Non-abelian extensions and Wells exact sequences of Lie-Yamaguti algebras

Qinxiu Sun Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023 [email protected] and Zhen Li Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023 [email protected]

Abstract.

The goal of the present paper is to investigate non-abelian extensions of Lie-Yamaguti algebras and explore extensibility of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras. First, we study non-abelian extensions of Lie-Yamaguti algebras and classify the non-abelian extensions in terms of non-abelian cohomology groups. Next, we characterize the non-abelian extensions in terms of Maurer-Cartan elements. Moreover, we discuss the equivalent conditions of the extensibility of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras, and derive the fundamental sequences of Wells in the context of Lie-Yamaguti algebras. Finally, we discuss the previous results in the case of abelian extensions of Lie-Yamaguti algebras.

Key words and phrases:

Lie-Yamaguti algebra, non-abelian extension, Maurer-Cartan element, extensibility, Wells exact sequence

2010 Mathematics Subject Classification:

17A30, 17A36, 17A40, 17B99

1. Introduction

A Lie-Yamaguti algebra first appeared in Nomizu’s work on the affine invariant connections on homogeneous spaces in 1950’s [23], which was a generalization of Lie algebras and Lie triple systems. In 1960’s, Yamaguti introduced an algebraic structure and named it a general Lie triple system or a Lie triple algebra [34, 33]. Later, the cohomology theory of this object was investigated in [34] by Yamaguti. In the study of Courant algebroids, Kinyon and Weinstein named general Lie triple systems by Lie–Yamaguti algebras [19]. Since then, Lie-Yamaguti algebras have attracted much attention and are widely explored. For example, the irreducible modules of Lie–Yamaguti algebras were considered in [3, 2], deformations and extensions of Lie-Yamaguti algebras were investigated in [20, 35]. The relative Rota-Bxter operators on Lie-Yamaguti algebras and their cohomologies were considered in [25, 31, 32].

Extensions are useful mathematical objects to understand the underlying structures. The non-abelian extension is a relatively general one among various extensions (e.g. central extensions, abelian extensions, non-abelian extensions etc.). Non-abelian extensions were first developed by Eilenberg and Maclane [10], which induced to the low dimensional non-abelian cohomology group. Then numerous works have been devoted to non-abelian extensions of various kinds of algebras, such as Lie (super)algebras, Leibniz algebras, Lie 2-algebras, Lie Yagamuti algebras, associative conformal algebras, Rota-Baxter groups, Rota-Baxter Lie algebras and Rota-Baxter Leibniz algebras, see [5, 7, 11, 13, 14, 17, 21, 22, 16] and their references. The abelian extensions of Lie Yagamuti algebras were considered in [13]. But little is known about the non-abelian extensions of Lie Yagamuti algebras. This is the first motivation for writing this paper.

Another interesting study related to extensions of algebraic structures is given by the extensibility or inducibility of a pair of automorphisms. When a pair of automorphisms is inducible? This problem was first considered by Wells [29] for abstract groups and further studied in [18, 24]. Since then, several authors have studied this subject further, see [16, 13, 14, 15, 22] and references therein. The extensibility problem of a pair of derivations on abelian extensions was investigated in [6, 30]. Recently, the extensibility problem of a pair of derivations and automorphisms was extended to the context of abelian extensions of Lie coalgebras [9]. As byproducts, the Wells short exact sequences were obtained for various kinds of algebras [7, 8, 13, 14, 15, 18, 22, 16], which connected the relative automorphism groups and the non-abelian second cohomology groups. Inspired by these results, we investigate extensibility of a pair of automorphisms on a non-abelian extension of Lie Yagamuti algebras. This is another motivation for writing the present paper. Moreover, we give necessary and sufficient conditions for a pair of automorphisms to be extensible, and derive the analogue of the Wells short exact sequences in the context of non-abelian extensions of Lie Yagamuti algebras.

The paper is organized as follows. In Section 2, we recall the definition of Lie-Yamaguti algebras and their representations. We also recall some basic information about the cohomology groups of Lie-Yamaguti algebras. In Section 3, we investigate non-abelian extensions and classify the non-abelian extensions using the non-abelian cohomology groups. In Section 4, we characterize equivalent non-abelian extensions using Maurer-Cartan elements. In Section 5, we study the extensibility problem of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras. In Section 6, we derive Wells short exact sequences in the context of non-abelian extensions of Lie-Yamaguti algebras. Finally, we discuss the previous results in the case of abelian extensions of Lie-Yamaguti algebras.

Throughout the paper, let $k$ be a field. Unless otherwise specified, all vector spaces and algebras are over $k$ .

2. Preliminaries on Lie-Yamaguti algebras

We recall the notions of Lie-Yamaguti algebras, representations and their cohomology theory. For the details see [34, 33].

Definition 2.1.

A Lie-Yamaguti algebra is a vector space $\mathfrak{g}$ with a bilinear map $[\cdot,\cdot]:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$ and a trilinear map $\{\ ,\ ,\ \}:\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$ , satisfying

(1)

[x_{1},x_{2}]+[x_{2},x_{1}]=0,~{}~{}\{x_{1},x_{2},x_{3}\}+\{x_{2},x_{1},x_{3}\}=0,

(2)

[[x_{1},x_{2}],x_{3}]+[[x_{2},x_{3}],x_{1}]+[[x_{3},x_{1}],x_{2}]+\{x_{1},x_{2},x_{3}\}+\{x_{2},x_{3},x_{1}\}+\{x_{3},x_{1},x_{2}\}=0,

(3)

\{[x_{1},x_{2}],x_{3},y_{1}\}+\{[x_{2},x_{3}],x_{1},y_{1}\}+\{[x_{3},x_{1}],x_{2},y_{1}\}=0,

(4)

\{x_{1},x_{2},[y_{1},y_{2}]\}=[\{x_{1},x_{2},y_{1}\},y_{2}]+[y_{1},\{x_{1},x_{2},y_{2}\}],

(5)

\{x_{1},x_{2},\{y_{1},y_{2},y_{3}\}\}=\{\{x_{1},x_{2},y_{1}\},y_{2},y_{3}\}+\{y_{1},\{x_{1},x_{2},y_{2}\},y_{3}\}+\{y_{1},y_{2},\{x_{1},x_{2},y_{3}\}\},

for all $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}\in\mathfrak{g}$ . Denote it by $(\mathfrak{g},[\ ,\ ],\{\ ,\ ,\ \})$ or simply by $\mathfrak{g}$ .

A subspace $L$ of $\mathfrak{g}$ is an ideal of $\mathfrak{g}$ if $[L,\mathfrak{g}]\subseteq L,\{L,\mathfrak{g},\mathfrak{g}\}\subseteq L$ and $\{\mathfrak{g},\mathfrak{g},L\}\subseteq L$ . An ideal $L$ of $\mathfrak{g}$ is said to be an abelian ideal of $\mathfrak{g}$ if $[L,L]=0$ and $\{L,L,\mathfrak{g}\}=\{\mathfrak{g},L,L\}=\{L,\mathfrak{g},L\}=0$ .

Definition 2.2.

A representation of a Lie-Yamaguti algebra $\mathfrak{g}$ consists of a vector space $V$ together with a linear map $\mu:\mathfrak{g}\longrightarrow{gl}(V)$ and bilinear maps $\theta,D:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow{gl}(V)$ satisfying

(6)

\theta([x_{1},x_{2}],x_{3})=\theta(x_{1},x_{3})\mu(x_{2})-\theta(x_{2},x_{3})\mu(x_{1}),

(7)

[D(x_{1},x_{2}),\mu(y_{1})]=\mu(\{x_{1},x_{2},y_{1}\}),

(8)

\theta(x_{1},[y_{1},y_{2}])=\mu(y_{1})\theta(x_{1},y_{2})-\mu(y_{2})\theta(x_{1},y_{1}),

(9)

[D(x_{1},x_{2}),\theta(y_{1},y_{2})]=\theta(\{x_{1},x_{2},y_{1}\},y_{2})+\theta(y_{1},\{x_{1},x_{2},y_{2}\}),

(10)

\theta(x_{1},\{y_{1},y_{2},y_{3}\})=\theta(y_{2},y_{3})\theta(x_{1},y_{1})-\theta(y_{1},y_{3})\theta(x_{1},y_{2})+D(y_{1},y_{2})\theta(x_{1},y_{3}),

(11)

D(x_{1},x_{2})-\theta(x_{2},x_{1})+\theta(x_{1},x_{2})+\mu([x_{1},x_{2}])-[\mu(x_{1}),\mu(x_{2})]=0,

for all $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}\in\mathfrak{g}$ . Denote the representation of $\mathfrak{g}$ by $(V,\mu,\theta,D)$ or simply by $V$ .

