{i}},\cdots,x_{n},x_{i})$
$\displaystyle-\sum_{i=1}^{n}(-1)^{i+1}\phi(x_{1},\cdots,\hat{x_{i}},\cdots,x_{n},x_{i}\cdot_{\mathfrak{g}}x_{n+1})$
$\displaystyle+\sum_{1\leq i<j\leq
n}(-1)^{i+j}\phi([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\cdots,\hat{x_{i}},\cdots,\hat{x_{j}},\cdots,x_{n+1}),$

for all $x_{i}\in\mathfrak{g},~{}i=1,\cdots,n+1$.

###  2.2. Relative Rota-Baxter operators on Lie algebras and their Maurer-
Cartan characterizations

######  Definition 2.3.

([27]) Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ be a Lie algebra and
$\rho:\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ a representation of
$\mathfrak{g}$ on a vector space $V$. A relative Rota-Baxter operator (or
${\mathcal{O}}$-operator) on $\mathfrak{g}$ with respect to the representation
$(V;\rho)$ is a linear map $T:V\longrightarrow\mathfrak{g}$ such that

(4) $[Tu,Tv]_{\mathfrak{g}}=T\big{(}\rho(Tu)(v)-\rho(Tv)(u)\big{)},\quad
u,v\in V.$

######  Definition 2.4.

Let $T_{1}$ and $T_{2}$ be two relative Rota-Baxter operators on a Lie algebra
$\mathfrak{g}$ with respect to the representation $(V;\rho)$. A homomorphism
from $T_{2}$ to $T_{1}$ consists of a Lie algebra homomorphism
$\phi:\mathfrak{g}\rightarrow\mathfrak{g}$ and a linear map $\psi:V\rightarrow
V$ satisfying

(5) $\displaystyle T_{1}\circ\psi$ $\displaystyle=$ $\displaystyle\phi\circ
T_{2},$ (6) $\displaystyle\psi(\rho(x)v)$ $\displaystyle=$
$\displaystyle\rho(\phi(x))\psi(v),\quad x\in\mathfrak{g},v\in V.$

Furthermore, if $\phi$ and $\psi$ are linear isomorphism, $(\phi,\psi)$ is
called an isomorphism from $T_{2}$ to $T_{1}$.

######  Theorem 2.5.

([4]) Let $T:V\to\mathfrak{g}$ be a relative Rota-Baxter operator on a Lie
algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ with respect to a
representation $(V;\rho)$. Define a multiplication $\cdot^{T}$ on $V$ by

(7) $u\cdot^{T}v=\rho(Tu)(v),\quad\forall u,v\in V.$

Then $(V,\cdot^{T})$ is a pre-Lie algebra.

We denote by $(V,[\cdot,\cdot]^{T})$ the sub-adjacent Lie algebra of the pre-
Lie algebra $(V,\cdot^{T})$. More precisely,

(8) $~{}[u,v]^{T}=\rho(Tu)(v)-\rho(Tv)(u).$

Moreover, $T$ is a Lie algebra homomorphism from $(V,[\cdot,\cdot]^{T})$ to
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$.

######  Definition 2.6.

Let
$(\mathfrak{g}=\oplus_{k=0}^{\infty}\mathfrak{g}_{i},[\cdot,\cdot],\mathrm{d})$
be a differential graded Lie algebra. A degree $1$ element
$\theta\in\mathfrak{g}_{1}$ is called a Maurer-Cartan element of
$\mathfrak{g}$ if it satisfies the following Maurer-Cartan equation:

(9) $\mathrm{d}\theta+\frac{1}{2}[\theta,\theta]=0.$

A graded Lie algebra is a differential graded Lie algebra with $d=0$.

Let $(V;\rho)$ be a representation of a Lie algebra $\mathfrak{g}$. Consider
the graded vector space

${\mathcal{C}}^{*}(V,\mathfrak{g}):=\oplus_{k=0}^{\dim(V)}\mathrm{Hom}(\wedge^{k}V,\mathfrak{g}).$

Define a skew-symmetric bracket operation

$\left\llbracket\cdot,\cdot\right\rrbracket:\mathrm{Hom}(\wedge^{n}V,\mathfrak{g})\times\mathrm{Hom}(\wedge^{m}V,\mathfrak{g})\longrightarrow\mathrm{Hom}(\wedge^{m+n}V,\mathfrak{g})$

by

$\displaystyle\left\llbracket P,Q\right\rrbracket(u_{1},u_{2},\cdots,u_{m+n})$
$\displaystyle=$
$\displaystyle\sum_{\sigma\in\mathbb{S}_{(m,1,n-1)}}(-1)^{\sigma}P(\rho(Q(u_{\sigma(1)},\cdots,u_{\sigma(m)}))u_{\sigma(m+1)},u_{\sigma(m+2)},\cdots,u_{\sigma(m+n)})$
$\displaystyle-(-1)^{mn}\sum_{\sigma\in\mathbb{S}_{(n,1,m-1)}}(-1)^{\sigma}Q(\rho(P(u_{\sigma(1)},\cdots,u_{\sigma(n)}))u_{\sigma(n+1)},u_{\sigma(n+2)},\cdots,u_{\sigma(m+n)})$
$\displaystyle+(-1)^{mn}\sum_{\sigma\in\mathbb{S}_{(n,m)}}(-1)^{\sigma}[P(u_{\sigma(1)},\cdots,u_{\sigma(n)}),Q(u_{\sigma(n+1)},\cdots,u_{\sigma(m+n)})]$

for all $P\in\mathrm{Hom}(\wedge^{n}V,\mathfrak{g})$ and
$Q\in\mathrm{Hom}(\wedge^{m}V,\mathfrak{g})$.

Furthermore, one has

######  Theorem 2.7.

([31])
$({\mathcal{C}}^{*}(V,\mathfrak{g}),\left\llbracket\cdot,\cdot\right\rrbracket)$
is a graded Lie algebra. Its Maurer-Cartan elements are precisely the relative
Rota-Baxter operators on $\mathfrak{g}$ with respect to the representation
$(V;\rho)$.

In the following, we recall cohomology for relative Rota-Baxter operators,
which corresponds to the Chevalley-Eilenberg cohomology.

######  Lemma 2.8.

([31]) Let $T$ be a relative Rota-Baxter operator on a Lie algebra
$\mathfrak{g}$ with respect to a representation $(V;\rho)$. Define

(11)
$\varrho:V\longrightarrow\mathfrak{gl}(\mathfrak{g}),\quad\varrho(u)(x):=[Tu,x]_{\mathfrak{g}}+T\rho(x)(u),\;\;\forall
x\in\mathfrak{g},u\in V.$

Then $\varrho$ is a representation of the sub-adjacent Lie algebra
$(V,[\cdot,\cdot]^{T})$ on the vector space $\mathfrak{g}$.

