# Nonabelian embedding tensors

Rong Tang  Department of Mathematics, Jilin University, Changchun 130012,
Jilin, China<EMAIL_ADDRESS>and  Yunhe Sheng  Department of Mathematics,
Jilin University, Changchun 130012, Jilin, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, first we introduce the notion of a nonabelian embedding tensor,
which is a nonabelian generalization of an embedding tensor. Then we introduce
the notion of a Leibniz-Lie algebra, which is the underlying algebraic
structure of a nonabelian embedding tensor, and can also be viewed as a
nonabelian generalization of a Leibniz algebra. Next using the derived
bracket, we construct a differential graded Lie algebra, whose Maurer-Cartan
elements are exactly nonabelian embedding tensors. Consequently, we obtain the
differential graded Lie algebra that governs deformations of a nonabelian
embedding tensor. Finally, we define the cohomology of a nonabelian embedding
tensor and use the second cohomology group to characterize linear
deformations.

###### Key words and phrases:

cohomology, deformation, nonabelian embedding tensor, Leibniz-Lie algebra

###### Contents

  1. 1 Introduction
  2. 2 Nonabelian embedding tensors and Leibniz algebras
  3. 3 Nonabelian embedding tensors and Leibniz-Lie algebras
  4. 4 The controlling differential graded Lie algebra of nonabelian embedding tensors
  5. 5 Cohomologies of nonabelian embedding tensors
  6. 6 Linear deformations of nonabelian embedding tensors

##  1\. Introduction

The notion of embedding tensors [23] can be traced back to the study of the
gauged supergravity theory. Embedding tensors were used to construct N =8
supersymmetric gauge theories and to study the Bagger-Lambert theory of
multiple M2-branes in [4]. Embedding tensors were also used to study $E_{11}$
[3] and deformations of maximal 9-dimensional supergravity [7]. Recently, this
topic has attracted much attention of the mathematical physics world [16, 20].
An embedding tensor on a Lie algebra $\mathfrak{g}$ with respect to a
representation $\rho:\mathfrak{g}\to\mathfrak{gl}(V)$ is a linear map
$T:V\,\rightarrow\,\mathfrak{g}$ satisfying

$[Tu,Tv]=T(\rho(Tu)v),\quad\forall u,v\in V.$

An embedding tensors gives rise to a tensor hierarchy algebra [26], while the
existing constructions [19, 21] are still mysterious.

Embedding tensors have been already studied in mathematics under the name of
averaging operators since 1960s. Luxuriant studies on averaging operators were
done for Banach algebras [17, 22, 27] from the analysis point of view.
Recently, there are more and more studies of averaging operators from the
algebraic point of view. Averaging operators on various types of algebras were
studied in [1]. In particular, an averaging operator on a Lie algebra gives
rise to a Leibniz algebra. More generally, averaging operators are studied on
the algebras over the general operad $\mathcal{P}$ and give rise to the
di-$\mathcal{P}$ algebras and tri-$\mathcal{P}$ algebras [12]. Averaging
operators are related to the procedure of replication, Hadamard products and
Manin white products in the operad theory [12, 24, 25, 31], and can be applied
to study double algebras, classical Yang-Baxter equations and Gröbner-Shirshov
bases [10, 11, 13, 33].

Note that the above vector space $V$ in the definition of an embedding tensor
is understood as the space that 1-form fields take values in. Since for the
classical gauge field theories, a 1-form field usually takes values in a Lie
algebra, so it is natural to replace the vector space $V$ by a Lie algebra
$\mathfrak{h}$. This will further need an action of the Lie algebra
$\mathfrak{g}$ on the Lie algebra $\mathfrak{h}$, and lead to the ‘nonabelian
embedding tensor’, which is the main subject that discussed in this paper.
Such generalization for averaging operators on associative algebra were
already considered in [12, 24], which can be understood as weighted averaging
operators.

We introduce a new algebraic structure, which is called a Leibniz-Lie algebra.
A Leibniz-Lie algebra contains a Lie algebra structure and a binary operation
such that some compatibility conditions hold (Definition 3.1). In particular,
if the Lie algebra structure is trivial, then it reduces to a Leibniz algebra.
So a Leibniz-Lie algebra can be viewed as a nonabelian generalization of a
Leibniz algebra. A nonabelian embedding tensor naturally induces a Leibniz-Lie
algebra. The concrete relations can be summarized by the following diagram:

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To justify nonabelian embedding tensors are indeed good mathematical
structures, we further construct a differential graded Lie algebra such that
its Maurer-Cartan elements are exactly nonabelian embedding tensors. As a
byproduct, we obtain the differential graded Lie algebra that governs
deformations of a nonabelian embedding tensor. The deformation of algebraic
structures were originated from the pioneer work [9] of Gerstenhaber. In
general, deformation theory was developed for algebras over binary quadratic
operads by Balavoine [2]. The deformation theory of embedding tensors were
studied in [28]. In the associative algebra context, the deformation theory of
averaging operators was studied in [6] via the derived brackets [15, 31], and
the deformation theory of averaging associative algebras was studied in [32].

A nonabelian embedding tensor induces a Leibniz-Lie algebra, which further
gives rise to a Leibniz algebra, which is called the descendent Leibniz
algebra of the nonabelian embedding tensor. We introduce a cohomology theory
of nonabelian embedding tensors via the Loday-Pirashvili cohomology of the
descendent Leibniz algebras. As applications, we use the second cohomology
group to characterize equivalent linear deformations of a nonabelian embedding
tensor. Moreover, we introduce the notion of a Nijenhuis element for a
nonabelian embedding tensor, and show that trivial linear deformations are
generated by Nijenhuis elements.

The paper is organized as follows. In Section 2, we introduce the notion of
nonabelian embedding tensors, and show that nonabelian embedding tensors can
be characterized by graphs of the nonabelian hemisemidirect product Leibniz
algebra. In Section 3, we define a new algebraic structure, which is called a
Leibniz-Lie algebra. We show that a nonabelian embedding tensor naturally
induces a Leibniz-Lie algebra. In Section 4, we construct a differential
graded Lie algebra whose Maurer-Cartan elements are nonabelian embedding
tensors. In Section 5, we introduce a cohomology theory of nonabelian
embedding tensors. In Section 6, we study linear deformations of nonabelian
embedding tensors using the established cohomology theory.

##  2\. Nonabelian embedding tensors and Leibniz algebras

In this section, we introduce the notion of nonabelian embedding tensors,
which are the nonabelian generalization of embedding tensors. We show that
nonabelian embedding tensors can be characterized by graphs of the nonabelian
hemisemidirect product Leibniz algebra.

A Leibniz algebra is a vector space $\mathfrak{G}$ together with a bilinear
operation
$[-,-]_{\mathfrak{G}}:\mathfrak{G}\otimes\mathfrak{G}\,\rightarrow\,\mathfrak{G}$
such that

$\displaystyle[x,[y,z]_{\mathfrak{G}}]_{\mathfrak{G}}=[[x,y]_{\mathfrak{G}},z]_{\mathfrak{G}}+[y,[x,z]_{\mathfrak{G}}]_{\mathfrak{G}},\quad\forall
x,y,z\in\mathfrak{G}.$

A representation of a Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ is
a triple $(V;\rho^{L},\rho^{R})$, where $V$ is a vector space,
$\rho^{L},\rho^{R}:\mathfrak{G}\,\rightarrow\,\mathfrak{gl}(V)$ are linear
maps such that for all $x,y\in\mathfrak{G}$,

$\displaystyle\rho^{L}([x,y]_{\mathfrak{G}})$ $\displaystyle=$
$\displaystyle[\rho^{L}(x),\rho^{L}(y)],$
$\displaystyle\rho^{R}([x,y]_{\mathfrak{G}})$ $\displaystyle=$
$\displaystyle[\rho^{L}(x),\rho^{R}(y)],$
$\displaystyle\rho^{R}(y)\circ\rho^{L}(x)$ $\displaystyle=$
$\displaystyle-\rho^{R}(y)\circ\rho^{R}(x).$

Here $[-,-]:\wedge^{2}\mathfrak{gl}(V)\,\rightarrow\,\mathfrak{gl}(V)$ is the
commutator Lie bracket on $\mathfrak{gl}(V)$.

Let $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ and
$(\mathfrak{h},[-,-]_{\mathfrak{h}})$ be Lie algebras. A Lie algebra
homomorphism $\rho:\mathfrak{g}\,\rightarrow\,\mathrm{Der}(\mathfrak{h})$ is
called an action of $\mathfrak{g}$ on $\mathfrak{h}$. An action $\rho$ of
$\mathfrak{g}$ on $\mathfrak{h}$ is called a coherent action if it satisfies

(1) $[\rho(x)u,v]_{\mathfrak{h}}=0,\quad\forall
x\in\mathfrak{g},u,v\in\mathfrak{h}.$

We denote a coherent action by $(\mathfrak{h};\rho)$. Denote by
$\overline{\mathrm{Der}}(\mathfrak{h})\subset\mathrm{Der}(\mathfrak{h})$ the
Lie subalgebra of the derivation Lie algebra $\mathrm{Der}(\mathfrak{h})$
satisfying

$\overline{\mathrm{Der}}(\mathfrak{h})=\\{D\in\mathrm{Der}(\mathfrak{h})|[Du,v]_{\mathfrak{h}}=0,~{}\forall
u,v\in\mathfrak{h}\\}.$

The Lie algebra $\overline{\mathrm{Der}}(\mathfrak{h})$ is called the coherent
derivation Lie algebra. A coherent action can be alternatively described by a
homomorphism from the Lie algebra $\mathfrak{g}$ to the coherent derivation
Lie algebra $\overline{\mathrm{Der}}(\mathfrak{h})$.

######  Example 2.1.

Recall that a two-step nilpotent Lie algebra [30] is a Lie algebra
$\mathfrak{g}$ such that

(2) $[[x,y],z]=0,\,\,\forall x,y,z\in\mathfrak{g}.$

For a two-step nilpotent Lie algebra, the adjoint action $\mathrm{ad}$ is a
coherent action and
$\overline{\mathrm{Der}}(\mathfrak{g})=\mathrm{Der}(\mathfrak{g})$.