When $(V,\mu,\theta,D)$ is a representation of $\mathfrak{g}$ , by a direct computation, the following conditions are also satisfied:

(12)

D([x_{1},x_{2}],x_{3})+D([x_{2},x_{3}],x_{1})+D([x_{3},x_{1}],x_{2})=0,

(13)

D(\{x_{1},x_{2},x_{3}\},x_{4})+D(x_{3},\{x_{1},x_{2},x_{4}\})=[D(x_{1},x_{2}),D(x_{3},x_{4})],

(14)

\theta(\{y_{1},y_{2},y_{3}\},x_{1})=\theta(y_{1},x_{1})\theta(y_{3},y_{2})-\theta(y_{2},x_{1})\theta(y_{3},y_{1})-\theta(y_{3},x_{1})D(y_{1},y_{2}).

Proposition 2.3.

Let $(\mathfrak{g},[\ ,\ ]_{\mathfrak{g}},\{\ ,\ ,\ \}_{\mathfrak{g}})$ be a Lie-Yamaguti algebra and $V$ a vector space. Assume that $\mu:\mathfrak{g}\longrightarrow{gl}(V)$ is a linear map and $\theta,D:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow{gl}(V)$ are bilinear maps. Then $(V,\mu,\theta,D)$ is a representation of $\mathfrak{g}$ if and only if $(\mathfrak{g}\oplus V,[\ ,\ ],\{\ ,\ ,\ \})$ is a Lie-Yamaguti algebra, where

\{x+u,y+v,z+w\}=\{x,y,z\}_{\mathfrak{g}}+\theta(y,z)u-\theta(x,z)v+D(x,y)w,

and

[x+u,y+v]=[x,y]_{\mathfrak{g}}+\mu(x)v-\mu(y)u,

for all $x,y,z\in T,u,v,w\in V$ . The Lie-Yamaguti algebra $(\mathfrak{g}\oplus V,[\ ,\ ],\{\ ,\ ,\ \})$ is called the semidirect product Lie-Yamaguti algebra. Denote it simply by $\mathfrak{g}\ltimes V$ .

Example 2.4.

Let $\mathfrak{g}$ be a Lie-Yamaguti algebra. Define ${ad}:\mathfrak{g}\longrightarrow{gl}(\mathfrak{g}),~{}R,L:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow{gl}(\mathfrak{g})$ respectively by ${ad}(x)(y)=[x,y],R(x,y)(z)=\{z,x,y\}$ and $L(x,y)(z)=\{x,y,z\}$ . Then $(\mathfrak{g},{ad},R,L)$ is a representation of $\mathfrak{g}$ , which is called the adjoint representation.

Next, we recall the cohomology of Lie-Yamaguti algebras following [33]. Let $\mathfrak{g}$ be a Lie-Yamaguti algebra and $(V,\mu,\theta,D)$ be its representation. The cochain complex is given as follows:

•

Set $C^{1}(\mathfrak{g},V)=\mathrm{Hom}(\mathfrak{g},V)$ and $C^{0}(\mathfrak{g},V)$ be the subspace spanned by the diagonal elements $(f,f)\in C^{1}(\mathfrak{g},V)\times C^{1}(\mathfrak{g},V)$ .

•

(For $n\geq 2$ ) Set $C^{n}(\mathfrak{g},V)=\mathrm{Hom}(\otimes^{n}\mathfrak{g},V)$ , where the space of all $n$ -linear maps $f\in C^{n}(\mathfrak{g},V)$ satisfying

f(x_{1},\cdot\cdot\cdot,x_{2i-1},x_{2i},\cdot\cdot\cdot,x_{n})=0,~{}~{}\hbox{if}~{}x_{2i-1}=x_{2i},~{}\forall~{}i=1,2,\cdot\cdot\cdot,[\frac{n}{2}].

Then, for all $n\geq 1$ , put the $(2n,2n+1)$ -cochain groups

C^{(2n,2n+1)}(\mathfrak{g},V)=C^{2n}(\mathfrak{g},V)\times C^{2n+1}(\mathfrak{g},V)

and

C^{(3,4)}(\mathfrak{g},V)=C^{3}(\mathfrak{g},V)\times C^{4}(\mathfrak{g},V).

Define the coboundary operator $\delta=(\delta_{I},\delta_{II})$ in the following cochain complex of $\mathfrak{g}$ with coefficents in $V$ :

For $n\geq 1$ , the coboundary operator $\delta=(\delta_{I},\delta_{II}):C^{(2n,2n+1)}(\mathfrak{g},V)\longrightarrow C^{(2n+2,2n+3)}(\mathfrak{g},V)$ is defined by

			$\displaystyle(\delta_{I}f)(x_{1},\cdot\cdot\cdot,x_{2n+2})$
		$\displaystyle=$	$\displaystyle\mu(x_{2n+1})g(x_{1},\cdot\cdot\cdot,x_{2n},x_{2n+2})-\mu(x_{2n+2})g(x_{1},\cdot\cdot\cdot,x_{2n+1})-g(x_{1},\cdot\cdot\cdot,x_{2n},[x_{2n+1},x_{2n+2}])$
			$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k+1}D(x_{2k-1},x_{2k})f(x_{1},\cdot\cdot\cdot,x_{2k-2},x_{2k+1},\cdot\cdot\cdot,x_{2n+2})$
			$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+2}(-1)^{n+k}f(x_{1},\cdot\cdot\cdot,x_{2k-2},x_{2k+1},\cdot\cdot\cdot,\{x_{2k-1},x_{2k},x_{j}\},\cdot\cdot\cdot,x_{2n+2})$

and

			$\displaystyle(\delta_{II}g)(x_{1},\cdot\cdot\cdot,x_{2n+3})$
		$\displaystyle=$	$\displaystyle\theta(x_{2n+2},x_{2n+3})g(x_{1},\cdot\cdot\cdot,x_{2n+1})-\theta(x_{2n+1},x_{2n+3})g(x_{1},\cdot\cdot\cdot,x_{2n},x_{2n+2})$
			$\displaystyle+\sum_{k=1}^{n+1}(-1)^{n+k+1}D(x_{2k-1},x_{2k})g(x_{1},\cdot\cdot\cdot,x_{2k+1},\cdot\cdot\cdot,x_{2n+3})$
			$\displaystyle+\sum_{k=1}^{n+1}\sum_{j=2k+1}^{2n+3}(-1)^{n+k}g(x_{1},\cdot\cdot\cdot,x_{2k+1},\cdot\cdot\cdot,\{x_{2k-1},x_{2k},x_{j}\},\cdot\cdot\cdot,x_{2n+3})$

for any pair $(f,g)\in C^{(2n,2n+1)}(\mathfrak{g},V)$ and $x_{1},\cdot\cdot\cdot,x_{2n+3}\in\mathfrak{g}$ . And when $n=0$ , the coboundary operator

\delta=(\delta_{I},\delta_{II}):C^{0}(\mathfrak{g},V)\longrightarrow C^{2}(\mathfrak{g},V)\times C^{3}(\mathfrak{g},V)

is defined as follows:

(\delta_{I}f)(x_{1},x_{2})=\mu(x_{1})f(x_{2})-\mu(x_{2})f(x_{1})-f([x_{1},x_{2}]),

(\delta_{II}f)(x_{1},x_{2},x_{3})=\theta(x_{2},x_{3})f(x_{1})-\theta(x_{1},x_{3})f(x_{2})+D(x_{1},x_{2})f(x_{3})-f(\{x_{1},x_{2},x_{3}\})

for any $f\in C^{1}(\mathfrak{g},V)$ and $x_{1},x_{2},x_{3}\in\mathfrak{g}$ .

Define $\delta^{*}=(\delta_{I}^{*},\delta_{II}^{*}):C^{(2,3)}(\mathfrak{g},V)\longrightarrow C^{(3,4)}(\mathfrak{g},V)$ by

	$\displaystyle\delta_{I}^{*}f(x,y,z)=$	$\displaystyle-\mu(x)f(y,z)-\mu(y)f(z,x)-\mu(z)f(x,y)+f([x,y],z)+f([y,z],x)$
		$\displaystyle+f([z,x],y)+g(x,y,z)+g(y,z,x)+g(z,x,y),$

	$\displaystyle\delta_{II}^{*}f(x,y,z,w)=$	$\displaystyle\theta(x,w)f(y,z)+\theta(y,w)f(z,x)+\theta(z,w)f(x,y)+g([x,y],z,w)$
		$\displaystyle+g([y,z],x,w)+g([z,x],y,w),$

for all $(f,g)\in C^{(2,3)}(\mathfrak{g},V)$ and $x,y,z,w\in\mathfrak{g}$ .