Let $\delta_{\rm
RB}:\mathrm{Hom}(\wedge^{k}V,\mathfrak{g})\longrightarrow\mathrm{Hom}(\wedge^{k+1}V,\mathfrak{g})$
be the corresponding Chevalley-Eilenberg coboundary operator. More precisely,
for all $f\in\mathrm{Hom}(\wedge^{k}V,\mathfrak{g})$ and
$u_{1},\cdots,u_{k+1}\in V$, we have

(12) $\displaystyle\delta_{\rm RB}f(u_{1},\cdots,u_{k+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k+1}(-1)^{i+1}[Tu_{i},f(u_{1},\cdots,\hat{u_{i}},\cdots,u_{k+1})]_{\mathfrak{g}}+\sum_{i=1}^{k+1}(-1)^{i+1}T\rho(f(u_{1},\cdots,\hat{u_{i}},\cdots,u_{k+1}))(u_{i})$
$\displaystyle+\sum_{1\leq i<j\leq
k+1}(-1)^{i+j}f(\rho(Tu_{i})(u_{j})-\rho(Tu_{j})(u_{i}),u_{1},\cdots,\hat{u_{i}},\cdots,\hat{u_{j}},\cdots,u_{k+1}).$

The coboundary operators $\delta_{\rm RB}$ can also be given in terms of the
graded Lie algebra
$({\mathcal{C}}^{*}(V,\mathfrak{g}),\left\llbracket\cdot,\cdot\right\rrbracket)$
by the following relation:

######  Proposition 2.9.

([31]) Let $T:V\longrightarrow\mathfrak{g}$ be a relative Rota-Baxter operator
on a Lie algebra $\mathfrak{g}$ with respect to a representation $(V;\rho)$.
Then we have

$\delta_{\rm RB}f=(-1)^{k}\left\llbracket T,f\right\rrbracket,\quad
f\in\mathrm{Hom}(\wedge^{k}V,\mathfrak{g}).$

##  3\. Maurer-Cartan characterizations of $\mathfrak{s}$-matrices on pre-Lie
algebras

In this section, we use the graded Lie algebra for relative Rota-Baxter
operators to give the graded Lie algebra whose Maurer-Cartan elements
characterize $\mathfrak{s}$-matrices on pre-Lie algebras.

Let $\mathfrak{g}$ be a vector space. For any
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$, the linear map
$r^{\sharp}:\mathfrak{g}^{*}\longrightarrow\mathfrak{g}$ is given by $\langle
r^{\sharp}(\alpha),\beta\rangle=r(\alpha,\beta),$ for all
$~{}\alpha,\beta\in\mathfrak{g}^{*}$. We say that
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$ is invertible, if the linear map
$r^{\sharp}$ is an isomorphism.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and
$r\in\mathfrak{g}\otimes\mathfrak{g}$. We introduce
$[r,r]\in\wedge^{2}\mathfrak{g}\otimes\mathfrak{g}$ as follows

(13)
$[r,r]=-r_{12}\cdot_{\mathfrak{g}}r_{13}+r_{12}\cdot_{\mathfrak{g}}r_{23}+[r_{13},r_{23}]_{\mathfrak{g}},$

where for $r=\sum_{i}x_{i}\otimes y_{i}$, set

$r_{12}=\sum_{i}x_{i}\otimes y_{i}\otimes 1,\quad r_{13}=\sum_{i}x_{i}\otimes
1\otimes y_{i},\quad r_{23}=\sum_{i}1\otimes x_{i}\otimes y_{i}$

and

$\displaystyle r_{12}\cdot_{\mathfrak{g}}r_{13}$ $\displaystyle=$
$\displaystyle\sum_{i,j}a_{i}\cdot_{\mathfrak{g}}a_{j}\otimes b_{i}\otimes
b_{j},\quad r_{12}\cdot_{\mathfrak{g}}r_{23}=\sum_{i,j}a_{i}\otimes
b_{i}\cdot_{\mathfrak{g}}a_{j}\otimes b_{j},$
$\displaystyle{[r_{13},r_{23}]_{\mathfrak{g}}}$ $\displaystyle=$
$\displaystyle\sum_{i,j}a_{i}\otimes
a_{j}\otimes[b_{i},b_{j}]_{\mathfrak{g}}.$

######  Definition 3.1.

([5]) Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra. If
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$ and satisfies $[r,r]=0$, then $r$ is
called an $\mathfrak{s}$-matrix in $(\mathfrak{g},\cdot_{\mathfrak{g}})$.

######  Remark 3.2.

An $\mathfrak{s}$-matrix in a pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ is a solution of the $S$-equation in
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ which is an analogue of the classical
Yang-Baxter equation in a Lie algebra. It plays an important role in the
theory of pre-Lie bialgebras, see [5] for more details.

######  Definition 3.3.

A pseudo-Hessian structure on a pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ is a symmetric nondegenerate $2$-cocycle
$\mathfrak{B}\in\mathrm{Sym}^{2}(\mathfrak{g}^{*})$, i.e.
$\mathrm{d}^{T}\mathfrak{B}=0$. More precisely,

$\mathfrak{B}(x\cdot_{\mathfrak{g}}y,z)-\mathfrak{B}(x,y\cdot_{\mathfrak{g}}z)=\mathfrak{B}(y\cdot_{\mathfrak{g}}x,z)-\mathfrak{B}(y,x\cdot_{\mathfrak{g}}z),\quad\forall~{}x,y,z\in\mathfrak{g}.$

A pre-Lie algebra equipped with a pseudo-Hessian structure is called a pseudo-
Hessian pre-Lie algebra, and denoted by
$(\mathfrak{g},\cdot_{\mathfrak{g}},\mathfrak{B})$.

######  Proposition 3.4.

([5]) Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and $r$
an invertible $\mathfrak{s}$-matrix. Define
$\mathfrak{B}\in\mathrm{Sym}^{2}(\mathfrak{g}^{*})$ by

(14) $\mathfrak{B}(x,y)=\langle
x,(r^{\sharp})^{-1}(y)\rangle,\quad\forall~{}x,y\in\mathfrak{g}.$

Then $(\mathfrak{g},\cdot_{\mathfrak{g}},\mathfrak{B})$ is a pseudo-Hessian
pre-Lie algebra.

######  Theorem 3.5.

([5]) Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and $r$
an $\mathfrak{s}$-matrix. Then $(\mathfrak{g}^{*},\cdot_{r})$ is a pre-Lie
algebra, where the multiplication
$\cdot_{r}:\otimes^{2}\mathfrak{g}^{*}\longrightarrow\mathfrak{g}^{*}$ is
given by

(15)
$\alpha\cdot_{r}\beta=\mathrm{ad}^{*}_{r^{\sharp}(\alpha)}\beta-R^{*}_{r^{\sharp}(\beta)}\alpha,\quad\alpha,\beta\in\mathfrak{g}^{*}.$

Furthermore, $r^{\sharp}$ is a pre-Lie algebra homomorphism from
$(\mathfrak{g}^{*},\cdot_{r})$ to $(\mathfrak{g},\cdot_{\mathfrak{g}})$.