######  Definition 2.2.

A nonabelian embedding tensor on a Lie algebra $\mathfrak{g}$ with respect to
a coherent action $(\mathfrak{h};\rho)$ is a linear map
$T:\mathfrak{h}\to\mathfrak{g}$ such that

(3)
$[Tu,Tv]_{\mathfrak{g}}=T\big{(}\rho(Tu)(v)+[u,v]_{\mathfrak{h}}\big{)},\quad\forall
u,v\in\mathfrak{h}.$

######  Definition 2.3.

Let $T$ and $T^{\prime}$ be two nonabelian embedding tensors on a Lie algebra
$\mathfrak{g}$ with respect to a coherent action $(\mathfrak{h};\rho)$. A
homomorphism from $T^{\prime}$ to $T$ consists of two Lie algebra
homomorphisms $\phi_{\mathfrak{g}}:\mathfrak{g}\to\mathfrak{g}$ and
$\phi_{\mathfrak{h}}:\mathfrak{h}\to\mathfrak{h}$ such that

(4) $\displaystyle T\circ\phi_{\mathfrak{h}}$ $\displaystyle=$
$\displaystyle\phi_{\mathfrak{g}}\circ T^{\prime},$ (5)
$\displaystyle\phi_{\mathfrak{h}}\rho(x)u$ $\displaystyle=$
$\displaystyle\rho(\phi_{\mathfrak{g}}(x))\phi_{\mathfrak{h}}(u),\quad\forall
x\in\mathfrak{g},u\in\mathfrak{h}.$

In particular, if both $\phi_{\mathfrak{g}}$ and $\phi_{\mathfrak{h}}$ are
invertible, $(\phi_{\mathfrak{g}},\phi_{\mathfrak{h}})$ is called an
isomorphism from $T^{\prime}$ to $T$.

######  Remark 2.4.

When $(\mathfrak{h},[-,-]_{\mathfrak{h}})$ is an abelian Lie algebra, we
obtain that $T$ is a usual embedding tensor [16, 28]. In addition, when $\rho$
is the trivial action of $\mathfrak{g}$ on $\mathfrak{h}$, $T$ is a Lie
algebra homomorphism from $\mathfrak{h}$ to $\mathfrak{g}$.

Consider the direct sum Lie algebra
$\overline{\mathrm{Der}}(\mathfrak{h})\oplus\mathfrak{h}$, in which the Lie
bracket is given by

$[A+u,B+v]=[A,B]+[u,v]_{\mathfrak{h}},\quad\forall
A,B\in\overline{\mathrm{Der}}(\mathfrak{h}),u,v\in\mathfrak{h}$

Define
$\rho:\overline{\mathrm{Der}}(\mathfrak{h})\to\mathfrak{gl}(\overline{\mathrm{Der}}(\mathfrak{h})\oplus\mathfrak{h})$
by

$\rho(A)(B+v)=Av,\quad\forall
A,B\in\overline{\mathrm{Der}}(\mathfrak{h}),~{}v\in\mathfrak{h}.$

Then it is straightforward to deduce that $\rho$ is a coherent action of the
Lie algebra $\overline{\mathrm{Der}}(\mathfrak{h})$ on the Lie algebra
$\overline{\mathrm{Der}}(\mathfrak{h})\oplus\mathfrak{h}$.

######  Proposition 2.5.

The projection
$\mathrm{pr}:\overline{\mathrm{Der}}(\mathfrak{h})\oplus\mathfrak{h}\to\overline{\mathrm{Der}}(\mathfrak{h})$
is a nonabelian embedding tensor on the Lie algebra
$\overline{\mathrm{Der}}(\mathfrak{h})$ with respect to the coherent action
$(\overline{\mathrm{Der}}(\mathfrak{h})\oplus\mathfrak{h};\rho).$

###### Proof.

It follows from

$\mathrm{pr}\Big{(}\rho(\mathrm{pr}(A+u))(B+v)+[A+u,B+v]\Big{)}=[A,B]=[\mathrm{pr}(A+u),\mathrm{pr}(B+v)],$

for all $A,B\in\overline{\mathrm{Der}}(\mathfrak{h}),u,v\in\mathfrak{h}$. ∎

######  Example 2.6.

The Heisenberg Lie algebra $H_{3}(\mathbb{C})$ is the three-dimensional
complex Lie algebra with the Lie bracket given by

$\displaystyle[e_{1},e_{2}]=e_{3},\quad[e_{1},e_{3}]=0,\quad[e_{2},e_{3}]=0,$

with respect to a basis $\\{e_{1},e_{2},e_{3}\\}$. It is obvious that the
Heisenberg Lie algebra $H_{3}(\mathbb{C})$ is a two-step nilpotent Lie
algebra. A linear transformation
$T=\left(\begin{array}[]{ccc}r_{11}&r_{12}&r_{13}\\\ r_{21}&r_{22}&r_{23}\\\
r_{31}&r_{32}&r_{33}\end{array}\right)$ is a nonabelian embedding tensor with
respect to the adjoint action if and only if
$[Te_{i},Te_{j}]=T([Te_{i},e_{j}]+[e_{i},e_{j}]),~{}~{}i,j\in\\{1,2,3\\}.$
First by

$\displaystyle 0$ $\displaystyle=$
$\displaystyle[Te_{1},Te_{1}]=T([Te_{1},e_{1}]+[e_{1},e_{1}])$
$\displaystyle=$ $\displaystyle
T[r_{11}e_{1}+r_{21}e_{2}+r_{31}e_{3},e_{1}]=-r_{21}r_{13}e_{1}-r_{21}r_{23}e_{2}-r_{21}r_{33}e_{3},$

we obtain $r_{13}r_{21}=0,~{}r_{21}r_{23}=0,~{}r_{21}r_{33}=0.$ Similarly, we
have

$\begin{array}[]{rcl rcl
rcl}r_{12}r_{13}&=&0,&r_{12}r_{23}&=&0,&r_{12}r_{33}&=&0,\\\
(r_{11}+1)r_{13}&=&0,&(r_{11}+1)r_{23}&=&0,&r_{11}r_{22}-r_{12}r_{21}&=&(r_{11}+1)r_{33},\\\
(r_{22}+1)r_{13}&=&0,&(r_{22}+1)r_{23}&=&0,&r_{12}r_{21}-r_{11}r_{22}&=&-(r_{22}+1)r_{33},\\\
r_{11}r_{23}&=&r_{13}r_{21},&r_{13}r_{23}&=&r_{23}^{2}=0,&r_{13}r_{21}-r_{11}r_{23}&=&-r_{23}r_{33},\\\
r_{12}r_{23}&=&r_{13}r_{22},&r_{13}r_{23}&=&r_{13}^{2}=0,&r_{13}r_{22}-r_{12}r_{23}&=&r_{13}r_{33}.\end{array}$

Therefore, we have

  * (i)

If $r_{13}=r_{23}=r_{33}=0$, then
$T=\left(\begin{array}[]{ccc}r_{11}&r_{12}&0\\\ r_{21}&r_{22}&0\\\
r_{31}&r_{32}&0\end{array}\right)$ is a nonabelian embedding tensor on
$H_{3}(\mathbb{C})$ if and only if $r_{11}r_{22}=r_{12}r_{21}$. Moreover, for
any $a,b,c,d,k\in\mathbb{C}$, we deduce that
$T=\left(\begin{array}[]{ccc}c&d&0\\\ kc&kd&0\\\ a&b&0\end{array}\right)$ and
$T=\left(\begin{array}[]{ccc}0&0&0\\\ c&d&0\\\ a&b&0\end{array}\right)$ are
nonabelian embedding tensors on $H_{3}(\mathbb{C})$.

  * (ii)

If $r_{13}=r_{23}=0$ and $r_{33}\not=0$, then $r_{12}=r_{21}=0$,
$r_{11}=r_{22}$ and $T=\left(\begin{array}[]{ccc}r_{11}&0&0\\\ 0&r_{11}&0\\\
r_{31}&r_{32}&r_{33}\end{array}\right)$ is a nonabelian embedding tensor on
$H_{3}(\mathbb{C})$ if and only if $r_{11}^{2}-r_{33}r_{11}-r_{33}=0$.
Moreover, for any $a,b\in\mathbb{C}$ and a nonzero complex number
$0\not=t\in\mathbb{C}$, we deduce that
$T=\left(\begin{array}[]{ccc}\frac{t+\sqrt{t^{2}+4t}}{2}&0&0\\\
0&\frac{t+\sqrt{t^{2}+4t}}{2}&0\\\ a&b&t\end{array}\right)$ and
$T=\left(\begin{array}[]{ccc}\frac{t-\sqrt{t^{2}+4t}}{2}&0&0\\\
0&\frac{t-\sqrt{t^{2}+4t}}{2}&0\\\ a&b&t\end{array}\right)$ are nonabelian
embedding tensors on $H_{3}(\mathbb{C})$.

These are all the possibilities of nonabelian embedding tensors on
$H_{3}(\mathbb{C})$ with respect to the adjoint representation.

Let $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ and
$(\mathfrak{h},[-,-]_{\mathfrak{h}})$ be Lie algebras. Let
$\rho:\mathfrak{g}\,\rightarrow\,\mathrm{Der}(\mathfrak{h})$ be a coherent
action of $\mathfrak{g}$ on $\mathfrak{h}$. On the direct sum vector space
$\mathfrak{g}\oplus\mathfrak{h},$ define a bilinear bracket operation
$[-,-]_{\rho}$ by

(6)
$[x+u,y+v]_{\rho}=[x,y]_{\mathfrak{g}}+\rho(x)v+[u,v]_{\mathfrak{h}},\quad\forall
x,y\in\mathfrak{g},~{}u,v\in\mathfrak{h}.$

######  Proposition 2.7.

With the above notations, $(\mathfrak{g}\oplus\mathfrak{h},[-,-]_{\rho})$ is a
Leibniz algebra, which is called the nonabelian hemisemidirect product, and
denoted by $\mathfrak{g}\ltimes^{h}_{\rho}\mathfrak{h}$.

###### Proof.

By the Jacobi identities for the Lie brackets $[-,-]_{\mathfrak{g}}$ and
$[-,-]_{\mathfrak{h}}$, and the fact that $\rho$ is a homomorphism from the
Lie algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ to the derivation Lie
algebra $\mathrm{Der}(\mathfrak{h})$, we have

$\displaystyle[x+u,[y+v,z+w]_{\rho}]_{\rho}-[[x+u,y+v]_{\rho},z+w]_{\rho}-[y+v,[x+u,z+w]_{\rho}]_{\rho}$
$\displaystyle=$
$\displaystyle[x,[y,z]_{\mathfrak{g}}]_{\mathfrak{g}}-[[x,y]_{\mathfrak{g}},z]_{\mathfrak{g}}-[y,[x,z]_{\mathfrak{g}}]_{\mathfrak{g}}+[u,[v,w]_{\mathfrak{h}}]_{\mathfrak{h}}-[[u,v]_{\mathfrak{h}},w]_{\mathfrak{h}}-[v,[u,w]_{\mathfrak{h}}]_{\mathfrak{h}}$
$\displaystyle+\rho(x)[v,w]_{\mathfrak{h}}-[\rho(x)v,w]_{\mathfrak{h}}-[v,\rho(x)w]_{h}+\rho(x)\rho(y)w-\rho([x,y]_{\mathfrak{g}})w-\rho(y)\rho(x)w$
$\displaystyle+[u,\rho(y)w]_{\mathfrak{h}}-\rho(y)[u,w]_{\mathfrak{h}}$
$\displaystyle=$
$\displaystyle[u,\rho(y)w]_{\mathfrak{h}}-\rho(y)[u,w]_{\mathfrak{h}}.$

By (1) and the fact that $\rho(y)\in\mathrm{Der}(\mathfrak{h})$, we deduce
that $[u,\rho(y)w]_{\mathfrak{h}}-\rho(y)[u,w]_{\mathfrak{h}}=0$. Therefore,
$(\mathfrak{g}\oplus\mathfrak{h},[-,-]_{\rho})$ is a Leibniz algebra. ∎

######  Remark 2.8.

In fact, the nonabelian hemisemidirect product
$\mathfrak{g}\ltimes^{h}_{\rho}\mathfrak{h}$ is a split Leibniz algebra
extension of the Lie algebra $\mathfrak{g}$ by the Lie algebra $\mathfrak{h}$
via the antisymmetric representation. See [29, Proposition 1] for more details
about nonabelian extensions of Leibniz algebras.