Denote the set of all the $(2n,2n+1)$ -cocycles and the $(2n,2n+1)$ -coboundaries, respectively by $Z^{(2n,2n+1)}(\mathfrak{g},V)$ and $B^{(2n,2n+1)}(\mathfrak{g},V)$ . In particular,

Z^{(2,3)}(\mathfrak{g},V)=\{(f,g)\in C^{(2,3)}(\mathfrak{g},V)|\delta(f,g)=0,\delta^{*}(f,g)=0\},

B^{(2,3)}(\mathfrak{g},V)=\{(\delta_{I}(f),\delta_{II}(f))|f\in C^{1}(\mathfrak{g},V)\}.

Define $H^{1}(\mathfrak{g},V)=\{f\in C^{1}(\mathfrak{g},V)|\delta_{I}(f)=0,\delta_{II}(f)=0\}$ , which is called the first cohomology group of $\mathfrak{g}$ with coefficients in the representation $V$ and define

H^{(2n,2n+1)}(\mathfrak{g},V)=Z^{(2n,2n+1)}(\mathfrak{g},V)/B^{(2n,2n+1)}(\mathfrak{g},V),~{}~{}n\geq 1

which is called the (2n,2n+1)-cohomology group of $\mathfrak{g}$ with coefficients in the representation $V$ .

3. Non-abelian extensions and non-abelian (2,3)-cocycles of Lie-Yamaguti algebras

In this section, we are devoted to considering non-abelian extensions and non-abelian (2,3)-cocycles of Lie-Yamaguti algebras.

Definition 3.1.

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie-Yamaguti algebras. A non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ is a Lie-Yamaguti algebra $\hat{\mathfrak{g}}$ , which fits into a short exact sequence of Lie-Yamaguti algebras

\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0.

When $\mathfrak{h}$ is an abelian ideal of $\hat{\mathfrak{g}}$ , the extension $\mathcal{E}$ is called an abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ . Denote an extension as above simply by $\hat{\mathfrak{g}}$ or $\mathcal{E}$ . A section of $p$ is a linear map $s:\mathfrak{g}\longrightarrow\hat{\mathfrak{g}}$ such that $ps=I_{\mathfrak{g}}$ .

Definition 3.2.

Let $\hat{\mathfrak{g}}_{1}$ and $\hat{\mathfrak{g}}_{2}$ be two non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ . They are said to be equivalent if there is a homomorphism of Lie-Yamaguti algebras $f:\hat{\mathfrak{g}}_{1}\longrightarrow\hat{\mathfrak{g}}_{2}$ such that the following commutative diagram holds:

(15)

Denote by $\mathcal{E}_{nab}(\mathfrak{g},\mathfrak{h})$ the set of all non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ .

Next, we define a non-abelian cohomology group and show that the non-abelian extensions are classified by the non-abelian cohomology groups.

Definition 3.3.

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie-Yamaguti algebras. A non-abelian (2,3)-cocycle on $\mathfrak{g}$ with values in $\mathfrak{h}$ is a septuple $(\chi,\omega,\mu,\theta,D,\rho,T)$ of maps such that $\omega:\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h}$ is trilinear, $\chi:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h},~{}\theta,D:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h})$ are bilinear and $\mu:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h}),~{}\rho,T:\mathfrak{g}\longrightarrow\mathrm{Hom}(\mathfrak{h}\wedge\mathfrak{h},\mathfrak{h})$ are linear, and the following five parts of identities are satisfied for all $x_{i},y_{i},x,y,z\in\mathfrak{g}~{}(i=1,2,3),a,b,c\in\mathfrak{h}$ ,

•

Those resembling Eq. (1):

(16) $\chi([x,y]_{\mathfrak{g}}+\chi(y,x)=0,~{}~{}\omega(x,y,z)+\omega(y,x,z)=0,$

(17) $D(x,y)a+D(y,x)a=0,~{}T(x)(a,b)+T(x)(b,a)=0.$

•

Those resembling Eq. (2):

		$\displaystyle\chi([x,y]_{\mathfrak{g}},z)-\mu(z)\chi(x,y)+\omega(x,y,z)+\chi([y,z]_{\mathfrak{g}},x)-\mu(x)\chi(y,z)+\omega(y,z,x)$
(18)			$\displaystyle+\chi([z,x]_{\mathfrak{g}},y)-\mu(y)\chi(z,x)+\omega(z,x,y)=0,$

(19)

\mu([x,y]_{\mathfrak{g}})a+[\chi(x,y),a]_{\mathfrak{h}}+D(x,y)a-\mu(x)\mu(y)a-\theta(y,x)a+\mu(y)\mu(x)a+\theta(x,y)a=0,

(20)

[\mu(x)a,b]_{\mathfrak{h}}+\rho(x)(a,b)-\mu(x)[a,b]_{\mathfrak{h}}+T(x)(a,b)-[\mu(x)b,a]_{\mathfrak{h}}-\rho(x)(b,a)=0.

•

Those resembling Eq. (3):

(21)

\theta(z,w)\chi(x,y)+\theta(x,w)\chi(y,z)+\theta(y,w)\chi(z,x)=0,

		$\displaystyle D([x,y]_{\mathfrak{g}},z)a-\rho(z)(\chi(x,y),a)+D([y,z]_{\mathfrak{g}},x)a-\rho(x)(\chi(y,z),a)$
(22)			$\displaystyle+D([z,x]_{\mathfrak{g}},y)a-\rho(y)(\chi(z,x),a)=0,$

(23)

\theta([x,y]_{\mathfrak{g}},z)a-T(z)(\chi(x,y),a)-\theta(x,z)\mu(y)a+\theta(y,z)\mu(x)a=0,

(24)

\rho([x,y]_{\mathfrak{g}})(a,b)+\{\chi(x,y),a,b\}_{\mathfrak{h}}-\rho(x)(\mu(y)a,b)+\rho(y)(\mu(x)a,b)=0,

(25)

T(y)(\mu(x)a,b)+\theta(x,y)[a,b]_{\mathfrak{h}}-T(y)(\mu(x)b,a)=0,

(26)

\{\mu(x)a,b,c\}_{\mathfrak{h}}-\rho(x)([a,b]_{\mathfrak{h}},c)-\{\mu(x)b,a,c\}_{\mathfrak{h}}=0,

(27)

T(x)([a,b]_{\mathfrak{h}},c)+T(x)([b,c]_{\mathfrak{h}},a)+T(x)([c,a]_{\mathfrak{h}},b)=0.

•

Those resembling Eq. (4):

(28)

D(x,y)\chi(z,w)=\chi(\{x,y,z\}_{\mathfrak{g}},w)-\mu(w)\omega(x,y,z)+\mu(z)\omega(x,y,w)+\chi(z,\{x,y,w\}_{\mathfrak{g}}),

(29)

D(x,y)\mu(z)a=\mu(\{x,y,z\}_{\mathfrak{g}})a+[\omega(x,y,z),a]_{\mathfrak{h}}+\mu(z)D(x,y)a,

(30)

\theta(x,[y,z]_{\mathfrak{g}})a-\rho(x)(a,\chi(y,z))=\mu(y)\theta(x,z)a-\mu(z)\theta(x,y)a,

(31)

D(x,y)[a,b]_{\mathfrak{h}}=[D(x,y)a,b]_{\mathfrak{h}}+[a,D(x,y)b]_{\mathfrak{h}},

(32)

\rho(x)(a,\mu(y)b)+[\theta(x,y)a,b]_{\mathfrak{h}}-\mu(y)\rho(x)(a,b)=0,

(33)

T([x,y]_{\mathfrak{g}})(a,b)+\{a,b,\chi(x,y)\}_{\mathfrak{h}}-\mu(x)T(y)(a,b)+\mu(y)T(x)(a,b)=0,

(34)

\{a,b,\mu(x)c\}_{\mathfrak{h}}=\mu(x)\{a,b,c\}_{\mathfrak{h}}-[c,T(x)(a,b)]_{\mathfrak{h}},

(35)

\rho(x)(a,[b,c]_{\mathfrak{h}})=[\rho(x)(a,b),c]_{\mathfrak{h}}+[b,\rho(x)(a,c)]_{\mathfrak{h}}.