The sub-adjacent Lie algebra of the pre-Lie algebra
$(\mathfrak{g}^{*},\cdot_{r})$ is $(\mathfrak{g}^{*},[\cdot,\cdot]_{r})$,
where $[\cdot,\cdot]_{r}$ is given by

(16)
$[\alpha,\beta]_{r}=L^{*}_{r^{\sharp}(\alpha)}\beta-L^{*}_{r^{\sharp}(\beta)}\alpha,\quad\alpha,\beta\in\mathfrak{g}^{*}.$

Furthermore, by a direct calculation, we have

######  Proposition 3.6.

For all $\alpha,\beta\in\mathfrak{g}^{*}$, we have

(17) $\displaystyle
r^{\sharp}({[\alpha,\beta]}_{r})-[r^{\sharp}(\alpha),r^{\sharp}(\beta)]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle[r,r](\alpha,\beta,\cdot).$

Note that $L^{*}:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g}^{*})$ is a
representation of the sub-adjacent Lie algebra $\mathfrak{g}^{c}$ on the dual
space $\mathfrak{g}^{*}$. Thus, by Proposition 3.6, for
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$, we have

######  Proposition 3.7.

$r$ is an $\mathfrak{s}$-matrix on a pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ if and only if
$r^{\sharp}:\mathfrak{g}^{*}\rightarrow\mathfrak{g}$ is a relative Rota-Baxter
operator on the Lie algebra $\mathfrak{g}^{c}$ with respect to the
representation $(\mathfrak{g}^{*};L^{*})$.

By Theorem 2.7 and Proposition 3.7, we have

######  Proposition 3.8.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$.

  * (i)

$({\mathcal{C}}^{*}(\mathfrak{g}^{*},\mathfrak{g}):=\oplus_{k\geq
0}\Gamma(\mathrm{Hom}(\wedge^{k}\mathfrak{g}^{*},\mathfrak{g})),\left\llbracket\cdot,\cdot\right\rrbracket)$
is a graded Lie algebra, where the bracket
$\left\llbracket\cdot,\cdot\right\rrbracket$ is given by (2.2) with
$\rho=L^{*}$.

  * (ii)

$r$ is an $\mathfrak{s}$-matrix on the pre-Lie algebra $\mathfrak{g}$ if and
only if $r^{\sharp}$ is a Maurer-Cartan element of the graded Lie algebra
$({\mathcal{C}}^{*}(\mathfrak{g}^{*},\mathfrak{g}),\left\llbracket\cdot,\cdot\right\rrbracket)$.

For $k\geq 0$, define
$\Psi:\wedge^{k}\mathfrak{g}\otimes\mathfrak{g}\longrightarrow{\mathcal{C}}^{k}(\mathfrak{g}^{*},\mathfrak{g})$
by

(18)
$\langle\Psi(\varphi)(\alpha_{1},\cdots,\alpha_{k}),\alpha_{k+1}\rangle=\langle\varphi,\alpha_{1}\wedge\cdots\wedge\alpha_{k}\otimes\alpha_{k+1}\rangle,\quad\alpha_{1},\cdots,\alpha_{k+1}\in\mathfrak{g}^{*},$

and
$\Upsilon:{\mathcal{C}}^{k}(\mathfrak{g}^{*},\mathfrak{g})\longrightarrow\wedge^{k}\mathfrak{g}\otimes\mathfrak{g}$
by

$\langle\Upsilon(P),\alpha_{1}\wedge\cdots\wedge\alpha_{k}\otimes\alpha_{k+1}\rangle=\langle
P(\alpha_{1},\cdots,\alpha_{k}),\alpha_{k+1}\rangle,\quad\alpha_{1},\cdots,\alpha_{k+1}\in\mathfrak{g}^{*}.$

Obviously we have
$\Psi\circ\Upsilon={\rm{Id}},~{}~{}\Upsilon\circ\Psi={\rm{Id}}.$

By Lemma 3.8 , we have

######  Theorem 3.9.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra. Then, there is
a graded Lie bracket
$\llbracket\cdot,\cdot\rrbracket_{\mathfrak{s}}:(\wedge^{k}\mathfrak{g}\otimes\mathfrak{g})\times(\wedge^{l}\mathfrak{g}\otimes\mathfrak{g})\longrightarrow\wedge^{k+l}\mathfrak{g}\otimes\mathfrak{g}$
on the graded vector space $C_{\mathfrak{s}}^{*}(\mathfrak{g}):=\oplus_{k\geq
1}C_{\rm\mathfrak{s}}^{k}(\mathfrak{g})$ with
$C_{\mathfrak{s}}^{k}(\mathfrak{g}):=\wedge^{k-1}\mathfrak{g}\otimes\mathfrak{g}$
given by

$\llbracket\varphi,\phi\rrbracket_{\mathfrak{s}}:=\Upsilon\llbracket\Psi(\varphi),\Psi(\phi)\rrbracket,\quad\varphi\in\wedge^{k}\mathfrak{g}\otimes\mathfrak{g},\phi\in\wedge^{l}\mathfrak{g}\otimes\mathfrak{g}.$

Furthermore, $r\in\mathrm{Sym}^{2}(\mathfrak{g})$ is an $\mathfrak{s}$-matrix
on the pre-Lie algebra $\mathfrak{g}$ if and only if $r$ is a Maurer-Cartan
element of the graded Lie algebra
$(C_{\mathfrak{s}}^{*}(\mathfrak{g}),\left\llbracket\cdot,\cdot\right\rrbracket_{\mathfrak{s}})$.
More precisely, we have

(19) $\llbracket
r,r\rrbracket_{\mathfrak{s}}(\alpha_{1},\alpha_{2},\alpha_{3})=2[r,r](\alpha_{1},\alpha_{2},\alpha_{3}),\quad\alpha_{1},\alpha_{2},\alpha_{3}\in\mathfrak{g}^{*}.$

##  4\. Cohomology and deformations of $\mathfrak{s}$-matrices on pre-Lie
algebras

In this section, we first give the cohomology of $\mathfrak{s}$-matrices on
pre-Lie algebras. Then we build the relationships between cohomology of an
$\mathfrak{s}$-matrix and the corresponding relative Rota-Baxter operator. At
last, we use this cohomology to study the one-parameter infinitesimal
deformations of $\mathfrak{s}$-matrices.

Let $r\in\mathrm{Sym}^{2}(\mathfrak{g})$ be an $\mathfrak{s}$-matrix on a pre-
Lie algebra $(\mathfrak{g},\cdot_{\mathfrak{g}})$. Set
$C_{\mathfrak{s}}^{k}(\mathfrak{g})=\wedge^{k-1}\mathfrak{g}\otimes\mathfrak{g}$
and $C_{\mathfrak{s}}^{*}(\mathfrak{g})=\sum_{k\geq
1}C_{\mathfrak{s}}^{k}(\mathfrak{g})$. Define
$\delta_{\mathfrak{s}}:C_{\mathfrak{s}}^{k}(\mathfrak{g})\longrightarrow
C_{\mathfrak{s}}^{k+1}(\mathfrak{g})$ by

(20) $\delta_{\mathfrak{s}}\varphi=(-1)^{k-1}\left\llbracket
r,\varphi\right\rrbracket_{\mathfrak{s}},\quad\varphi\in
C_{\mathfrak{s}}^{k}(\mathfrak{g}).$

By the graded Jacobi identity, we have
$\delta_{\mathfrak{s}}\circ\delta_{\mathfrak{s}}=0$. Thus
$(C_{\mathfrak{s}}^{*}(\mathfrak{g}),\delta_{\mathfrak{s}})$ is a cochain
complex. Denote by $H_{\mathfrak{s}}^{k}(\mathfrak{g})$ the $k$-th cohomology
group.