######  Remark 2.9.

Note that when $\mathfrak{h}$ is abelian, the bracket $[-,-]_{\rho}$ is the
hemisemidirect product of a Lie algebra and its representation introduced in
[14] in the study of Courant algebroids. This kind of Leibniz algebras are
very important, e.g. semisimple Leibniz algebras are of this form [8].

Let $T:\mathfrak{h}\to\mathfrak{g}$ be a linear map. Denote by $G_{T}$ the
graph of $T$, i.e. $G_{T}=\\{Tu+u|\forall u\in\mathfrak{h}\\}$.

######  Proposition 2.10.

Let $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ and
$(\mathfrak{h},[-,-]_{\mathfrak{h}})$ be Lie algebras. Let
$\rho:\mathfrak{g}\,\rightarrow\,\mathrm{Der}(\mathfrak{h})$ be a coherent
action of $\mathfrak{g}$ on $\mathfrak{h}$. Then a linear map
$T:\mathfrak{h}\to\mathfrak{g}$ is a nonabelian embedding tensor if and only
if its graph $G_{T}$ is a Leibniz subalgebra of the Leibniz algebra
$(\mathfrak{g}\oplus\mathfrak{h},[-,-]_{\rho})$.

###### Proof.

We have

$\displaystyle[Tu+u,Tv+v]_{\rho}=[Tu,Tv]_{\mathfrak{g}}+\rho(Tu)v+[u,v]_{\mathfrak{h}},\quad\forall
u,v\in\mathfrak{h},$

which implies that the graph $G_{T}$ is a Leibniz subalgebra if and only if
(3) holds, i.e. $T:\mathfrak{h}\to\mathfrak{g}$ is a nonabelian embedding
tensor. ∎

Since $\mathfrak{h}$ and $G_{T}$ are naturally isomorphic, we get the
following conclusion immediately.

######  Corollary 2.11.

Let $T:\mathfrak{h}\to\mathfrak{g}$ be a nonabelian embedding tensor on a Lie
algebra $\mathfrak{g}$ with respect to a coherent action
$(\mathfrak{h};\rho)$. Then $(\mathfrak{h},[-,-]_{T})$ is a Leibniz algebra,
where the Leibniz bracket $[-,-]_{T}$ is given by

(7) $[u,v]_{T}=\rho(Tu)v+[u,v]_{\mathfrak{h}},\quad\forall
u,v\in\mathfrak{h}.$

Moreover, $T$ is a Leibniz algebra homomorphism from the Leibniz algebra
$(\mathfrak{h},[-,-]_{T})$ to the Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$.

The Leibniz algebra $(\mathfrak{h},[-,-]_{T})$ is called the descendent
Leibniz algebra of the nonabelian embedding tensor $T$.

######  Example 2.12.

Consider the nonabelian embedding tensor given in Proposition 2.5. Then by
Corollary 2.11, one obtain a Leibniz algebra structure on
$\overline{\mathrm{Der}}(\mathfrak{h})\oplus\mathfrak{h}:$

$[A+u,B+v]_{\mathrm{pr}}=[A+u,B+v]+\rho(\mathrm{pr}(A+u))(B+v)=[A,B]+Av+[u,v]_{\mathfrak{h}},$

for all $A,B\in\overline{\mathrm{Der}}(\mathfrak{h}),~{}u,v\in\mathfrak{h}$,
which is exactly the nonabelian hemisemidirect product of
$\overline{\mathrm{Der}}(\mathfrak{h})$ and $\mathfrak{h}$ given in
Proposition 2.7.

The association of a Leibniz algebra from a nonabelian embedding tensor enjoys
the functorial property.

######  Proposition 2.13.

Let $T$ and $T^{\prime}$ be two nonabelian embedding tensors on a Lie algebra
$\mathfrak{g}$ with respect to a coherent action $(\mathfrak{h};\rho)$ and
$(\phi_{\mathfrak{g}},\phi_{\mathfrak{h}})$ a homomorphism from $T^{\prime}$
to $T$. Then $\phi_{\mathfrak{h}}$ is a homomorphism of Leibniz algebras from
$(\mathfrak{h},[-,-]_{T^{\prime}})$ to $(\mathfrak{h},[-,-]_{T})$.

###### Proof.

For all $u,v\in\mathfrak{h}$, we have

$\displaystyle\phi_{\mathfrak{h}}([u,v]_{T^{\prime}})$ $\displaystyle=$
$\displaystyle\phi_{\mathfrak{h}}(\rho(T^{\prime}u)v+[u,v]_{\mathfrak{h}})=\rho(\phi_{\mathfrak{g}}(T^{\prime}u))\phi_{\mathfrak{h}}(v)+[\phi_{\mathfrak{h}}(u),\phi_{\mathfrak{h}}(v)]_{\mathfrak{h}}$
$\displaystyle=$
$\displaystyle\rho(T\phi_{\mathfrak{h}}(u))\phi_{\mathfrak{h}}(v)+[\phi_{\mathfrak{h}}(u),\phi_{\mathfrak{h}}(v)]_{\mathfrak{h}}=[\phi_{\mathfrak{h}}(u),\phi_{\mathfrak{h}}(v)]_{T}.$

Then $\phi_{\mathfrak{h}}$ is a homomorphism of Leibniz algebras from
$(\mathfrak{h},[-,-]_{T^{\prime}})$ to $(\mathfrak{h},[-,-]_{T})$. ∎

##  3\. Nonabelian embedding tensors and Leibniz-Lie algebras

In this section, we introduce the notion of a Leibniz-Lie algebra as the
underlying algebraic structure of a nonabelian embedding tensor. A Leibniz-Lie
algebra gives rise to a Leibniz algebra, as well as nonabelian embedding
tensors via different approaches.

######  Definition 3.1.

A Leibniz-Lie algebra $(\mathfrak{h},[-,-]_{\mathfrak{h}},\rhd)$ consists of a
Lie algebra $(\mathfrak{h},[-,-]_{\mathfrak{h}})$ and a binary product
$\rhd:\mathfrak{h}\otimes\mathfrak{h}\,\rightarrow\,\mathfrak{h}$ such that
for all $x,y,z\in\mathfrak{h},$ one has

(8) $\displaystyle x\rhd(y\rhd z)$ $\displaystyle=$ $\displaystyle(x\rhd
y)\rhd z+y\rhd(x\rhd z)+[x,y]_{\mathfrak{h}}\rhd z,$ (9) $\displaystyle
x\rhd[y,z]_{\mathfrak{h}}$ $\displaystyle=$ $\displaystyle[x\rhd
y,z]_{\mathfrak{h}}=0.$

A homomorphism from a Leibniz-Lie algebra
$(\mathfrak{h},[-,-]_{\mathfrak{h}},\rhd)$ to a Leibniz-Lie algebra
$(\mathfrak{h}^{\prime},[-,-]_{\mathfrak{h}^{\prime}},\rhd^{\prime})$ is a
Leibniz algebra homomorphism $\phi_{\mathfrak{h}}$ such that
$\phi_{\mathfrak{h}}(x\rhd
y)=\phi_{\mathfrak{h}}(x)\rhd^{\prime}\phi_{\mathfrak{h}}(y)$.

######  Remark 3.2.

A Leibniz algebra $(\mathfrak{G},[-,-]_{\mathfrak{G}})$ is naturally a
Leibniz-Lie algebra, in which the Lie algebra structure is abelian.

######  Example 3.3.

Let $H_{3}(\mathbb{C})$ be the Heisenberg Lie algebra. We define a binary
product $\rhd:H_{3}(\mathbb{C})\otimes
H_{3}(\mathbb{C})\,\rightarrow\,H_{3}(\mathbb{C})$ by

$\begin{array}[]{rcl rcl rcl rcl}e_{1}\rhd e_{1}&=&-e_{3},&e_{1}\rhd
e_{2}&=&e_{3},&e_{2}\rhd e_{2}&=&e_{3},&e_{2}\rhd e_{1}&=&-e_{3}\\\ e_{1}\rhd
e_{3}&=&0,&e_{2}\rhd e_{3}&=&0,&e_{3}\rhd e_{3}&=&0,&e_{3}\rhd
e_{1}&=&e_{3}\rhd e_{2}=0.\end{array}$

Then $(H_{3}(\mathbb{C}),[-,-],\rhd)$ is a Leibniz-Lie algebra.

######  Theorem 3.4.

Let $(\mathfrak{h},[-,-]_{\mathfrak{h}},\rhd)$ be a Leibniz-Lie algebra. Then
the binary operation
$[-,-]:\mathfrak{h}\otimes\mathfrak{h}\,\rightarrow\,\mathfrak{h}$ given by

(10) $\displaystyle[x,y]=x\rhd y+[x,y]_{\mathfrak{h}},\,\,\,\,\forall
x,y\in\mathfrak{h},$

defines a Leibniz algebra structure on $\mathfrak{h}$, which is denoted by
$\mathfrak{h}^{\mathsf{Leib}}$ and called the subadjacent Leibniz algebra.

Moreover, define
$\mathcal{L}:\mathfrak{h}\,\rightarrow\,\mathfrak{gl}(\mathfrak{h})$ by

(11) $\mathcal{L}_{x}y=x\rhd y.$

Then $(\mathfrak{h};\mathcal{L},0)$ is a representation of the Leibniz algebra
$(\mathfrak{h},[-,-])$ on $\mathfrak{h}$.