•

Those resembling Eq. (5):

	$\displaystyle D(x_{1},x_{2})\omega(y_{1},y_{2},y_{3})+\omega(x_{1},x_{2},\{y_{1},y_{2},y_{3}\}_{\mathfrak{g}})=\omega(\{x_{1},x_{2},y_{1}\}_{\mathfrak{g}},y_{2},y_{3})$
	$\displaystyle+\theta(y_{2},y_{3})\omega(x_{1},x_{2},y_{1})+\omega(y_{1},\{x_{1},x_{2},y_{2}\}_{\mathfrak{g}},y_{3})-\theta(y_{1},y_{3})\omega(x_{1},x_{2},y_{2})$
(36)		$\displaystyle+\omega(y_{1},y_{2},\{x_{1},x_{2},y_{3}\}_{\mathfrak{g}})+D(y_{1},y_{2})\omega(x_{1},x_{2},y_{3}),$

		$\displaystyle D(x,y)\theta(z,w)a-\theta(z,w)D(x,y)a$
(37)		$\displaystyle=$	$\displaystyle\theta(\{x,y,z\}_{\mathfrak{g}},w)a+\theta(z,\{x,y,w\}_{\mathfrak{g}})a-T(w)(\omega(x,y,z),a)-\rho(z)(a,\omega(x,y,w)),$

(38)

\theta(x,\{y,z,w\}_{\mathfrak{g}})a-\rho(x)(a,\omega(y,z,w))=\theta(z,w)\theta(x,y)a-\theta(y,w)\theta(x,z)a+D(y,z)\theta(x,w)a,

		$\displaystyle D(x,y)D(z,w)a-D(z,w)D(x,y)a$
(39)		$\displaystyle=$	$\displaystyle D(\{x,y,z\}_{\mathfrak{g}},w)a+D(z,\{x,y,w\}_{\mathfrak{g}})a-\rho(w)(\omega(x,y,z),a)+\rho(z)(\omega(x,y,w),a),$

(40)

\displaystyle(D(x,y)\rho(z)-\rho(\{x,y,z\}_{\mathfrak{g}}))(a,b)

\displaystyle=\rho(z)((D(x,y)a,b)+(a,D(x,y)b))+\{\omega(x,y,z),a,b\}_{\mathfrak{h}},

(41)

D(y,z)(\rho(x)(a,b))=\rho(x)(a,D(y,z)b)-\rho(z)(\theta(x,y)a,b)+\rho(y)(\theta(x,z)a,b),

(42)

\theta(y,z)(\rho(x)(a,b))=\rho(x)(a,\theta(y,z)b)-T(z)(\theta(x,y)a,b)-\rho(y)(b,\theta(x,z)a),

(43)

\displaystyle(D(x,y)T(z)-T(\{x,y,z\}_{\mathfrak{g}}))(a,b)=T(z)((D(x,y)a,b)+(a,D(x,y)b))+\{a,b,\omega(x,y,z)\}_{\mathfrak{h}},

(44)

\displaystyle(T(\{x,y,z\}_{\mathfrak{g}})-D(x,y)T(z))(a,b)+\{a,b,\omega(x,y,z)\}_{\mathfrak{h}}=(\theta(y,z)T(x)-\theta(x,z)T(y))(a,b),

(45)

D(x,y)(\{a,b,c\}_{\mathfrak{h}})=\{D(x,y)a,b,c\}_{\mathfrak{h}}+\{a,D(x,y)b,c\}_{\mathfrak{h}}+\{a,b,D(x,y)c\}_{\mathfrak{h}},

(46)

\rho(x)(a,\rho(y)(b,c))=\rho(y)(\rho(x)(a,b),c)+\rho(y)(b,\rho(x)(a,c))-\{\theta(x,y)a,b,c\}_{\mathfrak{h}},

(47)

\{a,b,\theta(x,y)c\}_{\mathfrak{h}}=\theta(x,y)\{a,b,c\}_{\mathfrak{h}}-T(y)(T(x)(a,b),c)-\rho(x)(c,T(y)(a,b)),

(48)

\rho(x)(a,T(y)(b,c))=T(y)(\rho(x)(a,b),c)+T(y)(b,\rho(x)(a,c))-\{a,c,\theta(x,y)a\}_{\mathfrak{h}},

(49)

\{a,b,D(x,y)c\}_{\mathfrak{h}}=D(x,y)\{a,b,c\}_{\mathfrak{h}}-\rho(y)(T(x)(a,b),c),+\rho(x)(T(y)(a,b),c),

(50)

\{a,b,\rho(x)(c,d)\}_{\mathfrak{h}}=\rho(x)(\{a,b,c\}_{\mathfrak{h}},d)-\{c,T(x)(a,b),d\}_{\mathfrak{h}}+\rho(x)(c,\{a,b,d\}_{\mathfrak{h}}),

(51)

\rho(x)(a,\{b,c,d\}_{\mathfrak{h}})=\{\rho(x)(a,b),c,d\}_{\mathfrak{h}}+\{b,\rho(x)(a,c),d\}_{\mathfrak{h}}+\{b,c,\rho(x)(a,d)\}_{\mathfrak{h}}.

(52)

\{a,b,T(x)(c,d)\}_{\mathfrak{h}}=T(x)(\{a,b,c\}_{\mathfrak{h}},d)+T(x)(c,\{a,b,d\}_{\mathfrak{h}})+\{c,d,T(x)(a,b)\}_{\mathfrak{h}}.

Definition 3.4.

Let $(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})$ and $(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ be two non-abelian (2,3)-cocycles on $\mathfrak{g}$ with values in $\mathfrak{h}$ . They are said to be equivalent if there exists a linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ such that for all $x,y,z\in\mathfrak{g}$ and $a,b\in\mathfrak{h}$ , the following equalities hold:

(53)

\chi_{1}(x,y)-\chi_{2}(x,y)=[\varphi(x),\varphi(y)]_{\mathfrak{h}}+\varphi[x,y]_{\mathfrak{g}}-\mu_{2}(x)\varphi(y)+\mu_{2}(y)\varphi(x),

		$\displaystyle\omega_{1}(x,y,z)-\omega_{2}(x,y,z)=\theta_{2}(x,z)\varphi(y)-D_{2}(x,y)\varphi(z)+\rho_{2}(x)(\varphi(y),\varphi(z))-\theta_{2}(y,z)\varphi(x)$
(54)			$\displaystyle+T_{2}(z)(\varphi(x),\varphi(y))-\rho_{2}(y)(\varphi(x),\varphi(z))-\{\varphi(x),\varphi(y),\varphi(z)\}_{\mathfrak{h}}+\varphi\{x,y,z\}_{\mathfrak{g}},$

(55)

\mu_{1}(x)a-\mu_{2}(x)a=[a,\varphi(x)]_{\mathfrak{h}},

(56)

\theta_{1}(x,y)a-\theta_{2}(x,y)a=\rho_{2}(x)(a,\varphi(y))-T_{2}(y)(a,\varphi(x))+\{a,\varphi(x),\varphi(y)\}_{\mathfrak{h}},

(57)

D_{1}(x,y)a-D_{2}(x,y)a=\rho_{2}(y)(\varphi(x),a)-\rho_{2}(x)(\varphi(y),a)+\{\varphi(x),\varphi(y),a\}_{\mathfrak{h}},

(58)

\rho_{1}(x)(a,b)-\rho_{2}(x)(a,b)=\{a,\varphi(x),b\}_{\mathfrak{h}},~{}~{}T_{1}(x)(a,b)-T_{2}(x)(a,b)=\{b,a,\varphi(x)\}_{\mathfrak{h}}.

For convenience, we abbreviate a non-abelian (2,3)-cocycle $(\chi,\omega,\mu,\theta,D,\rho,T)$ as $(\chi,\omega)$ , denote the equivalent class of a non-abelian (2,3)-cocycle $(\chi,\omega,\mu,\theta,D,\rho,T)$ simply by $[(\chi,\omega)]$ . Furthermore, we denote the set of equivalent classes of non-abelian (2,3)-cocycles by $H_{nab}^{(2,3)}(\mathfrak{g},\mathfrak{h})$ .