Furthermore, we have

######  Proposition 4.1.

For $\varphi\in C_{\mathfrak{s}}^{k}(\mathfrak{g})$, we have

(21)
$\displaystyle\delta_{\mathfrak{s}}\varphi(\alpha_{1},\cdots,\alpha_{k+1})$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{k}(-1)^{i+1}\varphi(\alpha_{1},\cdots,\hat{\alpha_{i}},\cdots,\alpha_{k},\alpha_{i}\cdot_{r}\alpha_{k+1})$
$\displaystyle+\sum_{1\leq i<j\leq
k}(-1)^{i+j}\varphi([\alpha_{i},\alpha_{j}]_{r},\alpha_{1},\cdots,\hat{\alpha_{i}},\cdots,\hat{\alpha_{j}},\cdots,\alpha_{k+1}),$

where $\alpha_{1},\cdots,\alpha_{k+1}\in\mathfrak{g}^{*}$, $\cdot_{r}$ is
given by (15) and $[\cdot,\cdot]_{r}$ is given by (16).

It is clear that the coboundary operator $\delta_{\mathfrak{s}}$ is just the
coboundary operator given by (3) associated to the pre-Lie algebra
$(\mathfrak{g}^{*},\cdot_{r})$ with coefficients in trivial representations.

By Proposition 3.7, we have

######  Proposition 4.2.

Let $r$ be an $\mathfrak{s}$-matrix on a pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$. Then $(\mathfrak{g}^{*},\ast_{r})$ is a
pre-Lie algebra, where $\ast_{r}$ is given by

(22)
$\alpha\ast_{r}\beta=L^{*}_{r^{\sharp}(\alpha)}\beta,\quad\alpha,\beta\in\mathfrak{g}^{*}.$

It is obvious that the pre-Lie algebras $(\mathfrak{g}^{*},\ast_{r})$ and
$(\mathfrak{g}^{*},\cdot_{r})$ have the same sub-adjacent Lie algebra
$(\mathfrak{g}^{*},[\cdot,\cdot]_{r})$.

By Lemma 2.8, we have

######  Proposition 4.3.

Let $r$ be an $\mathfrak{s}$-matrix on a pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$. Then

(23)
$\varrho:\mathfrak{g}^{*}\longrightarrow\mathfrak{gl}(\mathfrak{g}),\quad\varrho(\alpha)(x)=[r^{\sharp}(\alpha),x]_{\mathfrak{g}}+r^{\sharp}(L^{*}_{x}\alpha),\quad
x\in\mathfrak{g},\alpha\in\mathfrak{g}^{*}$

is a representation of the sub-adjacent Lie algebra
$(\mathfrak{g}^{*},[\cdot,\cdot]_{r})$ on the vector space $\mathfrak{g}$.

Note that the representation $\varrho$ given by (23) is exactly the dual
representation of the left multiplication operation of the pre-Lie algebra
$(\mathfrak{g}^{*},\ast_{r})$.

Let $r$ be an $\mathfrak{s}$-matrix on a pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$. By (12), for
$P\in{\mathcal{C}}^{k}(\mathfrak{g}^{*},\mathfrak{g})$ and
$\alpha_{1},\cdots,\alpha_{k+1}\in\mathfrak{g}^{*}$, the coboundary operator
$\delta_{\rm
RB}:{\mathcal{C}}^{k}(\mathfrak{g}^{*},\mathfrak{g})\longrightarrow{\mathcal{C}}^{k+1}(\mathfrak{g}^{*},\mathfrak{g})$
of the relative Rota-Baxter operator $r^{\sharp}$ is given by

(24) $\displaystyle\delta_{\rm RB}P(\alpha_{1},\cdots,\alpha_{k+1})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k+1}(-1)^{i+1}[r^{\sharp}(\alpha_{i}),P(\alpha_{1},\alpha_{2},\cdots,\hat{\alpha_{i}},\cdots,\alpha_{k+1})]_{\mathfrak{g}}$
$\displaystyle+\sum_{i=1}^{k+1}(-1)^{i+1}r^{\sharp}\big{(}L^{*}_{P(\alpha_{1},\alpha_{2},\cdots,\hat{\alpha_{i}},\cdots,\alpha_{k+1})}\alpha_{i}\big{)}$
$\displaystyle+\sum_{1\leq
i<j\leq{k+1}}(-1)^{i+j}P([\alpha_{i},\alpha_{j}]_{r},\alpha_{1},\cdots,\hat{\alpha_{i}},\cdots,\hat{\alpha_{j}},\cdots,\alpha_{k+1}).$

Denote by $H^{k}(\mathfrak{g}^{*},\mathfrak{g})$ the $k$-th cohomology group,
called the  $k$-th cohomology group of the relative Rota-Baxter operator
$r^{\sharp}$.

######  Proposition 4.4.

With the above notations, the map $\Psi$ defined by (18) is a cochain
isomorphism between cochain complexes
$(C_{\mathfrak{s}}^{*}(\mathfrak{g}),\delta_{\mathfrak{s}})$ and
$({\mathcal{C}}^{*}(\mathfrak{g}^{*},\mathfrak{g}),\delta_{\rm RB})$, i.e. we
have the following commutative diagram:

$\textstyle{\cdots\longrightarrow
C_{\mathfrak{s}}^{k+1}(\mathfrak{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\quad\delta_{\mathfrak{s}}}$$\textstyle{C_{\mathfrak{s}}^{k+2}(\mathfrak{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{\cdots}$$\textstyle{\cdots\longrightarrow{\mathcal{C}}^{k}(\mathfrak{g}^{*},\mathfrak{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\delta_{\rm
RB}}$$\textstyle{{\mathcal{C}}^{k+1}(\mathfrak{g}^{*},\mathfrak{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.}$

Consequently, $\Psi$ induces an isomorphism map $\Psi_{\ast}$ between the
corresponding cohomology groups.