###### Proof.

For all $x,y,z\in\mathfrak{h}$, we have

$\displaystyle[x,[y,z]=[x,y\rhd z+[y,z]_{\mathfrak{h}}]$ $\displaystyle=$
$\displaystyle x\rhd(y\rhd z)+[x,y\rhd
z]_{\mathfrak{h}}+x\rhd[y,z]_{\mathfrak{h}}+[x,[y,z]_{\mathfrak{h}}]_{\mathfrak{h}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{Post-2}}}{{=}}$ $\displaystyle
x\rhd(y\rhd z)+[x,[y,z]_{\mathfrak{h}}]_{\mathfrak{h}}.$

On the other hand, we have

$\displaystyle[[x,y],z]+[y,[x,z]$ $\displaystyle=$ $\displaystyle[x\rhd
y+[x,y]_{\mathfrak{h}},z]+[y,x\rhd z+[x,z]_{\mathfrak{h}}]$ $\displaystyle=$
$\displaystyle(x\rhd y)\rhd z+[x\rhd
y,z]_{\mathfrak{h}}+[x,y]_{\mathfrak{h}}\rhd
z+[[x,y]_{\mathfrak{h}},z]_{\mathfrak{h}}$ $\displaystyle+y\rhd(x\rhd
z)+[y,x\rhd
z]_{\mathfrak{h}}+y\rhd[x,z]_{\mathfrak{h}}+[y,[x,z]_{\mathfrak{g}}]_{\mathfrak{h}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{Post-2}}}{{=}}$
$\displaystyle(x\rhd y)\rhd z+[x,y]_{\mathfrak{h}}\rhd
z+[[x,y]_{\mathfrak{h}},z]_{\mathfrak{h}}+y\rhd(x\rhd
z)+[y,[x,z]_{\mathfrak{h}}]_{\mathfrak{h}}.$

By (8), we deduce that $(\mathfrak{h},[-,-])$ is a Leibniz algebra.

Also by (8), we get $\mathcal{L}_{[x,y]}=[\mathcal{L}_{x},\mathcal{L}_{y}],$
which implies that $(\mathfrak{h};\mathcal{L},0)$ is a representation of the
Leibniz algebra $(\mathfrak{h},[-,-])$ on $\mathfrak{h}$. ∎

An embedding tensor naturally induces a Leibniz algebra. Now a nonabelian
embedding tensor naturally induces a Leibniz-Lie algebra.

######  Theorem 3.5.

Let $T:\mathfrak{h}\to\mathfrak{g}$ be a nonabelian embedding tensor on a Lie
algebra $\mathfrak{g}$ with respect to a coherent action
$(\mathfrak{h};\rho)$. Then there is a Leibniz-Lie algebra structure on the
Lie algebra $(\mathfrak{h},[-,-]_{\mathfrak{h}})$ given by

(12) $\displaystyle u\rhd v=\rho(Tu)v,\,\,\,\,\forall u,v\in\mathfrak{h}.$

###### Proof.

For all $u,v,w\in\mathfrak{h}$, we have

$\displaystyle u\rhd(v\rhd w)$ $\displaystyle=$
$\displaystyle\rho(Tu)\rho(Tv)w=\rho([Tu,Tv]_{\mathfrak{g}})w+\rho(Tv)\rho(Tu)w$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:nonET}}}{{=}}$
$\displaystyle\rho\big{(}T(\rho(Tu)v+[u,v]_{\mathfrak{h}})\big{)}w+\rho(Tv)\rho(Tu)w$
$\displaystyle=$ $\displaystyle(u\rhd v)\rhd w+[u,v]_{\mathfrak{h}}\rhd
w+v\rhd(u\rhd w),$

which implies that (8) in Definition 3.1 holds. Moreover, we have

$\displaystyle
u\rhd[v,w]_{\mathfrak{h}}=\rho(Tu)[v,w]_{\mathfrak{h}}=[\rho(Tu)v,w]_{\mathfrak{h}}+[v,\rho(Tu)w]_{\mathfrak{h}}.$

By (1), $u\rhd[v,w]_{\mathfrak{h}}=[u\rhd v,w]_{\mathfrak{h}}=0$. Thus,
$(\mathfrak{h},[-,-]_{\mathfrak{h}},\rhd)$ is a Leibniz-Lie algebra. ∎

######  Remark 3.6.

The Leibniz algebra determined by the Leibniz-Lie algebra
$(\mathfrak{h},[-,-]_{\mathfrak{h}},\rhd)$ according to Theorem 3.4 is exactly
the Leibniz algebra given in Corollary 2.11.

######  Example 3.7.

Consider the nonabelian embedding tensor given in Proposition 2.5. Then by
Theorem 3.5,
$(\overline{\mathrm{Der}}(\mathfrak{h})\oplus\mathfrak{h},[-,-],\rhd)$ is a
Leibniz-Lie algebra, where $[-,-]$ is the direct sum Lie bracket and $\rhd$ is
given by

$(A+u)\rhd(B+v)=\rho(\mathrm{pr}(A+u))(B+v)=Av,$

for all $A,B\in\overline{\mathrm{Der}}(\mathfrak{h}),~{}u,v\in\mathfrak{h}$.

Furthermore, by Theorem 3.4, the Leibniz-Lie algebra
$(\overline{\mathrm{Der}}(\mathfrak{h})\oplus\mathfrak{h},[-,-],\rhd)$ gives
rise to a Leibniz algebra which is exactly the nonabelian hemisemidirect
product Leibniz algebra given in Example 2.12.

It is straightforward to obtain the following result, which implies that the
association given in Theorem 3.5 is functorial.

######  Proposition 3.8.

Let $T$ and $T^{\prime}$ be two nonabelian embedding tensors on a Lie algebra
$\mathfrak{g}$ with respect to a coherent action $(\mathfrak{h};\rho)$ and
$(\phi_{\mathfrak{g}},\phi_{\mathfrak{h}})$ a homomorphism from $T^{\prime}$
to $T$. Then $\phi_{\mathfrak{h}}$ is a homomorphism of Leibniz-Lie algebras
from $(\mathfrak{h},[\cdot,\cdot]_{\mathfrak{h}},\rhd_{T^{\prime}})$ to
$(\mathfrak{h},[\cdot,\cdot]_{\mathfrak{h}},\rhd_{T})$.

At the end of this section, we show that a Leibniz-Lie algebra
$(\mathfrak{h},[-,-]_{\mathfrak{h}},\rhd)$ can give rise to a nonabelian
embedding tensor via two different ways such that the subadjacent Leibniz
algebra is exactly the descendent Leibniz algebra of the nonabelian embedding
tensor. First we make some preparations. Let
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ be a Leibniz algebra. Denote by
$\mathsf{Leib}(\mathfrak{G})$ the ideal of squares generated by
$[x,x]_{\mathfrak{G}}$ for all $x\in\mathfrak{G}$. Then

$\mathfrak{G}_{\mathrm{Lie}}:=\mathfrak{G}/\mathsf{Leib}(\mathfrak{G})$

is naturally a Lie algebra in which we denote the Lie bracket by
$[-,-]_{\mathfrak{G}_{\mathrm{Lie}}}$. The equivalence class of
$x\in\mathfrak{G}$ in $\mathfrak{G}_{\mathrm{Lie}}$ will be denoted by
$\bar{x}$. Moreover, $\mathsf{Leib}(\mathfrak{G})$ is contained in the left
center of the Leibniz algebra $\mathfrak{G}.$ We denote by

(13) $\mathrm{pr}:\mathfrak{G}\,\rightarrow\,\mathfrak{G}_{\mathrm{Lie}}$

the natural projection from the Leibniz algebra $\mathfrak{G}$ to the Lie
algebra $\mathfrak{G}_{\mathrm{Lie}}$. Obviously, the projection $\mathrm{pr}$
preserves the bracket operation, i.e. the following equality holds:

(14)
$\mathrm{pr}[x,y]_{\mathfrak{G}}=[\mathrm{pr}(x),\mathrm{pr}(y)]_{\mathfrak{G}_{\mathrm{Lie}}},\quad\forall
x,y\in\mathfrak{G}.$

Let $(\mathfrak{h},[-,-]_{\mathfrak{h}},\rhd)$ be a Leibniz-Lie algebra. Since
$[-,-]_{\mathfrak{h}}$ is skewsymmetric, the ideal
$\mathsf{Leib}(\mathfrak{h}^{\mathsf{Leib}})$ is generated by $x\rhd x$ for
all $x\in\mathfrak{h}$. Consider the quotient Lie algebra

$(\mathfrak{h}^{\mathsf{Leib}})_{\mathrm{Lie}}=\mathfrak{h}^{\mathsf{Leib}}/\mathsf{Leib}(\mathfrak{h}^{\mathsf{Leib}}).$

Define an action of $(\mathfrak{h}^{\mathsf{Leib}})_{\mathrm{Lie}}$ on the Lie
algebra $(\mathfrak{h},[-,-]_{\mathfrak{h}})$ by

$\rho(\bar{u})v=u\rhd v,\quad\forall u,v\in\mathfrak{h}.$

It is well defined. In fact, if $\bar{u}=\bar{u^{\prime}}$, then
$u-u^{\prime}\in\mathsf{Leib}(\mathfrak{h}^{\mathsf{Leib}})$. By (8), we
obtain $(u-u^{\prime})\rhd v=0.$ Moreover, by (9), we have

$[\rho(\bar{u})v,w]_{\mathfrak{h}}=[u\rhd
v,w]_{\mathfrak{h}}=0,\quad\rho(\bar{u})[v,w]_{\mathfrak{h}}={u}\rhd[v,w]_{\mathfrak{h}}=0.$

######  Proposition 3.9.

Let $(\mathfrak{h},[-,-]_{\mathfrak{h}},\rhd)$ be a Leibniz-Lie algebra. Then
the natural projection
$\mathrm{pr}:\mathfrak{h}\,\rightarrow\,(\mathfrak{h}^{\mathsf{Leib}})_{\mathrm{Lie}}$
is a nonabelian embedding tensor on the Lie algebra
$(\mathfrak{h}^{\mathsf{Leib}})_{\mathrm{Lie}}$ with respect to the coherent
action $(\mathfrak{h};\rho)$ such that the induced descendent Leibniz algebra
is exactly the subadjacent Leibniz algebra $\mathfrak{h}^{\mathsf{Leib}}$.

###### Proof.

It follows from

$\displaystyle[\mathrm{pr}(u),\mathrm{pr}(v)]_{(\mathfrak{h}^{\mathsf{Leib}})_{\mathrm{Lie}}}=\mathrm{pr}[u,v]=\mathrm{pr}([u,v]_{\mathfrak{h}}+u\rhd
v)=\mathrm{pr}([u,v]_{\mathfrak{h}}+\rho(\mathrm{pr}({u}))v)$

directly. ∎

There is also another approach to construct a nonabelian embedding tensor from
a Leibniz-Lie algebra. Let $(\mathfrak{h},[-,-]_{\mathfrak{h}},\rhd)$ be a
Leibniz-Lie algebra and $\mathrm{Der}(\mathfrak{h})$ the derivation Lie
algebra of the Lie algebra $(\mathfrak{h},[-,-]_{\mathfrak{h}})$. By (9),
$\mathcal{L}_{u}\in\overline{\mathrm{Der}}(\mathfrak{h})$, where $\mathcal{L}$
is given by (11).