Using the above notations, we define multilinear maps $[\ ,\ ]_{\chi}$ and $[\ ,\ ,\ ]_{\omega}$ on $\mathfrak{g}\oplus\mathfrak{h}$ by

(59)

\displaystyle[x+a,y+b]_{\chi}

\displaystyle=[x,y]_{\mathfrak{g}}+\chi(x,y)+\mu(x)b-\mu(y)a+[a,b]_{\mathfrak{h}},

	$\displaystyle\{x+a,y+b,z+c\}_{\omega}=$	$\displaystyle\{x,y,z\}_{\mathfrak{g}}+\omega(x,y,z)+D(x,y)c+\theta(y,z)a-\theta(x,z)b$
(60)			$\displaystyle+T(z)(a,b)+\rho(x)(b,c)-\rho(y)(a,c)+\{a,b,c\}_{\mathfrak{h}}$

for all $x,y,z\in\mathfrak{g}$ and $a,b,c\in\mathfrak{h}$ .

Proposition 3.5.

With the above notions, $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\{\ ,\ ,\ \}_{\omega})$ is a Lie-Yamaguti algebra if and only if the septuple $(\chi,\omega,\mu,\theta,D,\rho,T)$ is a non-abelian (2,3)-cocycle. Denote this Lie-Yamaguti algebra $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\{\ ,\ ,\ \}_{\omega})$ simply by $\mathfrak{g}\oplus_{(\chi,\omega)}\mathfrak{h}$ .

Proof.

$(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\{\ ,\ ,\ \}_{\omega})$ is a Lie-Yamaguti algebra if and only if Eqs. (1)-(5) hold for $[\ ,\ ]_{\chi},\{\ ,\ ,\ \}_{\omega}$ . In fact, it is easy to check that Eq. (1) holds for $[\ ,\ ]_{\chi},\{\ ,\ ,\ \}_{\omega}$ if and only if (16)-(17) hold. In the following, we always assume that $x,y,z,w\in\mathfrak{g}$ and $a,b,c,d\in\mathfrak{h}$ .

For the Eq. (2), we discuss it for the following cases: for all $x_{1},x_{2},x_{3}\in\mathfrak{g}\oplus\mathfrak{h}$ ,

$(I)$

when all of the three elements $x_{1},x_{2},x_{3}$ belong to $\mathfrak{g}$ , (2) holds if and only if (• ‣ 3.3) holds.
$(II)$
when $x_{1},x_{2},x_{3}$ equal to:
1. $(i)$
  
  $x,y,a$ or $a,x,y$ or $x,a,y$ respectively, (2) holds if and only if (19) holds.
2. $(ii)$
  
  $a,b,x$ or $a,x,b$ or $x,a,b$ respectively, (2) holds for if and only if (20) holds.

For the Eq. (3), we discuss it for the following cases: for all $x_{1},x_{2},x_{3},y_{1}\in\mathfrak{g}\oplus\mathfrak{h}$ ,

$(I)$

when all of the four elements $x_{1},x_{2},x_{3},y_{1}$ belong to $\mathfrak{g}$ , (3) holds if and only if (21) holds.
$(II)$
when $x_{1},x_{2},x_{3},y_{1}$ equal to:
1. $(i)$
  
  $x,y,z,a$ , (3) holds if and only if (• ‣ 3.3) holds.
2. $(ii)$
  
  $x,y,a,z$ or $x,a,y,z$ or $a,x,y,z$ respectively, (3) holds if and only if (23) holds.
3. $(iii)$
  
  $x,y,a,b$ or $x,a,y,b$ or $a,x,y,b$ respectively, (3) holds if and only if (24) holds.
4. $(iv)$
  
  $x,a,b,y$ or $a,x,b,y$ or $a,b,x,y$ respectively, (3) holds if and only if (25) holds.
5. $(v)$
  
  $x,a,b,c$ or $a,x,b,c$ or $a,b,x,y,c$ respectively, (3) holds if and only if (26) holds.
6. $(vi)$
  
  $a,b,c,x$ , (3) holds if and only if (27) holds.

For the Eq. (4), we discuss it for the following cases: for all $x_{1},x_{2},y_{1},y_{2}\in\mathfrak{g}\oplus\mathfrak{h}$ ,

$(I)$

when all of the four elements $x_{1},x_{2},y_{1},y_{2}$ belong to $\mathfrak{g}$ , (4) holds if and only if (28) holds.
$(II)$
when $x_{1},x_{2},y_{1},y_{2}$ equal to:
1. $(i)$
  
  $x,y,z,a$ or $x,y,a,z$ , (4) holds if and only if (29) holds.
2. $(ii)$
  
  $x,a,y,z$ or $a,x,y,z$ respectively, (4) holds if and only if (30) holds.
3. $(iii)$
  
  $x,y,a,b$ , (4) holds if and only if (31) holds.
4. $(iv)$
  
  $x,a,y,b$ or $a,x,y,b$ or $x,a,b,y$ or $a,x,b,y$ respectively, (4) holds if and only if (32) holds.
5. $(v)$
  
  $a,b,x,y$ , (4) holds if and only if (33) holds.
6. $(vi)$
  
  $a,b,c,x$ or $a,b,x,c$ respectively, (4) holds if and only if (34) holds.
7. $(vii)$
  
  $a,x,b,c$ or $x,a,b,c$ respectively, (4) holds if and only if (35) holds.

For the Eq. (5), we discuss it for the following cases: for all $x_{1},x_{2},y_{1},y_{2},y_{3}\in\mathfrak{g}\oplus\mathfrak{h}$ ,

$(I)$

when all of the five elements $x_{1},x_{2},y_{1},y_{2},y_{3}$ belong to $\mathfrak{g}$ , (5) holds if and only if (• ‣ 3.3) holds.
$(II)$
when $x_{1},x_{2},y_{1},y_{2},y_{3}$ equal to:
1. $(i)$
  
  $x,y,z,w,a$ , (5) holds if and only if (• ‣ 3.3) holds.
2. $(ii)$
  
  $x,y,z,a,w$ or $x,y,a,z,w$ respectively, (5) holds if and only if (• ‣ 3.3) holds.
3. $(iii)$
  
  $x,a,y,z,w$ or $a,x,y,z,w$ respectively, (5) holds if and only if (38) holds.
4. $(iv)$
  
  $x,y,z,a,b$ or $x,y,a,y,b$ , (5) holds if and only if (40) holds.
5. $(v)$
  
  $x,a,y,z,b$ or $a,x,y,z,b$ respectively, (5) holds if and only if (41) holds.
6. $(vi)$
  
  $x,y,a,b,z$ , (5) holds if and only if (43) holds.
7. $(vii)$
  
  $x,a,y,b,z$ or $a,x,y,b,z$ or $x,a,b,y,z$ or $a,x,b,y,z$ respectively, (4) holds if and only if (42) holds.
8. $(viii)$
  
  $a,b,x,y,z$ , (5) holds if and only if (44) holds.
9. $(ix)$
  
  $x,y,a,b,c$ , (5) holds if and only if (45) holds.
10. $(x)$
  
  $x,a,y,b,c$ or $a,x,y,b,c$ or $a,x,b,y,c$ or $x,a,b,y,c$ respectively, (5) holds if and only if (46) holds.
11. $(xi)$
  
  $x,a,b,c,y$ or $a,x,b,c,y$ respectively, (5) holds if and only if (48) holds.
12. $(xii)$
  
  $a,b,x,c,y$ or $a,b,c,x,y$ respectively, (5) holds if and only if (47) holds.
13. $(xiii)$
  
  $a,b,x,y,c$ , (5) holds if and only if (49) holds.
14. $(xiv)$
  
  $a,b,c,d,x$ , (5) holds if and only if (52) holds.
15. $(xv)$
  
  $a,b,c,x,d$ or $a,b,x,c,d$ respectively, (5) holds if and only if (50) holds.
16. $(xvi)$
  
  $a,x,b,c,d$ or $x,a,b,c,d$ respectively, (5) holds if and only if (51) holds.

This completes the proof. ∎

Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$ . Define $\chi_{s}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h},~{}\omega_{s}:\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h},~{}\mu_{s}:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h}),~{}\theta_{s},D_{s}:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h}),~{}\rho_{s},T_{s}:\mathfrak{g}\longrightarrow\mathrm{Hom}(\mathfrak{h}\wedge\mathfrak{h},\mathfrak{h})$ respectively by

(61)

\chi_{s}(x,y)=[s(x),s(y)]_{\hat{\mathfrak{g}}}-s[x,y]_{\mathfrak{g}},

(62)

\omega_{s}(x,y,z)=\{s(x),s(y),s(z)\}_{\hat{\mathfrak{g}}}-s\{x,y,z\}_{\mathfrak{g}},

(63)

\theta_{s}(x,y)a=\{a,s(x),s(y)\}_{\hat{\mathfrak{g}}},~{}~{}~{}~{}\rho_{s}(x)(a,b)=\{s(x),a,b\}_{\hat{\mathfrak{g}}},

(64)

D_{s}(x,y)a=\{s(x),s(y),a\}_{\hat{\mathfrak{g}}},~{}~{}~{}~{}T_{s}(x)(a,b)=\{a,b,s(x)\}_{\hat{\mathfrak{g}}}

for any $x,y,z\in\mathfrak{g},a,b\in\mathfrak{h}$ .