###### Proof.

It follows by a direct calculation. ∎

Now we introduce a new cochain complex, whose cohomology groups control
deformations of $\mathfrak{s}$-matrices. Let $r$ be an $\mathfrak{s}$-matrix
on a pre-Lie algebra. For all
$\alpha_{1},\alpha_{2},\alpha_{3}\in\mathfrak{g}^{*}$, define

$\displaystyle\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})$
$\displaystyle=$
$\displaystyle\\{x\in{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})\mid(R_{x}\otimes
1+1\otimes R_{x})r=0\\},$
$\displaystyle\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{2}(\mathfrak{g})$
$\displaystyle=$
$\displaystyle\\{\varphi\in{{\mathcal{C}}}_{\mathfrak{s}}^{2}(\mathfrak{g})\mid\varphi(\alpha_{1},\alpha_{2})=\varphi(\alpha_{2},\alpha_{1})\\},$
$\displaystyle\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{3}(\mathfrak{g})$
$\displaystyle=$
$\displaystyle\\{\varphi\in{{\mathcal{C}}}_{\mathfrak{s}}^{3}(\mathfrak{g})\mid\varphi(\alpha_{1},\alpha_{2},\alpha_{3})+c.p.=0\\},$
$\displaystyle\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{k}(\mathfrak{g})$
$\displaystyle=$
$\displaystyle{{\mathcal{C}}}_{\mathfrak{s}}^{k}(\mathfrak{g}),\quad k\geq 4.$

It is straightforward to verify that the cochain complex
$(\tilde{C}_{\mathfrak{s}}^{*}(\mathfrak{g}),\delta_{\mathfrak{s}})$ is a
subcomplex of the cochain complex
$(C_{\mathfrak{s}}^{*}(\mathfrak{g}),\delta_{\mathfrak{s}})$. Denote by
$\tilde{H}_{\mathfrak{s}}^{k}(\mathfrak{g})$ the $k$-th cohomology group,
called the $k$-th cohomology group of the $\mathfrak{s}$-matrix $r$

######  Definition 4.5.

Let $\mathfrak{g}$ be a pre-Lie algebra and
$r_{1},r_{2}\in\mathrm{Sym}^{2}(\mathfrak{g})$ two $\mathfrak{s}$-matrices. A
weak homomorphism from $r_{2}$ to $r_{1}$ is a pair $(\phi,\varphi)$ such that
$\phi:\mathfrak{g}^{c}\rightarrow\mathfrak{g}^{c}$ is a Lie algebra
homomorphism and $\varphi:\mathfrak{g}\rightarrow\mathfrak{g}$ is a linear map
satisfying

(25) $\displaystyle(\varphi\otimes{\rm{Id}_{\mathfrak{g}}})r_{1}$
$\displaystyle=$ $\displaystyle({\rm{Id}_{\mathfrak{g}}}\otimes\phi)r_{2},$
(26) $\displaystyle\varphi(\phi(x)\cdot_{\mathfrak{g}}y)$ $\displaystyle=$
$\displaystyle x\cdot_{\mathfrak{g}}\varphi(y),\quad x,y\in\mathfrak{g}.$

If $\phi$ and $\varphi$ are linear isomorphisms, then $(\phi,\varphi)$ is
called a weak isomorphism from $r_{2}$ to $r_{1}$.

The relation between a weak homomorphism for $\mathfrak{s}$-matrices and the
homomorphism for relative Rota-Baxter operators is given by the following
proposition.

######  Proposition 4.6.

Let $\mathfrak{g}$ be a pre-Lie algebra and
$r_{1},r_{2}\in\mathrm{Sym}^{2}(\mathfrak{g})$ two $\mathfrak{s}$-matrices.
Then $(\phi,\varphi)$ is a weak homomorphism from $r_{2}$ to $r_{1}$ if and
only if $(\phi,\varphi^{*})$ is a homomorphism from the relative Rota-Baxter
operator $r^{\sharp}_{2}$ to $r^{\sharp}_{1}$ with respect to the
representation $(\mathfrak{g}^{*};L^{*})$.

######  Definition 4.7.

Let $\mathfrak{g}$ be a pre-Lie algebra and
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$ an $\mathfrak{s}$-matrix. If $r+t\kappa$
is also an $\mathfrak{s}$-matrix for any $t\in\mathbb{R}$, then we say that
$\kappa$ generates a one-parameter infinitesimal deformation of $r$.

######  Proposition 4.8.

If $\kappa$ generates a one-parameter infinitesimal deformation of $r$, then
$\kappa$ is a $2$-cocycle.

###### Proof.

Since $r+t\kappa$ is an $\mathfrak{s}$-matrix for any $t\in\mathbb{R}$, we
have

$\left\llbracket
r+t\kappa,r+t\kappa\right\rrbracket_{\mathfrak{s}}=\left\llbracket
r,r\right\rrbracket_{\mathfrak{s}}+2t\left\llbracket
r,\kappa\right\rrbracket_{\mathfrak{s}}+t^{2}\left\llbracket\kappa,\kappa\right\rrbracket_{\mathfrak{s}}=0,$

which implies that $\left\llbracket r,\kappa\right\rrbracket_{\mathfrak{s}}=0$
and thus $\delta_{\mathfrak{s}}\kappa=0$. ∎

######  Definition 4.9.

Two one-parameter infinitesimal deformations $r^{1}_{t}=r+t\kappa_{1}$ and
$r^{2}_{t}=r+t\kappa_{2}$ of an $\mathfrak{s}$-matrix $r$ are called
equivalent if there exists
$x\in\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})$, such that
$({\rm{Id}_{\mathfrak{g}}}+t\mathrm{ad}_{x},{\rm{Id}_{\mathfrak{g}}}-tL_{x}):r^{2}_{t}\rightarrow
r^{1}_{t}$ is a weak homomorphism.

######  Proposition 4.10.

If two one-parameter infinitesimal deformations $r^{1}_{t}=r+t\kappa_{1}$ and
$r^{2}_{t}=r+t\kappa_{2}$ of an $\mathfrak{s}$-matrix $r$ are equivalent, then
$\kappa_{1}$ and $\kappa_{2}$ are in the same cohomology class of
$\tilde{H}_{\mathfrak{s}}^{2}(\mathfrak{g})$.

###### Proof.

Since
$({\rm{Id}_{\mathfrak{g}}}+\mathrm{ad}_{x},{\rm{Id}_{\mathfrak{g}}}-tL_{x}):r^{2}_{t}\rightarrow
r^{1}_{t}$ is a weak homomorphism, there exists
$x\in\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})$ such that

${\rm{Id}_{\mathfrak{g}}}\otimes({\rm{Id}_{\mathfrak{g}}}+t\mathrm{ad}_{x})(r^{2}_{t})=({\rm{Id}_{\mathfrak{g}}}-tL_{x})\otimes{\rm{Id}_{\mathfrak{g}}}(r^{1}_{t}),$

which implies that
$\kappa_{2}-\kappa_{1}=-({\rm{Id}_{\mathfrak{g}}}\otimes\mathrm{ad}_{x}+L_{x}\otimes{\rm{Id}_{\mathfrak{g}}})(r)=\delta_{\mathfrak{s}}(x).$
∎

######  Definition 4.11.

A one-parameter infinitesimal deformation $r_{t}=r+t\kappa$ is called trivial
if there exists $x\in\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})$,
such that
$({\rm{Id}_{\mathfrak{g}}}+\mathrm{ad}_{x},{\rm{Id}_{\mathfrak{g}}}-tL_{x}):r^{2}_{t}\rightarrow
r$ is a weak homomorphism.