######  Proposition 3.10.

Let $(\mathfrak{h},[-,-]_{\mathfrak{h}},\rhd)$ be a Leibniz-Lie algebra. Then
$\mathcal{L}:\mathfrak{h}\,\rightarrow\,\overline{\mathrm{Der}}(\mathfrak{h})$
is a nonabelian embedding tensor on the Lie algebra
$\overline{\mathrm{Der}}(\mathfrak{h})$ via the natural coherent action on the
Lie algebra $(\mathfrak{h},[-,-]_{\mathfrak{h}})$ such that the induced
descendent Leibniz algebra is exactly the subadjacent Leibniz algebra
$\mathfrak{h}^{\mathsf{Leib}}$.

###### Proof.

By (8), we have

$\displaystyle[\mathcal{L}_{u},\mathcal{L}_{v}]=\mathcal{L}_{[u,v]_{\mathfrak{h}}+\mathcal{L}_{u}v},\quad\forall
u,v\in\mathfrak{h},$

which implies that
$\mathcal{L}:\mathfrak{h}\,\rightarrow\,\overline{\mathrm{Der}}(\mathfrak{h})$
is a nonabelian embedding tensor. The other conclusion is obvious. ∎

##  4\. The controlling differential graded Lie algebra of nonabelian
embedding tensors

In this section first we recall the controlling graded Lie algebra that
characterize Leibniz algebras as Maurer-Cartan elements. Then we construct the
controlling differential graded Lie algebra of nonabelian embedding tensors.

A permutation $\sigma\in\mathbb{S}_{n}$ is called an $(i,n-i)$-shuffle if
$\sigma(1)<\cdots<\sigma(i)$ and $\sigma(i+1)<\cdots<\sigma(n)$. If $i=0$ or
$n$, we assume $\sigma={\rm{Id}}$. The set of all $(i,n-i)$-shuffles will be
denoted by $\mathbb{S}_{(i,n-i)}$.

Let $\mathfrak{g}$ be a vector space. We consider the graded vector space

$C^{\bullet}(\mathfrak{g},\mathfrak{g})=\oplus_{n=0}^{+\infty}C^{n}(\mathfrak{g},\mathfrak{g})=\oplus_{n=0}^{+\infty}\mathrm{Hom}(\otimes^{n+1}\mathfrak{g},\mathfrak{g}).$

It is known that $C^{\bullet}(\mathfrak{g},\mathfrak{g})$ equipped with the
Balavoine bracket [2]:

(15)
$\displaystyle[P,Q]_{\mathsf{B}}=P\bar{\circ}Q-(-1)^{pq}Q\bar{\circ}P,\,\,\,\,\forall
P\in C^{p}(\mathfrak{g},\mathfrak{g}),Q\in C^{q}(\mathfrak{g},\mathfrak{g}),$

is a graded Lie algebra, where $P\bar{\circ}Q\in
C^{p+q}(\mathfrak{g},\mathfrak{g})$ is defined by
$P\bar{\circ}Q=\sum_{k=1}^{p+1}P\circ_{k}Q,$ and $\circ_{k}$ is defined by

$\displaystyle(P\circ_{k}Q)(x_{1},\cdots,x_{p+q+1})$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in\mathbb{S}_{(k-1,q)}}(-1)^{(k-1)q}(-1)^{\sigma}P(x_{\sigma(1)},\cdots,x_{\sigma(k-1)},Q(x_{\sigma(k)},\cdots,x_{\sigma(k+q-1)},x_{k+q}),x_{k+q+1},\cdots,x_{p+q+1}).$

The following conclusion is straightforward.

######  Lemma 4.1.

For $\Omega\in
C^{1}(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}(\otimes^{2}\mathfrak{g},\mathfrak{g})$,
we have

$\displaystyle\frac{1}{2}[\Omega,\Omega]_{\mathsf{B}}(x_{1},x_{2},x_{3})=\Omega\bar{\circ}\Omega(x_{1},x_{2},x_{3})=\Omega(\Omega(x_{1},x_{2}),x_{3})-\Omega(x_{1},\Omega(x_{2},x_{3}))+\Omega(x_{2},\Omega(x_{1},x_{3})).$

Thus, $\Omega$ defines a Leibniz algebra structure if and only if
$[\Omega,\Omega]_{\mathsf{B}}=0$, i.e. $\Omega$ is a Maurer-Cartan element of
the graded Lie algebra
$(C^{\bullet}(\mathfrak{g},\mathfrak{g}),[-,-]_{\mathsf{B}})$.

Let $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ and
$(\mathfrak{h},[-,-]_{\mathfrak{h}})$ be Lie algebras. Let
$\rho:\mathfrak{g}\,\rightarrow\,\mathrm{Der}(\mathfrak{h})$ be a coherent
action of $\mathfrak{g}$ on $\mathfrak{h}$. Consider the graded vector space
$C^{*}(\mathfrak{h},\mathfrak{g}):=\oplus_{n=1}^{+\infty}\mathrm{Hom}(\otimes^{n}\mathfrak{h},\mathfrak{g})$.
Define a linear map
$\mathrm{d}:\mathrm{Hom}(\otimes^{n}\mathfrak{h},\mathfrak{g})\to\mathrm{Hom}(\otimes^{n+1}\mathfrak{h},\mathfrak{g})$
by

(16) $\displaystyle(\mathrm{d}f)(v_{1},\cdots,v_{n+1})$ $\displaystyle=$
$\displaystyle\sum_{1\leq i<j\leq
n+1}(-1)^{n-1+i}f(v_{1},\cdots,\hat{v}_{i},\cdots,v_{j-1},[v_{i},v_{j}]_{\mathfrak{h}},v_{j+1},\cdots,v_{n+1}),$

for all $f\in\mathrm{Hom}(\otimes^{n}\mathfrak{h},\mathfrak{g})$ and
$v_{1},\cdots,v_{n+1}\in\mathfrak{h}$. Also define a graded bracket operation

$\left\llbracket-,-\right\rrbracket:\mathrm{Hom}(\otimes^{m}\mathfrak{h},\mathfrak{g})\times\mathrm{Hom}(\otimes^{n}\mathfrak{h},\mathfrak{g})\to\mathrm{Hom}(\otimes^{m+n}\mathfrak{h},\mathfrak{g}),$
$\displaystyle\left\llbracket\theta,\phi\right\rrbracket(v_{1},\cdots,v_{m+n})$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{m}\sum_{\sigma\in\mathbb{S}_{(k-1,n)}}(-1)^{(k-1)n+1}(-1)^{\sigma}\theta(v_{\sigma(1)},\cdots,v_{\sigma(k-1)},\rho(\phi(v_{\sigma(k)},\cdots,v_{\sigma(k+n-1)}))v_{k+n},v_{k+n+1},\cdots,v_{m+n})$
$\displaystyle+\sum_{\sigma\in\mathbb{S}_{(m,n)}}(-1)^{mn+1}(-1)^{\sigma}[\theta(v_{\sigma(1)},\cdots,v_{\sigma(m)}),\phi(v_{\sigma(m+1)},\cdots,v_{\sigma(m+n-1)},v_{\sigma(m+n)})]_{\mathfrak{g}}$
$\displaystyle+\sum_{k=1}^{n}\sum_{\sigma\in\mathbb{S}_{(k-1,m)}}(-1)^{m(k+n-1)}(-1)^{\sigma}\phi(v_{\sigma(1)},\cdots,v_{\sigma(k-1)},\rho(\theta(v_{\sigma(k)},\cdots,v_{\sigma(k+m-1)}))v_{k+m},v_{k+m+1},\cdots,v_{m+n}),$

for all
$\theta\in\mathrm{Hom}(\otimes^{m}\mathfrak{h},\mathfrak{g}),~{}\phi\in\mathrm{Hom}(\otimes^{n}\mathfrak{h},\mathfrak{g})$
and $v_{1},\cdots,v_{m+n}\in\mathfrak{h}$.

######  Theorem 4.2.

Let $\rho:\mathfrak{g}\,\rightarrow\,\mathrm{Der}(\mathfrak{h})$ be a coherent
action of a Lie algebra $(\mathfrak{g},[-,-]_{\mathfrak{g}})$ on a Lie algebra
$(\mathfrak{h},[-,-]_{\mathfrak{h}})$. Then the triple
$(C^{*}(\mathfrak{h},\mathfrak{g}),\left\llbracket-,-\right\rrbracket,\mathrm{d})$
is a differential graded Lie algebra, whose Maurer-Cartan elements are
nonabelian embedding tensors.

###### Proof.

In short, the graded Lie algebra
$(C^{*}(\mathfrak{h},\mathfrak{g}),\left\llbracket-,-\right\rrbracket)$ is
obtained via the derived bracket [15]. In fact, the Balavoine bracket
$[-,-]_{\mathsf{B}}$ associated to the direct sum vector space
$\mathfrak{g}\oplus\mathfrak{h}$ gives rise to a graded Lie algebra
$(\oplus_{k=0}^{+\infty}\mathrm{Hom}(\otimes^{k+1}(\mathfrak{g}\oplus\mathfrak{h}),\mathfrak{g}\oplus\mathfrak{h}),[-,-]_{\mathsf{B}})$.
Obviously
$\oplus_{k=0}^{+\infty}\mathrm{Hom}(\otimes^{k+1}\mathfrak{h},\mathfrak{g})$
is an abelian subalgebra. We denote the Lie brackets $[-,-]_{\mathfrak{g}}$
and $[-,-]_{\mathfrak{h}}$ by $\mu_{\mathfrak{g}}$ and $\mu_{\mathfrak{h}}$
respectively, and denote the hemisemidirect product Leibniz algebra structure
on $\mathfrak{g}\oplus\mathfrak{h}$ by $\mu_{\mathfrak{g}}\boxplus\rho$. Thus
$\mu_{\mathfrak{g}}\boxplus\rho$ and $\mu_{\mathfrak{h}}$ are Maurer-Cartan
elements of the graded Lie algebra
$(\oplus_{k=0}^{+\infty}\mathrm{Hom}(\otimes^{k+1}(\mathfrak{g}\oplus\mathfrak{h}),\mathfrak{g}\oplus\mathfrak{h}),[-,-]_{\mathsf{B}})$.
Define the derived bracket on the graded vector space
$\oplus_{k=1}^{+\infty}\mathrm{Hom}(\otimes^{k}\mathfrak{h},\mathfrak{g})$ by

$\left\llbracket\theta,\phi\right\rrbracket:=(-1)^{m-1}[[\mu_{\mathfrak{g}}\boxplus\rho,\theta]_{\mathsf{B}},\phi]_{\mathsf{B}},\quad\forall\theta\in\mathrm{Hom}(\otimes^{m}\mathfrak{h},\mathfrak{g}),~{}\phi\in\mathrm{Hom}(\otimes^{n}\mathfrak{h},\mathfrak{g}),$

which is exactly the bracket given by (4). By
$[\mu_{\mathfrak{g}}\boxplus\rho,\mu_{\mathfrak{g}}\boxplus\rho]_{\mathsf{B}}=0,$
we deduce that
$(C^{*}(\mathfrak{h},\mathfrak{g}),\left\llbracket-,-\right\rrbracket)$ is a
graded Lie algebra.