By direct computations, we have

Proposition 3.6.

With the above notions, $(\chi_{s},\omega_{s},\mu_{s},\theta_{s},D_{s},\rho_{s},T_{s})$ is a non-abelian (2,3)-cocycle on $\mathfrak{g}$ with values in $\mathfrak{h}$ . We call it the non-abelian (2,3)-cocycle corresponding to the extension $\mathcal{E}$ induced by $s$ . Naturally, $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi_{s}},\{\ ,\ ,\ \}_{\omega_{s}})$ is a Lie-Yamaguti algebra. Denote this Lie-Yamaguti algebra simply by $\mathfrak{g}\oplus_{(\chi_{s},\omega_{s})}\mathfrak{h}$ .

In the following, we denote $(\chi_{s},\omega_{s},\mu_{s},\theta_{s},D_{s},\rho_{s},T_{s})$ by $(\chi,\omega,\mu,\theta,D,\rho,T)$ without ambiguity.

Lemma 3.7.

Let $(\chi_{i},\omega_{i},\mu_{i},\theta_{i},D_{i},\rho_{i},T_{i})$ be the non-abelian (2,3)-cocycle corresponding to the extension $\mathcal{E}$ induced by $s_{i}$ (i=1,2). Then $(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})$ and $(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ are equivalent, that is, the equivalent classes of non-abelian (2,3)-cocycles corresponding to a non-abelian extension induced by a section are independent on the choice of sections.

Proof.

Let $\hat{\mathfrak{g}}$ be a non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ . Assume that $s_{1}$ and $s_{2}$ are two different sections of $p$ , $(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})$ and $(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ are the corresponding non-abelian (2,3)-cocycles. Define a linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ by $\varphi(x)=s_{2}(x)-s_{1}(x)$ . Since $p\varphi(x)=ps_{2}(x)-ps_{1}(x)=0$ , $\varphi$ is well defined. Thanks to Eqs. (62)-(64), we get

			$\displaystyle w_{1}(x,y,z)=\{s_{1}(x),s_{1}(y),s_{1}(z)\}_{\hat{\mathfrak{g}}}-s_{1}\{x,y,z\}_{\mathfrak{g}}$
		$\displaystyle=$	$\displaystyle\{s_{2}(x)-\varphi(x),s_{2}(y)-\varphi(y),s_{2}(z)-\varphi(z)\}_{\hat{\mathfrak{g}}}-(s_{2}\{x,y,z\}_{\mathfrak{g}}-\varphi(\{x,y,z\}_{\mathfrak{g}})$
		$\displaystyle=$	$\displaystyle\{s_{2}(x),s_{2}(y),s_{2}(z)\}_{\hat{\mathfrak{g}}}-\{s_{2}(x),\varphi(y),s_{2}(z)\}_{\hat{\mathfrak{g}}}-\{s_{2}(x),s_{2}(y),\varphi(z)\}_{\hat{\mathfrak{g}}}+\{s_{2}(x),\varphi(y),\varphi(z)\}_{\hat{\mathfrak{g}}}$
			$\displaystyle-\{\varphi(x),s_{2}(y),s_{2}(z)\}_{\hat{\mathfrak{g}}}+\{\varphi(x),\varphi(y),s_{2}(z)\}_{\hat{\mathfrak{g}}}+\{\varphi(x),s_{2}(y),\varphi(z)\}_{\hat{\mathfrak{g}}}-\{\varphi(x),\varphi(y),\varphi(z)\}_{\hat{\mathfrak{g}}}$
			$\displaystyle-s_{2}\{x,y,z\}_{\mathfrak{g}}+\varphi\{x,y,z\}_{\mathfrak{g}}$
		$\displaystyle=$	$\displaystyle w_{2}(x,y,z)+\theta_{2}(x,z)\varphi(y)-D_{2}(x,y)\varphi(z)+\rho_{2}(x)(\varphi(y),\varphi(z))-\theta_{2}(y,z)\varphi(x)+T_{2}(z)(\varphi(x),\varphi(y))$
			$\displaystyle-\rho_{2}(y)(\varphi(x),\varphi(z))-\{\varphi(x),\varphi(y),\varphi(z)\}_{\hat{\mathfrak{g}}}+\varphi\{x,y,z\}_{\mathfrak{g}},$

which yields that Eq. (3.4) holds. Similarly, Eqs. (53) and (55)-(58) hold. This finishes the proof. ∎

According to Proposition 3.5 and Proposition 3.6, given a non-abelian extension $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$ , we have a non-abelian (2,3)-cocycle $(\chi_{s},\omega_{s},\mu_{s},\theta_{s},D_{s},\rho_{s},T_{s})$ and a Lie-Yamaguti algebra $\mathfrak{g}\oplus_{(\chi_{s},\omega_{s})}\mathfrak{h}$ . It follows that $\mathcal{E}_{(\chi_{s},\omega_{s})}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\mathfrak{g}\oplus_{(\chi_{s},\omega_{s})}\mathfrak{h}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ is a non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ . Since any element $\hat{w}\in\hat{\mathfrak{g}}$ can be written as $\hat{w}=a+s(x)$ with $a\in\mathfrak{h},x\in\mathfrak{g}$ , define a linear map

f:\hat{\mathfrak{g}}\longrightarrow\mathfrak{g}\oplus_{(\chi_{s},\omega_{s})}\mathfrak{h},~{}f(\hat{w})=f(a+s(x))=a+x.

It is easy to check that $f$ is an isomorphism of Lie-Yamaguti algebras such that the following commutative diagram holds:

which indicates that the non-abelian extensions $\mathcal{E}$ and $\mathcal{E}_{(\chi_{s},\omega_{s})}$ of $\mathfrak{g}$ by $\mathfrak{h}$ are equivalent. On the other hand, if $(\chi,\omega,\mu,\theta,D,\rho,T)$ is a non-abelian (2,3)-cocycle on $\mathfrak{g}$ with values in $\mathfrak{h}$ , there is a Lie-Yamaguti algebra $\mathfrak{g}\oplus_{(\chi,\omega)}\mathfrak{h}$ , which yields the following non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ :

\mathcal{E}_{(\chi,\omega)}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\mathfrak{g}\oplus_{(\chi,\omega)}\mathfrak{h}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0,

where $i$ is the inclusion and $\pi$ is the projection.

In the following, we focus on the relationship between non-abelian (2,3)-cocycles and extensions.

Proposition 3.8.

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie-Yamaguti algebras. Then the equivalent classes of non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ are classified by the non-abelian cohomology group, that is, $\mathcal{E}_{nab}(\mathfrak{g},\mathfrak{h})\simeq H_{nab}^{(2,3)}(\mathfrak{g},\mathfrak{h})$ .

Proof.

Define a linear map

\Theta:\mathcal{E}_{nab}(\mathfrak{g},\mathfrak{h})\rightarrow H_{nab}^{(2,3)}(\mathfrak{g},\mathfrak{h}),~{}

where $\Theta$ assigns an equivalent class of non-abelian extensions to the class of corresponding non-abelian (2,3)-cocycles. First, we check that $\Theta$ is well-defined. Assume that $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are two equivalent non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ via the map $f$ , that is, the commutative diagram (15) holds. Let $s_{1}:\mathfrak{g}\rightarrow\hat{\mathfrak{g}}_{1}$ be a section of $p_{1}$ . Then $p_{2}fs_{1}=p_{1}s_{1}=I_{\mathfrak{g}}$ , which follows that $s_{2}=fs_{1}$ is a section of $p_{2}$ . Let $(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})$ and $(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ be two non-abelian (2,3)-cocycles induced by the sections $s_{1},s_{2}$ respectively. Then we have,

$\displaystyle\theta_{1}(x,y)a$	$\displaystyle=$	$\displaystyle f(\theta_{1}(x,y)a)=f(\{a,s_{1}(x),s_{1}(y)\}_{\hat{\mathfrak{g}}_{1}})$
	$\displaystyle=$	$\displaystyle\{f(a),fs_{1}(x),fs_{1}(y)\}_{\hat{\mathfrak{g}}_{2}}$
	$\displaystyle=$	$\displaystyle\{a,s_{2}(x),s_{2}(y)\}_{\hat{\mathfrak{g}}_{2}}$
	$\displaystyle=$	$\displaystyle\theta_{2}(x,y)a.$

By the same token, we have

D_{1}(x,y)a=D_{2}(x,y)a,\chi_{1}(x,y)=\chi_{2}(x,y),\omega_{1}(x,y,z)=\omega_{2}(x,y,z),

\rho_{1}(x)(a,b)=\rho_{2}(x)(a,b),T_{1}(x)(a,b)=T_{2}(x)(a,b).