Let $r_{t}=r+t\kappa$ be a trivial deformation of an $\mathfrak{s}$-matrix
$r$. Then there exists
$x\in\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})$ such that
$({\rm{Id}_{\mathfrak{g}}}+\mathrm{ad}_{x},{\rm{Id}_{\mathfrak{g}}}-tL_{x}):r^{2}_{t}\rightarrow
r$ is a weak homomorphism. First, by the fact that
${\rm{Id}_{\mathfrak{g}}}+\mathrm{ad}_{x}:\mathfrak{g}^{c}\rightarrow\mathfrak{g}^{c}$
is a Lie algebra homomorphism, we have

(27) $[[x,y]_{\mathfrak{g}},[x,z]_{\mathfrak{g}}]_{\mathfrak{g}}=0,\quad
y,z\in\mathfrak{g}.$

By (25), we have

(28)
$({\rm{Id}}\otimes\mathrm{ad}_{x})({\rm{Id}}\otimes\mathrm{ad}_{x}+L_{x}\otimes{\rm{Id}})r=0.$

By (26), we have

(29) $x\cdot_{\mathfrak{g}}([x,y]_{\mathfrak{g}}\cdot_{\mathfrak{g}}z)=0,\quad
y,z\in\mathfrak{g}.$

######  Definition 4.12.

Let $\mathfrak{g}$ be a pre-Lie algebra and
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$ an $\mathfrak{s}$-matrix. An element
$x\in\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})$ is called a
Nijenhuis element if $x$ satisfies (27)-(29).

Thus, a trivial deformation of an $\mathfrak{s}$-matrix gives rise to a
Nijenhuis element. Conversely, we have

######  Proposition 4.13.

Let $\mathfrak{g}$ be a pre-Lie algebra and
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$ an $\mathfrak{s}$-matrix. Then for any
Nijenhuis element
$x\in\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})$,
$r_{t}=r+t\left\llbracket r,x\right\rrbracket_{\mathfrak{s}}$ is a trivial
one-parameter infinitesimal deformation of $r$.

###### Proof.

Since $r\in\mathrm{Sym}^{2}(\mathfrak{g})$ is an $\mathfrak{s}$-matrix and
$x\in\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})$, by graded Jacobi
identity, we have

$\left\llbracket\left\llbracket
r,x\right\rrbracket_{\mathfrak{s}},\left\llbracket
r,x\right\rrbracket_{\mathfrak{s}}\right\rrbracket_{\mathfrak{s}}=0,$

which implies that $\left\llbracket
r,x\right\rrbracket_{\mathfrak{s}}\in\mathrm{Sym}^{2}(\mathfrak{g})$ is also
an $\mathfrak{s}$-matrix. Also, we have $\left\llbracket r,\left\llbracket
r,x\right\rrbracket_{\mathfrak{s}}\right\rrbracket_{\mathfrak{s}}=0$. Thus
$r_{t}=r+t\left\llbracket r,x\right\rrbracket_{\mathfrak{s}}$ is a one-
parameter infinitesimal deformation of $r$.

By (27),
${\rm{Id}_{\mathfrak{g}}}+\mathrm{ad}_{x}:\mathfrak{g}^{c}\rightarrow\mathfrak{g}^{c}$
is a Lie algebra homomorphism. By (28), (25) follows. By (29), (26) follows.
Therefore, the one-parameter infinitesimal deformation
$r_{t}=r+t\left\llbracket r,x\right\rrbracket_{\mathfrak{s}}$ is trivial. ∎

##  5\. Deformations of phase spaces of Lie algebras via
$\mathfrak{s}$-matrices

In this section, we apply the deformations of $\mathfrak{s}$-matrices into the
study of deformations of phase spaces of Lie algebras.

Recall that a Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ is
called a symplectic Lie algebra if there exists a nondegenerate skew-symmetric
$2$-cocycle $\omega$ on $\mathfrak{g}$, that is,

(30)
$\omega([x,y]_{\mathfrak{g}},z)+\omega([y,z]_{\mathfrak{g}},x)+\omega([z,x]_{\mathfrak{g}},y)=0,\quad
x,y,z\in\mathfrak{g}.$

Let $\mathfrak{h}$ be a subalgebra of a symplectic Lie algebra
$(\mathfrak{g},\omega)$ and

$\mathfrak{h}^{\bot}=\\{x\in\mathfrak{g}|\omega(x,y)=0,~{}\forall~{}y\in\mathfrak{g}\\}.$

If $\mathfrak{h}=\mathfrak{h}^{\bot}$, then $\mathfrak{h}$ is called
Lagrangian.

######  Lemma 5.1.

([18]) Let $(\mathfrak{g},\omega)$ be a symplectic Lie algebra. Then there
exists a compatible pre-Lie algebra structure $\cdot_{\mathfrak{g}}$ on
$\mathfrak{g}$ given by

(31) $\omega(x\cdot_{\mathfrak{g}}y,z)=-\omega(y,[x,z]_{\mathfrak{g}}),\quad
x,y,z\in\mathfrak{g}.$

Let $(\mathfrak{g},\omega)$ be a symplectic Lie algebra. Recall that
$\mathfrak{g}$ is called a para-Kähler Lie algebra if $\mathfrak{g}$ is a
direct sum of the underlying vector spaces of two Lagrangian subalgebras
$\mathfrak{g}_{+}$ and $\mathfrak{g}_{-}$.

A linear map $\varphi:\mathfrak{g}\rightarrow\mathfrak{h}$ is called a
homomorphism from para-kähler Lie algebra
$(\mathfrak{g},\mathfrak{g}_{+},\mathfrak{g}_{-},\omega_{1})$ to para-kähler
Lie algebra $(\mathfrak{h},\mathfrak{h}_{+},\mathfrak{h}_{-},\omega_{2})$ if
$\phi$ is a Lie algebra homomorphism such that

(32)
$\phi(\mathfrak{g}_{+})=\mathfrak{h}_{+},\quad\phi(\mathfrak{g}_{-})=\mathfrak{h}_{-},\quad\omega_{1}(x,y)=\omega_{2}(\phi(x),\phi(y)),\quad
x,y\in\mathfrak{g}.$

If $\phi$ is also linear isomorphism, then $\phi$ is call a isomorphism from
$(\mathfrak{g},\mathfrak{g}_{+},\mathfrak{g}_{-},\omega_{1})$ to
$(\mathfrak{h},\mathfrak{h}_{+},\mathfrak{h}_{-},\omega_{2})$.

######  Definition 5.2.

([25]) Let $\mathfrak{g}$ be a Lie algebra. If there exists a Lie algebra
structure on a direct sum of the underlying vector space of $\mathfrak{g}$ and
$\mathfrak{g}^{*}$ such that $\mathfrak{g}$ and $\mathfrak{g}^{*}$ are Lie
subalgebras and the natural skew-symmetric bilinear form on
$\mathfrak{g}\oplus\mathfrak{g}^{*}$

(33) $\omega_{p}(x+\alpha,y+\beta)=\langle\alpha,y\rangle-\langle
x,\beta\rangle,\quad x,y\in\mathfrak{g},\alpha,\beta\in\mathfrak{g}^{*}$

is a $2$-cocycle, then it is called a phase space of the Lie algebra
$\mathfrak{g}$.

The relationship between a phase space of a Lie algebra and a para-kähler Lie
algebra is given by the following proposition.