Define a linear map $\mathrm{d}=:[\mu_{\mathfrak{h}},-]_{\mathsf{B}}$ on the
graded space
$\oplus_{k=1}^{+\infty}\mathrm{Hom}(\otimes^{k}(\mathfrak{g}\oplus\mathfrak{h}),\mathfrak{g}\oplus\mathfrak{h})$.
We obtain that $\mathrm{d}$ is closed on the subspace
$\oplus_{k=1}^{+\infty}\mathrm{Hom}(\otimes^{k}\mathfrak{h},\mathfrak{g})$,
and is given by (16).

By $[\mu_{\mathfrak{h}},\mu_{\mathfrak{h}}]_{\mathsf{B}}=0$, we obtain that
$\mathrm{d}\circ\mathrm{d}=0.$ Moreover, by
$\mathrm{Im}(\rho)\subset\mathrm{Der}(\mathfrak{h})$ and
$[\rho(x)u,v]_{\mathfrak{h}}=0$, we obtain that
$[\mu_{\mathfrak{g}}\boxplus\rho,\mu_{\mathfrak{h}}]_{\mathsf{B}}=0.$ Then, we
deduce that $\mathrm{d}$ is a derivation of
$(C^{*}(\mathfrak{h},\mathfrak{g}),\left\llbracket-,-\right\rrbracket)$.
Therefore,
$(C^{*}(\mathfrak{h},\mathfrak{g}),\left\llbracket-,-\right\rrbracket,\mathrm{d})$
is a differential graded Lie algebra.

Finally, for a degree 1 element $T\in\mathrm{Hom}(\mathfrak{h},\mathfrak{g})$,
we have

$\displaystyle\big{(}\mathrm{d}T+\frac{1}{2}\left\llbracket
T,T\right\rrbracket\big{)}(u,v)=-T[u,v]_{\mathfrak{h}}-T(\rho(Tu)v)+[Tu,Tv]_{\mathfrak{g}}.$

Thus, Maurer-Cartan elements are precisely nonabelian embedding tensors. ∎

Let $T:\mathfrak{h}\to\mathfrak{g}$ be a nonabelian embedding tensor. Since
$T$ is a Maurer-Cartan element of the differential graded Lie algebra
$(C^{*}(\mathfrak{h},\mathfrak{g}),\left\llbracket-,-\right\rrbracket,\mathrm{d})$
by Theorem 4.2, we deduce that

$d_{T}:=\mathrm{d}+\left\llbracket T,-\right\rrbracket$

is a graded derivation on the graded Lie algebra
$(C^{*}(\mathfrak{h},\mathfrak{g}),\left\llbracket-,-\right\rrbracket)$
satisfying $d^{2}_{T}=0$. Therefore,
$(C^{*}(\mathfrak{h},\mathfrak{g}),\left\llbracket-,-\right\rrbracket,d_{{T}})$
is a differential graded Lie algebra. This differential graded Lie algebra can
control deformations of nonabelian embedding tensors.

######  Theorem 4.3.

Let $T:\mathfrak{h}\to\mathfrak{g}$ be a nonabelian embedding tensor on a Lie
algebra $\mathfrak{g}$ with respect to a coherent action $(\mathfrak{h};\rho)$
and $T^{\prime}:\mathfrak{h}\to\mathfrak{g}$ a linear map. Then $T+T^{\prime}$
is still a nonabelian embedding tensor on $\mathfrak{g}$ with respect to the
coherent action $(\mathfrak{h};\rho)$ if and only if $T^{\prime}$ is a Maurer-
Cartan element of the differential graded Lie algebra
$(C^{*}(\mathfrak{h},\mathfrak{g}),\left\llbracket-,-\right\rrbracket,d_{{T}})$.

###### Proof.

By Theorem 4.2, $T+T^{\prime}$ is a nonabelian embedding tensor if and only if

$\displaystyle\mathrm{d}(T+T^{\prime})+\frac{1}{2}\left\llbracket
T+T^{\prime},T+T^{\prime}\right\rrbracket=0,$

which is equivalent to

$d_{T}T^{\prime}+\frac{1}{2}\left\llbracket
T^{\prime},T^{\prime}\right\rrbracket=0.$

That is, $T^{\prime}$ is a Maurer-Cartan element of
$(C^{*}(\mathfrak{h},\mathfrak{g}),\left\llbracket-,-\right\rrbracket,d_{{T}})$.
∎

##  5\. Cohomologies of nonabelian embedding tensors

In this section, we establish a cohomology theory for nonabelian embedding
tensors via the Loday-Pirashvili cohomology of the descendent Leibniz algebra
given in Corollary 2.11.

######  Definition 5.1.

([18]) Let $(V;\rho^{L},\rho^{R})$ be a representation of a Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$. The Loday-Pirashvili cohomology of
$\mathfrak{G}$ with coefficients in $V$ is the cohomology of the cochain
complex
$(C^{\bullet}(\mathfrak{G},V)=\oplus_{k=0}^{+\infty}C^{k}(\mathfrak{G},V),\partial)$,
where $C^{k}(\mathfrak{G},V)=\mathrm{Hom}(\otimes^{k}\mathfrak{G},V)$ and the
coboundary operator $\partial:C^{k}(\mathfrak{G},V)\to
C^{k+1}(\mathfrak{G},V)$ is defined by

$\displaystyle(\partial f)(x_{1},\cdots,x_{k+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(-1)^{i+1}\rho^{L}(x_{i})f(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1})+(-1)^{k+1}\rho^{R}(x_{k+1})f(x_{1},\cdots,x_{k})$
$\displaystyle+\sum_{1\leq i<j\leq
k+1}(-1)^{i}f(x_{1},\cdots,\hat{x_{i}},\cdots,x_{j-1},[x_{i},x_{j}]_{\mathfrak{G}},x_{j+1},\cdots,x_{k+1}),$

for all $x_{1},\cdots,x_{k+1}\in\mathfrak{G}$.

Let $T:\mathfrak{h}\to\mathfrak{g}$ be a nonabelian embedding tensor on a Lie
algebra $\mathfrak{g}$ with respect to a coherent action
$(\mathfrak{h};\rho)$. By Corollary 2.11,
$[u,v]_{T}:=[u,v]_{\mathfrak{h}}+\rho(Tu)v$ defines a Leibniz algebra
structure on $\mathfrak{h}$. Furthermore, define
$\rho^{L}:\mathfrak{h}\,\rightarrow\,\mathfrak{gl}(\mathfrak{g})$ and
$\rho^{R}:\mathfrak{h}\,\rightarrow\,\mathfrak{gl}(\mathfrak{g})$ by

(18)
$\rho^{L}(u)y:=[Tu,y]_{\mathfrak{g}},\quad\rho^{R}(v)x:=[x,Tv]_{\mathfrak{g}}-T(\rho(x)v).$

######  Lemma 5.2.

With above notations, $(\mathfrak{g};\rho^{L},\rho^{R})$ is a representation
of the Leibniz algebra $(\mathfrak{h},[-,-]_{T})$.

###### Proof.

By (3), for all $u,v\in\mathfrak{h}$ and $x\in\mathfrak{g}$, we have

$\displaystyle\Big{(}\rho^{L}([u,v]_{T})-[\rho^{L}(u),\rho^{L}(v)]\Big{)}(x)$
$\displaystyle=$
$\displaystyle[T[u,v]_{T},x]_{\mathfrak{g}}-[Tu,[Tv,x]_{\mathfrak{g}}]_{\mathfrak{g}}+[Tv,[Tu,x]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle=$
$\displaystyle[[Tu,Tv]_{\mathfrak{g}},x]_{\mathfrak{g}}-[Tu,[Tv,x]_{\mathfrak{g}}]_{\mathfrak{g}}+[Tv,[Tu,x]_{\mathfrak{g}}]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle 0,$

which implies that $\rho^{L}([u,v]_{T})=[\rho^{L}(u),\rho^{L}(v)]$.

By (1) and (3), for all $u,v\in\mathfrak{h}$ and $x\in\mathfrak{g}$, we have

$\displaystyle\Big{(}\rho^{R}([u,v]_{T})-[\rho^{L}(u),\rho^{R}(v)]\Big{)}(x)$
$\displaystyle=$
$\displaystyle[x,T[u,v]_{T}]_{\mathfrak{g}}-T(\rho(x)[u,v]_{T})-[Tu,[x,Tv]_{\mathfrak{g}}-T(\rho(x)v)]_{\mathfrak{g}}+[[Tu,x]_{\mathfrak{g}},Tv]_{\mathfrak{g}}-T(\rho([Tu,x]_{\mathfrak{g}})v)$
$\displaystyle=$
$\displaystyle[x,[Tu,{T}v]_{\mathfrak{g}}]_{\mathfrak{g}}-[Tu,[x,Tv]_{\mathfrak{g}}]_{\mathfrak{g}}+[[Tu,x]_{\mathfrak{g}},Tv]_{\mathfrak{g}}$
$\displaystyle-T[u,\rho(x)v]_{\mathfrak{g}}-T\rho(x)\rho(Tu)v+[Tu,T(\rho(x)v)]_{\mathfrak{g}}-T\rho(Tu)\rho(x)v+T\rho(x)\rho(Tu)v$
$\displaystyle=$ $\displaystyle 0,$

which implies that $\rho^{R}([u,v]_{T})=[\rho^{L}(u),\rho^{R}(v)]$.