Thus, $(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})=(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ , which means that $\Theta$ is well-defined.

Next, we verify that $\Theta$ is injective. Indeed, suppose that $\Theta([\mathcal{E}_{1}])=[(\chi_{1},\omega_{1})]$ and $\Theta([\mathcal{E}_{2}])=[(\chi_{2},\omega_{2})]$ . If the equivalent classes $[(\chi_{1},\omega_{1})]=[(\chi_{2},\omega_{2})]$ , we obtain that the non-abelian (2,3)-cocycles $(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})$ and $(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ are equivalent via the linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ , satisfying Eqs. (53)-(58). Define a linear map $f:\mathfrak{g}\oplus_{(\chi_{1},\omega_{1})}\mathfrak{h}\longrightarrow\mathfrak{g}\oplus_{(\chi_{2},\omega_{2})}\mathfrak{h}$ by

f(x+a)=x-\varphi(x)+a,~{}~{}\forall~{}x\in\mathfrak{g},a\in\mathfrak{h}.

According to Eq. (3), for all $x,y,z\in\mathfrak{g},a,b,c\in\mathfrak{h}$ , we get

			$\displaystyle f(\{x+a,y+b,z+c\}_{\omega_{1}})$
		$\displaystyle=$	$\displaystyle f(\{x,y,z\}_{\mathfrak{g}}+\omega_{1}(x,y,z)+D_{1}(x,y)c+\theta_{1}(y,z)a-\theta_{1}(x,z)b$
			$\displaystyle+T_{1}(z)(a,b)+\rho_{1}(x)(b,c)-\rho_{1}(y)(a,c)+\{a,b,c\}_{\mathfrak{h}})$
		$\displaystyle=$	$\displaystyle\{x,y,z\}_{\mathfrak{g}}-\varphi(\{x,y,z\}_{\mathfrak{g}})+\omega_{1}(x,y,z)+D_{1}(x,y)c+\theta_{1}(y,z)a-\theta_{1}(x,z)b$
			$\displaystyle+T_{1}(z)(a,b)+\rho_{1}(x)(b,c)-\rho_{1}(y)(a,c)+\{a,b,c\}_{\mathfrak{h}},$

and

			$\displaystyle\{f(x+a),f(y+b),f(z+c)\}_{\omega_{2}}$
		$\displaystyle=$	$\displaystyle\{x-\varphi(x)+a,y-\varphi(y)+b,z-\varphi(z)+c\}_{\omega_{2}}$
		$\displaystyle=$	$\displaystyle\{x,y,z\}_{\mathfrak{g}}+\omega_{2}(x,y,z)+D_{2}(x,y)(c-\varphi(z))+\theta_{2}(y,z)(a-\varphi(x))-\theta_{2}(x,z)(b-\varphi(y))$
			$\displaystyle+T_{2}(z)(a-\varphi(x),b-\varphi(y))+\rho_{2}(x)(b-\varphi(y),c-\varphi(z))-\rho_{2}(y)(a-\varphi(x),c-\varphi(z))$
			$\displaystyle+\{a-\varphi(x),b-\varphi(y),c-\varphi(z)\}_{\mathfrak{h}})$
		$\displaystyle=$	$\displaystyle\{x,y,z\}_{\mathfrak{g}}+\omega_{2}(x,y,z)+D_{2}(x,y)(c-\varphi(z))+\theta_{2}(y,z)(a-\varphi(x))-\theta_{2}(x,z)(b-\varphi(y))$
			$\displaystyle+T_{2}(z)(\varphi(x),\varphi(y))-T_{2}(z)(a,\varphi(y))-T_{2}(z)(\varphi(x),b)+T_{2}(z)(a,b)$
			$\displaystyle+\rho_{2}(x)(\varphi(y),\varphi(z))-\rho_{2}(x)(\varphi(y),c)-\rho_{2}(x)(b,\varphi(z))+\rho_{2}(x)(b,c)-\rho_{2}(y)(\varphi(x),\varphi(z))$
			$\displaystyle+\rho_{2}(y)(\varphi(x),c)+\rho_{2}(y)(a,\varphi(z))-\rho_{2}(y)(a,c)-\{\varphi(x),\varphi(y),\varphi(z)\}_{\mathfrak{h}}+\{\varphi(x),\varphi(y),c\}_{\mathfrak{h}}$
			$\displaystyle+\{\varphi(x),b,\varphi(z)\}_{\mathfrak{h}}-\{\varphi(x),b,c\}_{\mathfrak{h}}+\{a,\varphi(y),\varphi(z)\}_{\mathfrak{h}}-\{a,b,\varphi(z)\}_{\mathfrak{h}}-\{a,\varphi(y),c\}_{\mathfrak{h}}+\{a,b,c\}_{\mathfrak{h}}.$

In view of Eqs. (53)-(58), we have $f(\{x+a,y+b,z+c\}_{\omega_{1}})=\{f(x+a),f(y+b),f(z+c)\}_{\omega_{2}}$ . Similarly, $f([x+a,y+b]_{\chi_{1}})=[f(x+a),f(y+b)]_{\chi_{2}}$ . Hence, $f$ is a homomorphism of Lie-Yamaguti algebras. Clearly, the following commutative diagram holds:

(65)

Thus $\mathcal{E}_{(\chi_{1},\omega_{1})}$ and $\mathcal{E}_{(\chi_{2},\omega_{2})}$ are equivalent non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ , which means that $[\mathcal{E}_{(\chi_{1},\omega_{1})}]=[\mathcal{E}_{(\chi_{2},\omega_{2})}]$ . Thus, $\Theta$ is injective.

Finally, we claim that $\Theta$ is surjective. For any equivalent class of non-abelian (2,3)-cocycles $[(\chi,\omega)]$ , by Proposition 3.5, there is a non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ :

\mathcal{E}_{(\chi,\omega)}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\mathfrak{g}\oplus_{(\chi,\omega)}\mathfrak{h}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0.

Therefore, $\Theta([\mathcal{E}_{(\chi,\omega)}])=[(\chi,\omega)]$ , which follows that $\Theta$ is surjective. In all, $\Theta$ is bijective. This finishes the proof.

∎

4. Non-abelian extensions in terms of Maurer-Cartan elements

In this section, we classify the non-abelian extensions using Maurer-Cartan elements. We start with recalling the Maurer-Cartan elements from [12].

Let $(L=\oplus_{i}L_{i},[\ ,\ ],d)$ be a differential graded Lie algebra. The set $\mathrm{MC}(L)$ of Maurer-Cartan elements of $(L,[\ ,\ ],d)$ is defined by

\mathrm{MC}(L)=\{\eta\in L_{1}|d\eta+\frac{1}{2}[\eta,\eta]=0\}.

Moreover, $\eta_{0},\eta_{1}\in\mathrm{MC}(L)$ are called gauge equivalent if and only if there exists an element $\varphi\in L_{0}$ such that

\eta_{1}=e^{ad_{\varphi}}\eta_{0}-\frac{e^{ad_{\varphi}}-1}{ad_{\varphi}}d\varphi.

Let $\mathfrak{g}$ be a vector space. Denote by

\mathbb{C}(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}(\wedge^{2}\mathfrak{g}\otimes\mathfrak{g},\mathfrak{g})\times\mathrm{Hom}(\wedge^{2}\mathfrak{g}\otimes\wedge^{2}\mathfrak{g},\mathfrak{g})

and

(66)

\mathcal{C}^{n}(\mathfrak{g},\mathfrak{g})=\left\{\begin{aligned} &\mathrm{Hom}(\mathfrak{g},\mathfrak{g}),&n=0,\\ &\mathrm{Hom}(\underbrace{\wedge^{2}\mathfrak{g}\otimes\cdot\cdot\cdot\otimes\wedge^{2}\mathfrak{g}}_{n},\mathfrak{g})\times\mathrm{Hom}(\underbrace{\wedge^{2}\mathfrak{g}\otimes\cdot\cdot\cdot\otimes\wedge^{2}\mathfrak{g}}_{n}\otimes\mathfrak{g},\mathfrak{g}),&n\geq 1,\end{aligned}\right.