######  Proposition 5.3.

([5]) Every para-kähler Lie algebra
$(\mathfrak{g},\mathfrak{g}_{+},\mathfrak{g}_{-},\omega)$ is isomorphic to a
phase space of $\mathfrak{g}_{+}$.

Every phase space of a Lie algebra can be equivalently described by a pair of
pre-Lie algebras.

######  Proposition 5.4.

([3]) Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra. Assume
that there exists another pre-Lie algebra structure
$``\ast_{\mathfrak{g}^{*}}"$ on $\mathfrak{g}^{*}$. Then
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p})$ is the phase space of
the Lie algebra $\mathfrak{g}^{c}$ if and only if
$({\mathfrak{g}}\oplus{\mathfrak{g}^{*}},[\cdot,\cdot]_{p},\omega_{p})$ is a
para-kähler Lie algebra, where the bracket $[\cdot,\cdot]_{p}$ is defined by

(34)
$\displaystyle[x+\alpha,y+\beta]_{p}=[\alpha,\beta]_{\mathfrak{g}^{*}}+L^{*}_{x}\beta-L^{*}_{y}\alpha+L^{*}_{\alpha}y-L^{*}_{\beta}x+[x,y]_{\mathfrak{g}},\quad
x,y\in\mathfrak{g},\alpha,\beta\in\mathfrak{g}^{*}.$

On the other hand, every phase space can be constructed from the above way.

In order to study the deformation of phase spaces of Lie algebras via
$\mathfrak{s}$-matrices, the notion of a weak homomorphism between para-kähler
Lie algebras is needed.

######  Definition 5.5.

Let $(\mathfrak{g},\mathfrak{g}^{*},\omega_{p})$ and
$(\mathfrak{h},\mathfrak{h}^{*},\omega_{p})$ be the phase spaces of Lie
algebras $\mathfrak{g}$ and $\mathfrak{h}$, respectively. A weak homomorphism
from $(\mathfrak{g},\mathfrak{g}^{*},\omega_{p})$ to
$(\mathfrak{h},\mathfrak{h}^{*},\omega_{p})$ consists of a Lie algebra
homomorphism $\phi:\mathfrak{g}\rightarrow\mathfrak{g}$ and a linear map
$\varphi:\mathfrak{h}\rightarrow\mathfrak{g}$ such that
$\varphi^{\ast}:\mathfrak{g}^{*}\rightarrow\mathfrak{h}^{*}$ and
$\phi+\varphi^{*}:(\mathfrak{g}\oplus\mathfrak{g}^{*},[\cdot,\cdot]_{p})\rightarrow(\mathfrak{h}\oplus\mathfrak{h}^{*},[\cdot,\cdot]_{p})$
are Lie algebra homomorphisms.

Furthermore, if $\phi$ and $\varphi$ are also linear isomorphisms, then
$(\phi,\varphi)$ is call a weak isomorphism from
$(\mathfrak{g},\mathfrak{g}^{*},\omega_{p})$ to
$(\mathfrak{h},\mathfrak{h}^{*},\omega_{p})$.

Let $\mathfrak{g}$ be a pre-Lie algebra and
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$ an $\mathfrak{s}$-matrix. Then
$(\mathfrak{g}^{*},[\cdot,\cdot]_{r})$ is a Lie algebra, where the bracket
$[\cdot,\cdot]_{r}$ is given by (16). Furthermore, one has

######  Proposition 5.6.

([5]) $({\mathfrak{g}}\oplus{\mathfrak{g}^{*}},[\cdot,\cdot]_{p},\omega_{p})$
is a para-kähler Lie algebra, where the bracket $[\cdot,\cdot]_{p}$ is defined
by

(35)
$\displaystyle[x+\alpha,y+\beta]_{p}=[\alpha,\beta]_{r}+L^{*}_{x}\beta-L^{*}_{y}\alpha+L^{*}_{\alpha}y-L^{*}_{\beta}x+[x,y]_{\mathfrak{g}},\quad
x,y\in\mathfrak{g},\alpha,\beta\in\mathfrak{g}^{*},$

where $L_{\alpha}:\mathfrak{g}\rightarrow\mathfrak{g}$ is given by

(36) $\langle L^{*}_{\alpha}x,\beta\rangle=-\langle
x,\alpha\cdot_{r}\beta\rangle,\quad\beta\in\mathfrak{g}^{*}.$

We call this para-kähler Lie algebra the phase space of the Lie algebra
$\mathfrak{g}^{c}$ associated to the $\mathfrak{s}$-matrix $r$ and denote it
by $(\mathfrak{g}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p},r)$.

######  Proposition 5.7.

Let $\mathfrak{g}$ be a pre-Lie algebra and
$r_{1},r_{2}\in\mathrm{Sym}^{2}(\mathfrak{g})$ two $\mathfrak{s}$-matrices. If
$(\phi,\varphi)$ is a weak homomorphism from $r_{2}$ to $r_{1}$, then
$(\phi,\varphi)$ is a weak homomorphism from the phase space
$(\mathfrak{g}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p},r_{2})$ to
$(\mathfrak{g}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p},r_{1})$.

###### Proof.

Since $\phi$ is a pre-Lie algebra homomorphism from $\mathfrak{g}$ to
$\mathfrak{g}$, $\phi$ is a Lie algebra homomorphism from $\mathfrak{g}^{c}$
to $\mathfrak{g}^{c}$. By (25) and (26), we have

$\displaystyle\langle\varphi^{*}[\alpha,\beta]_{r_{2}},x\rangle$
$\displaystyle=$
$\displaystyle-\langle\beta,\varphi(\phi(r_{2}^{\sharp}(\alpha))\cdot_{\mathfrak{g}}x)\rangle+\langle\alpha,\varphi(\phi(r_{2}^{\sharp}(\beta))\cdot_{\mathfrak{g}}x)\rangle$
$\displaystyle=$
$\displaystyle-\langle\beta,\varphi(r_{1}^{\sharp}\varphi^{*}(\alpha)\cdot_{\mathfrak{g}}x)\rangle+\langle\alpha,\varphi(r_{1}^{\sharp}\phi(\varphi^{*}(\beta))\cdot_{\mathfrak{g}}x)\rangle$
$\displaystyle=$
$\displaystyle-\langle\varphi^{*}\beta,r_{1}^{\sharp}\varphi^{*}(\alpha)\cdot_{\mathfrak{g}}x\rangle+\langle\varphi^{*}\alpha,r_{1}^{\sharp}\phi(\varphi^{*}(\beta))\cdot_{\mathfrak{g}}x\rangle$
$\displaystyle=$ $\displaystyle\langle
L_{r_{1}^{\sharp}\varphi^{*}(\alpha)}\varphi^{*}\beta-
L_{r_{1}^{\sharp}\varphi^{*}(\beta)}\varphi^{*}\alpha,x\rangle$
$\displaystyle=$
$\displaystyle\langle[\varphi^{*}(\alpha),\varphi^{*}(\beta)]_{r_{1}},x\rangle,$

which implies that $\varphi^{*}$ is a Lie algebra homomorphism from
$(\mathfrak{g}^{*},[\cdot,\cdot]_{2})$ to
$(\mathfrak{g}^{*},[\cdot,\cdot]_{1})$. Next, we show that
$(\phi+\varphi^{*})[x,\alpha]_{2}=[\phi(x),\varphi^{*}(\alpha)]_{1},$ which is
equivalent to