Finally, also by (1) and (3), we have

$\displaystyle\Big{(}\rho^{R}(v)\circ\rho^{L}(u)+\rho^{R}(v)\circ\rho^{R}(u)\Big{)}(x)$
$\displaystyle=$
$\displaystyle\rho^{R}(v)(-T\rho(x))=-[T\rho(x)u,Tv]_{\mathfrak{g}}+T\rho(T\rho(x)u)v$
$\displaystyle=$ $\displaystyle-[\rho(x)u,v]_{\mathfrak{g}}=0,$

which implies that
$\rho^{R}(v)\circ\rho^{L}(u)+\rho^{R}(v)\circ\rho^{R}(u)=0.$ Therefore,
$(\mathfrak{g};\rho^{L},\rho^{R})$ is a representation of the Leibniz algebra
$(\mathfrak{h},[-,-]_{T})$. ∎

Let
$\partial_{T}:\mathrm{Hom}(\otimes^{k}\mathfrak{h},\mathfrak{g})\to\mathrm{Hom}(\otimes^{k+1}\mathfrak{h},\mathfrak{g})$
be the corresponding Loday-Pirashvili coboundary operator of the Leibniz
algebra $(\mathfrak{h},[-,-]_{T})$ with coefficients in
$(\mathfrak{g};\rho^{L},\rho^{R})$. More precisely,
$\partial_{T}:\mathrm{Hom}(\otimes^{k}\mathfrak{h},\mathfrak{g})\to\mathrm{Hom}(\otimes^{k+1}\mathfrak{h},\mathfrak{g})$
is given by

$\displaystyle\partial_{T}\theta(u_{1},\cdots,u_{k+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}(-1)^{i+1}[Tu_{i},\theta(u_{1},\cdots,\hat{u}_{i},\cdots,u_{k+1})]_{\mathfrak{g}}$
$\displaystyle+(-1)^{k+1}[\theta(u_{1},\cdots,u_{k}),Tu_{k+1}]_{\mathfrak{g}}+(-1)^{k}T(\rho(\theta(u_{1},\cdots,u_{k}))u_{k+1})$
$\displaystyle+\sum_{1\leq i<j\leq
k+1}(-1)^{i}\theta(u_{1},\cdots,\hat{u}_{i},\cdots,u_{j-1},\rho(Tu_{i})(u_{j})+[u_{i},u_{j}]_{\mathfrak{h}},u_{j+1},\cdots,u_{k+1}).$

Now we define a cohomology theory governing deformations of a nonabelian
embedding tensor $T:\mathfrak{h}\,\rightarrow\,\mathfrak{g}.$ Define the space
of $0$-cochains $\mathfrak{C}^{0}(T)$ to be $0$ and the space of $1$-cochains
$\mathfrak{C}^{1}(T)$ to be $\mathfrak{g}$. For $n\geq 2$, define the space of
$n$-cochains $\mathfrak{C}^{n}(T)$ as
$\mathfrak{C}^{n}(T)=\mathrm{Hom}(\otimes^{n-1}\mathfrak{h},\mathfrak{g})$.

######  Definition 5.3.

Let $T:\mathfrak{h}\,\rightarrow\,\mathfrak{g}$ be a nonabelian embedding
tensor on a Lie algebra $\mathfrak{g}$ with respect to a coherent action
$(\mathfrak{h};\rho)$. We define the cohomology of the nonabelian embedding
tensor $T$ to be the cohomology of the cochain complex
$(\mathfrak{C}^{\bullet}(T)=\oplus_{k=0}^{+\infty}\mathfrak{C}^{k}(T),\partial_{T})$.

Denote the set of $k$-cocycles by $\mathcal{Z}^{k}(T)$ and the set of
$k$-coboundaries by $\mathcal{B}^{k}(T)$. Denote by

(20) $\mathcal{H}^{k}(T)=\mathcal{Z}^{k}(T)/\mathcal{B}^{k}(T),\quad k\geq 0,$

the corresponding $k$-th cohomology group.

Comparing the coboundary operators $\partial_{T}$ given above and the
operators $d_{{T}}=\mathrm{d}+\left\llbracket T,-\right\rrbracket$ defined by
the Maurer-Cartan element $T$, we have

######  Proposition 5.4.

Let ${T}:\mathfrak{h}\to\mathfrak{g}$ be a nonabelian embedding tensor. Then
we have

(21)
$\displaystyle\partial_{T}\theta=(-1)^{k-1}d_{T}\theta,\quad\forall\theta\in\mathrm{Hom}(\otimes^{k}\mathfrak{h},\mathfrak{g}),\,\,k=1,2,\cdots,$

###### Proof.

For all $u_{1},u_{2},\cdots,u_{k+1}\in\mathfrak{h}$ and
$\theta\in\mathrm{Hom}(\otimes^{k}\mathfrak{h},\mathfrak{g})$, we have

$\displaystyle(d_{T}\theta)(u_{1},u_{2},\cdots,u_{k+1})$ $\displaystyle=$
$\displaystyle(\mathrm{d}\theta+\left\llbracket
T,\theta\right\rrbracket)(u_{1},u_{2},\cdots,u_{k+1})$ $\displaystyle=$
$\displaystyle\sum_{1\leq i<j\leq
k+1}(-1)^{k-1+i}\theta(u_{1},\cdots,\hat{u}_{i},\cdots,u_{j-1},[u_{i},u_{j}]_{\mathfrak{h}},u_{j+1},\cdots,u_{k+1})-T\big{(}\rho(\theta(u_{1},\cdots,u_{k}))u_{k+1}\big{)}$
$\displaystyle+\sum_{i=1}^{k+1}(-1)^{k+1}(-1)^{i-1}[Tu_{i},\theta(u_{1},\cdots,\hat{u}_{i},\cdots,u_{k+1})]_{\mathfrak{g}}$
$\displaystyle+\sum_{1\leq i<j\leq
k+1}(-1)^{(j-1+k-1)}(-1)^{j-i-1}\theta(u_{1},\cdots,\hat{u}_{i},\cdots,u_{j-1},\rho(T(u_{i}))u_{j},u_{j+11},\cdots,u_{k+1}),$

which implies that $\partial_{T}\theta=(-1)^{k-1}d_{T}\theta$. The proof is
finished. ∎

##  6\. Linear deformations of nonabelian embedding tensors

In this section, we study linear deformations of a nonabelian embedding tensor
using the cohomology theory given in the previous section. In particular, we
introduce the notion of a Nijenhuis element associated to a nonabelian
embedding tensor, which gives rise to a trivial linear deformation of the
nonabelian embedding tensor.

######  Definition 6.1.

Let $T$ be a nonabelian embedding tensor on a Lie algebra $\mathfrak{g}$ with
respect to a coherent action $(\mathfrak{h};\rho)$ and
$\mathfrak{T}:\mathfrak{h}\to\mathfrak{g}$ a linear map. If
$T_{t}=T+t\mathfrak{T}$ is still a nonabelian embedding tensor on the Lie
algebra $\mathfrak{g}$ with respect to the coherent action
$(\mathfrak{h};\rho)$ for all $t\in\mathbb{C}$, we say that $\mathfrak{T}$
generates a linear deformation of the nonabelian embedding tensor $T$.

By considering the coefficients of $t$ and $t^{2}$, one deduce that
$T_{t}=T+t\mathfrak{T}$ is a linear deformation of a nonabelian embedding
tensor $T$ if and only if for any $u,v\in\mathfrak{h}$,

(22)
$\displaystyle~{}[Tu,\mathfrak{T}v]_{\mathfrak{g}}+[\mathfrak{T}u,Tv]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
T(\rho(\mathfrak{T}u)v)+\mathfrak{T}(\rho(Tu)v+[u,v]_{\mathfrak{h}}),$ (23)
$\displaystyle~{}[\mathfrak{T}u,\mathfrak{T}v]_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle\mathfrak{T}(\rho(\mathfrak{T}u)v).$

Note that Eq. (22) means that $\mathfrak{T}$ is a 1-cocycle of the Leibniz
algebra $(\mathfrak{h},[\cdot,\cdot]_{T})$ with coefficients in the
representation $(\mathfrak{g};\rho^{L},\rho^{R})$ given by Lemma 5.2 and Eq.
(23) means that $\mathfrak{T}$ is an embedding tensor on the Lie algebra
$\mathfrak{g}$ with respect to the representation $(\mathfrak{h};\rho)$.

Let $(\mathfrak{G},[-,-])$ be a Leibniz algebra and
$\omega:\otimes^{2}\mathfrak{G}\to\mathfrak{G}$ be a linear map. If for any
$t\in\mathbb{C}$, the multiplication $[-,-]_{t}$ defined by

$[u,v]_{t}=[u,v]+t\omega(u,v),\;\forall u,v\in\mathfrak{G},$

also gives a Leibniz algebra structure, we say that $\omega$ generates a
linear deformation of the Leibniz algebra $(\mathfrak{G},[-,-])$.

The two types of linear deformations are related as follows.

######  Proposition 6.2.

If $\mathfrak{T}$ generates a linear deformation of a nonabelian embedding
tensor $T$ on a Lie algebra $\mathfrak{g}$ with respect to a coherent action
$(\mathfrak{h};\rho)$, then the product $\omega_{\mathfrak{T}}$ on
$\mathfrak{h}$ defined by

$\omega_{\mathfrak{T}}(u,v)=\rho(\mathfrak{T}u)v,\quad\forall
u,v\in\mathfrak{h},$

generates a linear deformation of the associated Leibniz algebra
$(\mathfrak{h},[-,-]_{T})$.

###### Proof.

We denote by $[-,-]_{t}$ the corresponding Leibniz algebra structure
associated to the nonabelian embedding tensor $T+t\mathfrak{T}$. Then we have

$\displaystyle[u,v]_{t}$ $\displaystyle=$
$\displaystyle\rho((T+t\mathfrak{T})u)v+[u,v]_{\mathfrak{h}}$ $\displaystyle=$
$\displaystyle\rho(Tu)v+[u,v]_{\mathfrak{h}}+t\rho(\mathfrak{T}u)v$
$\displaystyle=$
$\displaystyle[u,v]_{T}+t\omega_{\mathfrak{T}}(u,v),\quad\forall u,v\in V,$

which implies that $\omega_{\mathfrak{T}}$ generates a linear deformation of
$(\mathfrak{h},[-,-]_{T})$.∎

######  Definition 6.3.

Let $T$ be a nonabelian embedding tensor on a Lie algebra $\mathfrak{g}$ with
respect to a coherent action $(\mathfrak{h};\rho)$. Two linear deformations
$T^{1}_{t}=T+t\mathfrak{T}_{1}$ and $T^{2}_{t}=T+t\mathfrak{T}_{2}$ are said
to be equivalent if there exists an $x\in\mathfrak{g}$ such that
$({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x},{{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))$
is a homomorphism from $T^{2}_{t}$ to $T^{1}_{t}$. In particular, a linear
deformation $T_{t}=T+t\mathfrak{T}$ of a nonabelian embedding tensor $T$ is
said to be trivial if there exists an $x\in\mathfrak{g}$ such that
$({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x},{{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))$
is a homomorphism from $T_{t}$ to $T$.