(37) $\displaystyle\phi(L_{\alpha}^{*}x)$ $\displaystyle=$ $\displaystyle
L_{\varphi^{*}(\alpha)}\phi(x),$ (38)
$\displaystyle\varphi^{*}(L^{*}_{x}\alpha)$ $\displaystyle=$ $\displaystyle
L^{*}_{\phi(x)}\varphi^{*}(\alpha),\quad
x\in\mathfrak{g},\alpha\in\mathfrak{g}^{*}.$

One can check that (37) follows from $\phi$ is a Lie algebra homomorphism,
(25) and (26), and (38) follows from (26). ∎

######  Definition 5.8.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ and
$(\mathfrak{g}^{*},\ast_{\mathfrak{g}^{*}})$ be pre-Lie algebras. Assume that
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p})$ is the phase space of
the Lie algebra $\mathfrak{g}^{c}$. For
$\pi\in\mathrm{Hom}(\mathfrak{g}^{*}\otimes\mathfrak{g}^{*},\mathfrak{g}^{*})$,
define an operation
$\ast_{t}:\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}\rightarrow\mathfrak{g}^{*}$
by

$\alpha\ast_{t}\beta=\alpha\ast_{\mathfrak{g}^{*}}\beta+t\pi(\alpha,\beta),\quad\alpha,\beta\in\mathfrak{g}^{*}.$

If ${\mathfrak{g}^{*}_{t}}=(\mathfrak{g}^{*},\ast_{t})$ is a pre-Lie algebra
structure on $\mathfrak{g}^{*}$ for any $t$, such that
$({\mathfrak{g}}^{c},{\mathfrak{g}_{t}^{*}}^{c},\omega_{p})$ is the phase
space of the Lie algebra $\mathfrak{g}^{c}$, then we say $\pi$ generates a
one-parameter infinitesimal deformation of the phase space
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}},\omega_{p})$ of the Lie algebra
$\mathfrak{g}^{c}$ and denote it by
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi}},\omega_{p})$.

By Proposition 5.6, we have

######  Proposition 5.9.

Let $\mathfrak{g}$ be a pre-Lie algebra, $r\in\mathrm{Sym}^{2}(\mathfrak{g})$
an $\mathfrak{s}$-matrix and
$(\mathfrak{g}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p},r)$ the phase space
associated to the $\mathfrak{s}$-matrix $r$. If $\kappa$ generates a one-
parameter infinitesimal deformation of $r$, then
$\pi\in\mathrm{Hom}(\mathfrak{g}^{*}\otimes\mathfrak{g}^{*},\mathfrak{g}^{*})$
defined by

$\pi(\alpha,\beta)=\mathrm{ad}^{\ast}_{\kappa(\alpha)}\beta-R^{\ast}_{\kappa(\beta)}\alpha,\quad\alpha,\beta\in\mathfrak{g}^{*}$

generates a one-parameter infinitesimal deformation of the phase space
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p},r)$.

######  Definition 5.10.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ and
$(\mathfrak{g}^{*},\ast_{\mathfrak{g}^{*}})$ be pre-Lie algebras. Assume that
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p})$ is the phase space of
the Lie algebra $\mathfrak{g}^{c}$. Two one-parameter infinitesimal
deformations $({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{1}}},\omega_{p})$
and $({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{2}}},\omega_{p})$ are said
to be equivalent if there exists $x\in\mathfrak{g}$, such that
$({\rm{Id}}+t\mathrm{ad}_{x},{\rm{Id}}-tL_{x})$ is a weak homomorphism from
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{2}}},\omega_{p})$ to
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{1}}},\omega_{p})$.

In particular, a one-parameter infinitesimal deformation
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi}},\omega_{p})$ is said to be
trivial if there exists $x\in\mathfrak{g}$, such that
$({\rm{Id}}+t\mathrm{ad}_{x},{\rm{Id}}-tL_{x})$ is a weak homomorphism from
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{2}}},\omega_{p})$ to
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}},\omega_{p})$.

######  Proposition 5.11.

Let $\mathfrak{g}$ be a pre-Lie algebra, $r\in\mathrm{Sym}^{2}(\mathfrak{g})$
an $\mathfrak{s}$-matrix and
$(\mathfrak{g}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p},r)$ the phase space
associated to the $\mathfrak{s}$-matrix $r$. Assume that
$r_{t}^{1}=r+t\kappa_{1}$ and $r_{t}^{2}=r+t\kappa_{2}$ are one-parameter
infinitesimal deformations of $r$, and
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{1}}},\omega_{p})$ and
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{2}}},\omega_{p})$ are the
corresponding one-parameter infinitesimal deformations, respectively. If
$r+t\kappa_{1}$ and $r+t\kappa_{2}$ are equivalent, then
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{1}}},\omega_{p})$ and
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{2}}},\omega_{p})$ are
equivalent.

###### Proof.

Since
$({\rm{Id}_{\mathfrak{g}}}+\mathrm{ad}_{x},{\rm{Id}_{\mathfrak{g}}}-tL_{x}):r^{2}_{t}\rightarrow
r^{1}_{t}$ is a weak homomorphism with
$x\in\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})$, by Proposition
5.7,
$({\rm{Id}_{\mathfrak{g}}}+\mathrm{ad}_{x},{\rm{Id}_{\mathfrak{g}}}-tL_{x})$
is a weak homomorphism from
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{2}}},\omega_{p})$ to
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{1}}},\omega_{p})$. Thus
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{1}}},\omega_{p})$ and
$({\mathfrak{g}}^{c},{\mathfrak{g}^{*c}_{\pi_{2}}},\omega_{p})$ are
equivalent. ∎

By Proposition 4.13, we have

######  Corollary 5.12.

Let $\mathfrak{g}$ be a pre-Lie algebra, $r\in\mathrm{Sym}^{2}(\mathfrak{g})$
an $\mathfrak{s}$-matrix and
$(\mathfrak{g}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p},r)$ the phase space
associated to the $\mathfrak{s}$-matrix $r$. Then, for any Nijenhuis element
$x\in\tilde{{\mathcal{C}}}_{\mathfrak{s}}^{1}(\mathfrak{g})$, $\pi$ defined by

$\pi(\alpha,\beta)=\mathrm{ad}^{*}_{[r^{\sharp}(\alpha),x]}\beta+\mathrm{ad}^{*}_{r^{\sharp}(L^{*}_{x}\alpha)}\beta-R^{*}_{[r^{\sharp}(\beta),x]}\alpha-R^{*}_{r^{\sharp}(L^{*}_{x}\beta)}\alpha,\quad\alpha,\beta\in\mathfrak{g}^{*}$

generates a trivial one-parameter infinitesimal deformations of
$(\mathfrak{g}^{c},{\mathfrak{g}^{*}}^{c},\omega_{p},r)$.

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