Let
$({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x},{{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))$
be a homomorphism from $T^{2}_{t}$ to $T^{1}_{t}$. Then
${{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x}$ and
${{\rm{Id}}}_{\mathfrak{h}}+t\rho(x)$ are Lie algebra endomorphisms of
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ and
$(\mathfrak{h},[-,-]_{\mathfrak{h}})$ respectively. Thus, we have

$\displaystyle({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})[y,z]_{\mathfrak{g}}$
$\displaystyle=$
$\displaystyle[({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})(y),({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})(z)]_{\mathfrak{g}},\;\forall
y,z\in\mathfrak{g},$
$\displaystyle({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))[u,v]_{\mathfrak{h}}$
$\displaystyle=$
$\displaystyle[({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))(u),({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))(v)]_{\mathfrak{h}},\;\forall
u,v\in\mathfrak{h},$

which implies that $x$ satisfies

(24)
$[[x,y]_{\mathfrak{g}},[x,z]_{\mathfrak{g}}]_{\mathfrak{g}}=0,\quad\forall
y,z\in\mathfrak{g}.$

Then by Eq. (4), we get

$(T+t\mathfrak{T}_{1})({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))(u)=({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})(T+t\mathfrak{T}_{2})(u),\quad\forall
u\in\mathfrak{h},$

which implies

(25) $\displaystyle(\mathfrak{T}_{2}-\mathfrak{T}_{1})(u)$ $\displaystyle=$
$\displaystyle T\rho(x)u-[x,Tu]_{\mathfrak{g}},$ (26)
$\displaystyle\mathfrak{T}_{1}\rho(x)(u)$ $\displaystyle=$
$\displaystyle[x,\mathfrak{T}_{2}u]_{\mathfrak{g}},\;\forall
u\in\mathfrak{h}.$

Finally, Eq. (5) gives

$({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))\rho(y)(u)=\rho(({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})(y))({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))(u),\quad\forall
y\in\mathfrak{g},u\in\mathfrak{h},$

which implies that $x$ satisfies

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

Note that Eq. (25) means that
$\mathfrak{T}_{2}-\mathfrak{T}_{1}=\partial_{T}x$. Thus, we have the following
result:

######  Theorem 6.4.

Let $T$ be a nonabelian embedding tensor on a Lie algebra $\mathfrak{g}$ with
respect to a coherent action $(\mathfrak{h};\rho)$. If two linear deformations
$T^{1}_{t}=T+t\mathfrak{T}_{1}$ and $T^{2}_{t}=T+t\mathfrak{T}_{2}$ are
equivalent, then $\mathfrak{T}_{1}$ and $\mathfrak{T}_{2}$ are in the same
cohomology class of $\mathcal{H}^{2}(T)$.

######  Definition 6.5.

Let $T$ be a nonabelian embedding tensor on a Lie algebra $\mathfrak{g}$ with
respect to a coherent action $(\mathfrak{h};\rho)$. An element
$x\in\mathfrak{g}$ is called a Nijenhuis element associated to $T$ if $x$
satisfies Eqs. (24), (27) and the equation

(28)
$\displaystyle~{}[x,T\rho(x)u-[x,Tu]_{\mathfrak{g}}]_{\mathfrak{g}}=0,\quad\forall
u\in\mathfrak{h}.$

Denote by $\mathrm{Nij}(T)$ the set of Nijenhuis elements associated to a
nonabelian embedding tensor $T$.

By Eqs. (24)-(27), it is obvious that a trivial linear deformation gives rise
to a Nijenhuis element. Conversely, a Nijenhuis element can also generate a
trivial linear deformation as the following theorem shows.

######  Theorem 6.6.

Let $T$ be a nonabelian embedding tensor on a Lie algebra $\mathfrak{g}$ with
respect to a coherent action $(\mathfrak{h};\rho)$. Then for any
$x\in\mathrm{Nij}(T)$, $T_{t}=T+t\mathfrak{T}$ with
$\mathfrak{T}=\partial_{T}x$ is a trivial linear deformation of the nonabelian
embedding tensor $T$.

We need the following lemma to prove this theorem.

######  Lemma 6.7.

Let $T$ be a nonabelian embedding tensor on a Lie algebra $\mathfrak{g}$ with
respect to a coherent action $(\mathfrak{h};\rho)$. Let
$\phi_{\mathfrak{g}}:\mathfrak{g}\to\mathfrak{g}$ and
$\phi_{\mathfrak{h}}:\mathfrak{h}\to\mathfrak{h}$ be two Lie algebra
isomorphisms such that Eq. (4) and Eq. (5) hold. Then
$\phi_{\mathfrak{g}}^{-1}\circ{T}\circ\phi_{\mathfrak{h}}$ is a nonabelian
embedding tensor on the Lie algebra $\mathfrak{g}$ with respect to the
coherent action $(\mathfrak{h};\rho)$.

###### Proof.

It follows from straightforward computations. ∎

The proof of Theorem 6.6: For any Nijenhuis element $x\in\mathrm{Nij}({T})$,
we define a linear map $\mathfrak{T}:\mathfrak{h}\to\mathfrak{g}$ by

(29)
$\displaystyle\mathfrak{T}(u)=(\partial_{T}x)(u)=T\rho(x)u-[x,Tu]_{\mathfrak{g}},~{}\forall
u\in\mathfrak{h}.$

Let $T_{t}=T+t\mathfrak{T}$. By the definition of Nijenhuis elements, for all
$y,z\in\mathfrak{g},u,v\in\mathfrak{h}$, we have

$\displaystyle({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})[y,z]_{\mathfrak{g}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Nij1}}}{{=}}$
$\displaystyle[({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})(y),({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})(z)]_{\mathfrak{g}},$
$\displaystyle({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))[u,v]_{\mathfrak{h}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:extracon}}}{{=}}$
$\displaystyle[({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))(u),({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))(v)]_{\mathfrak{h}},$
$\displaystyle({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})\circ T_{t}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Nijenhuis}}}{{=}}$
$\displaystyle T\circ({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x)),$
$\displaystyle({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))\rho(y)u$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Nij3}}}{{=}}$
$\displaystyle\rho(({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})y)({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))(u).$

Since $\mathrm{ad}_{x}$ and $\rho(x)$ are linear transformations of finite-
dimensional $\mathbb{C}$-vector spaces, it follows that
${{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x}$ and
${{\rm{Id}}}_{\mathfrak{h}}+t\rho(x)$ are Lie algebra isomorphisms for $|t|$
sufficiently small. Thus, by Lemma 6.7,

$\displaystyle({{\rm{Id}}}_{\mathfrak{g}}+t\mathrm{ad}_{x})^{-1}\circ{T}\circ({{\rm{Id}}}_{\mathfrak{h}}+t\rho(x))$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{+\infty}(-t\mathrm{ad}_{x})^{i}\circ(T+tT\rho(x))$
$\displaystyle=$ $\displaystyle T+t(T\rho(x)-\mathrm{ad}_{x}\circ
T)+\sum_{i=1}^{+\infty}(-1)^{i}t^{i+1}\mathrm{ad}_{x}^{i}\circ(T\rho(x)-\mathrm{ad}_{x}\circ
T)$ $\displaystyle\stackrel{{\scriptstyle\eqref{eq:Nijenhuis}}}{{=}}$
$\displaystyle T+t(T\rho(x)-\mathrm{ad}_{x}\circ T)=T+t\mathfrak{T}=T_{t}$

is a nonabelian embedding tensor on the Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ with respect to the coherent action
$(\mathfrak{h};\rho)$ for $|t|$ sufficiently small. Thus, $\mathfrak{T}$ given
by Eq. (29) satisfies the conditions (22) and (23). Therefore, ${T}_{t}$ is a
nonabelian embedding tensor for all $t$, which means that $\mathfrak{T}$ given
by Eq. (29) generates a linear deformation of $T$. It is straightforward to
see that this linear deformation is trivial.∎

Now we recall the notion of a Nijenhuis operator on a Leibniz algebra, which
gives rise to a trivial linear deformation of a Leibniz algebra.

######  Definition 6.8.

([5]) A linear map $N:\mathfrak{G}\to\mathfrak{G}$ on a Leibniz algebra
$(\mathfrak{G},[-,-]_{\mathfrak{G}})$ is called a Nijenhuis operator if

(30)
$[Nu,Nv]_{\mathfrak{G}}=N([Nu,v]_{\mathfrak{G}}+[u,Nv]_{\mathfrak{G}}-N[u,v]_{\mathfrak{G}}),\quad
u,v\in\mathfrak{G}.$

For its connection with a Nijenhuis element associated to a nonabelian
embedding tensor, we have the following result.

######  Proposition 6.9.

Let $x\in\mathfrak{g}$ be a Nijenhuis element associated to a nonabelian
embedding tensor $T$ on a Lie algebra $\mathfrak{g}$ with respect to a
coherent action $(\mathfrak{h};\rho)$. Then $\rho(x)$ is a Nijenhuis operator
on the descendent Leibniz algebra $(\mathfrak{h},[-,-]_{T})$.

###### Proof.

For all $u,v\in\mathfrak{h}$, we have

$\displaystyle\rho(x)([\rho(x)u,v]_{T}+[u,\rho(x)v]_{T}-\rho(x)[u,v]_{T})-[\rho(x)u,\rho(x)v]_{T}$
$\displaystyle\stackrel{{\scriptstyle\eqref{new-leibniz}}}{{=}}$
$\displaystyle\rho(x)\Big{(}\rho(T\rho(x)u)v+[\rho(x)u,v]_{\mathfrak{h}}+\rho(Tu)\rho(x)v+[u,\rho(x)v]_{\mathfrak{h}}-\rho(x)(\rho(Tu)v+[u,v]_{\mathfrak{h}})\Big{)}$
$\displaystyle-\rho(T\rho(x)u)\rho(x)v-[\rho(x)u,\rho(x)v]_{\mathfrak{h}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:extracon}}}{{=}}$
$\displaystyle\rho(x)\big{(}\rho(T\rho(x)u)v+\rho(Tu)\rho(x)v-\rho(x)\rho(Tu)v\big{)}-\rho(T\rho(x)u)\rho(x)v$
$\displaystyle=$
$\displaystyle\rho([x,T\rho(x)u]_{\mathfrak{g}})v+\rho(x)\rho([Tu,x]_{\mathfrak{g}})v$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Nijenhuis}}}{{=}}$
$\displaystyle\rho([x,[x,Tu]_{\mathfrak{g}}]_{\mathfrak{g}})v+\rho(x)\rho([Tu,x]_{\mathfrak{g}})v{=}-\rho([x,Tu]_{\mathfrak{g}})\rho(x)v$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Nij3}}}{{=}}$ $\displaystyle
0.$

Thus, we deduce that $\rho(x)$ is a Nijenhuis operator on the Leibniz algebra
$(\mathfrak{h},[-,-]_{T})$. ∎

We end this section by giving a concrete example of a linear deformation of a
nonabelian embedding tensor $T$ on the Heisenberg Lie algebra
$H_{3}(\mathbb{C})$.

######  Example 6.10.

Let $H_{3}(\mathbb{C})$ be the Heisenberg Lie algebra. Then
$T_{t}=\left(\begin{array}[]{ccc}\frac{t+\sqrt{t^{2}+4t}}{2}&0&0\\\
0&\frac{t+\sqrt{t^{2}+4t}}{2}&0\\\ a&b&t\end{array}\right)$ is a linear
deformation of the nonabelian embedding tensor
$T=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ a&b&0\end{array}\right)$ on
$H_{3}(\mathbb{C})$ with respect to the adjoint action for all
$a,b\in\mathbb{C}$.

A conflict of interest statement. On behalf of all authors, the corresponding
author states that there is no conflict of interest.

Data availability. Data sharing not applicable to this article as no datasets
were generated or analysed during the current study.

Acknowledgements. This research is supported by NSFC (11922110,12001228). We
give warmest thank to Jim Stasheff, Thomas Strobl and Chenchang Zhu for
helpful comments.

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