Then $\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g})=\mathcal{C}^{*}(\mathfrak{g},\mathfrak{g})\oplus\mathbb{C}(\mathfrak{g},\mathfrak{g})=\oplus_{n\geq 0}\mathcal{C}^{n}(\mathfrak{g},\mathfrak{g})\oplus\mathbb{C}(\mathfrak{g},\mathfrak{g})$ , where the degree of elements in $\mathcal{C}^{n}(\mathfrak{g},\mathfrak{g})$ is $n$ , the degree of elements in $\mathbb{C}(\mathfrak{g},\mathfrak{g})$ is $1$ and $f\in\mathrm{Hom}(\otimes^{2n+1}\mathfrak{g},\mathfrak{g})$ satisfying

(67)

\displaystyle f(x_{1},\cdot\cdot\cdot,x_{2i-1},x_{2i},\cdot\cdot\cdot,x_{n})=0,~{}~{}\hbox{if}~{}x_{2i-1}=x_{2i},~{}\forall~{}i=1,2,\cdot\cdot\cdot,[\frac{n}{2}].

For all $P=(P_{I},P_{II})\in\mathcal{C}^{p}(\mathfrak{g},\mathfrak{g}),Q=(Q_{I},Q_{II})\in\mathcal{C}^{q}(\mathfrak{g},\mathfrak{g})~{}(p,q\geq 1)$ , denote by

P\circ Q=((P\circ Q)_{I},(P\circ Q)_{II})\in\mathcal{C}^{p+q}(\mathfrak{g},\mathfrak{g}).

The definition of $P\circ Q$ is given in [31]. In detail,

		$\displaystyle(P\circ Q)_{I}(X_{1},\cdot\cdot\cdot,X_{p+q})$
	$\displaystyle=$	$\displaystyle\sum_{\begin{subarray}{c}\sigma\in sh(p,q),\\ \sigma(p+q)=p+q\end{subarray}}(-1)^{pq}sgn(\sigma)P_{II}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(p)},Q_{I}(X_{\sigma(p+1)},\cdot\cdot\cdot,X_{\sigma(p+q)})$
		$\displaystyle+\sum_{\begin{subarray}{c}k=1,\\ \sigma\in sh(k-1,q)\end{subarray}}^{p}(-1)^{q(k-1)}sgn(\sigma)P_{I}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(k-1)},x_{q+k}\wedge Q_{II}(X_{\sigma(k)},\cdot\cdot\cdot,X_{\sigma(k+q-1)},y_{k+q}),X_{k+q+1},\cdot\cdot\cdot,X_{p+q})$
		$\displaystyle+\sum_{\begin{subarray}{c}k=1,\\ \sigma\in sh(k-1,q)\end{subarray}}^{p}(-1)^{q(k-1)}sgn(\sigma)P_{I}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(k-1)},Q_{II}(X_{\sigma(k)},\cdot\cdot\cdot,X_{\sigma(k+q-1)},x_{k+q})\wedge y_{q+k},X_{k+q+1},\cdot\cdot\cdot,X_{p+q}),$

and

		$\displaystyle(P\circ Q)_{II}(X_{1},\cdot\cdot\cdot,X_{p+q},z)$
	$\displaystyle=$	$\displaystyle\sum_{\sigma\in sh(p,q)}(-1)^{pq}sgn(\sigma)P_{II}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(p)},Q_{II}(X_{\sigma(p+1)},\cdot\cdot\cdot,X_{\sigma(p+q)},z)$
		$\displaystyle+\sum_{\begin{subarray}{c}k=1,\\ \sigma\in sh(k-1,q)\end{subarray}}^{p}(-1)^{q(k-1)}sgn(\sigma)P_{II}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(k-1)},x_{q+k}\wedge Q_{II}(X_{\sigma(k)},\cdot\cdot\cdot,X_{\sigma(k+q-1)},y_{k+q}),X_{k+q+1},\cdot\cdot\cdot,X_{p+q},z)$
		$\displaystyle+\sum_{\begin{subarray}{c}k=1,\\ \sigma\in sh(k-1,q)\end{subarray}}^{p}(-1)^{q(k-1)}sgn(\sigma)P_{II}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(k-1)},Q_{II}(X_{\sigma(k)},\cdot\cdot\cdot,X_{\sigma(k+q-1)},x_{k+q})\wedge y_{q+k},X_{k+q+1},\cdot\cdot\cdot,X_{p+q},z).$

In particular, for $f\in\mathcal{C}^{0}(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}(\mathfrak{g},\mathfrak{g})$ and $P=(P_{I},P_{II})\in\mathcal{C}^{p}(\mathfrak{g},\mathfrak{g})$ , define

	$\displaystyle(P\circ f)_{I}(X_{1},\cdot\cdot\cdot,X_{p})=$	$\displaystyle\sum_{k=1}^{p}P_{I}(X_{1},\cdot\cdot\cdot,X_{k-1},x_{k}\wedge f(y_{k}),X_{k+1},\cdot\cdot\cdot,X_{p})$
(68)			$\displaystyle+\sum_{k=1}^{p}P_{I}(X_{1},\cdot\cdot\cdot,X_{k-1},f(x_{k})\wedge y_{k},X_{k+1},\cdot\cdot\cdot,X_{p}),$

(69)

(f\circ P)_{I}(X_{1},\cdot\cdot\cdot,X_{p})=f(P_{I}(X_{1},\cdot\cdot\cdot,X_{p})),

	$\displaystyle(P\circ f)_{II}(X_{1},\cdot\cdot\cdot,X_{p},z)=$	$\displaystyle\sum_{k=1}^{p}P_{II}(X_{1},\cdot\cdot\cdot,X_{k-1},x_{k}\wedge f(y_{k}),X_{k+1},\cdot\cdot\cdot,X_{p},z)$
(70)			$\displaystyle+\sum_{k=1}^{p}P_{I}(X_{1},\cdot\cdot\cdot,X_{k-1},f(x_{k})\wedge y_{k},X_{k+1},\cdot\cdot\cdot,X_{p},z),$

(71)

(f\circ P)_{II}(X_{1},\cdot\cdot\cdot,X_{p},z)=f(P_{II}(X_{1},\cdot\cdot\cdot,X_{p},z)).

In order to derive sufficient and necessary conditions of Lie-Yamaguti algebras in terms of Maurer-Cartan element of some graded Lie algebras, we define $P\bullet Q$ as follows: for all $P=(P_{I},P_{II})\in\mathcal{C}^{n}(\mathfrak{g},\mathfrak{g}),~{}Q=(Q_{I},Q_{II})\in\mathcal{C}^{n}(\mathfrak{g},\mathfrak{g})$ ,

(72)

P\bullet Q=\left\{\begin{aligned} &0,&p,q\neq 1,\\ &((P\bullet Q)_{I},(P\bullet Q)_{II}),&p=q=1,\end{aligned}\right.

where

		$\displaystyle(P\bullet Q)_{I}(x_{1},x_{2},x_{3})$
	$\displaystyle=$	$\displaystyle\frac{1}{2}\sum_{\sigma\in S_{3}}sgn(\sigma)P_{I}(Q_{I}(x_{\sigma(1)},x_{\sigma(2)}),x_{\sigma(3)})+\frac{1}{2}\sum_{\sigma\in S_{3}}sgn(\sigma)Q_{II}(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}),$

\displaystyle(P\bullet Q)_{II}(x_{1},x_{2},x_{3},x_{4})=\frac{1}{2}\sum_{\sigma\in S_{3}}sgn(\sigma)P_{II}(Q_{I}(x_{\sigma(1)},x_{\sigma(2)}),x_{\sigma(3)},x_{4}).

Let $\mathfrak{g}$ and $V$ be vector spaces. For any $(\chi,\omega),(\chi^{\prime},\omega^{\prime})\in\mathcal{C}^{1}(\mathfrak{g},\mathfrak{g})$ , put $\Pi=(\chi,\omega),~{}\Pi^{\prime}=(\chi^{\prime},\omega^{\prime})$ and $\Pi+\Pi^{\prime}=(\chi+\chi^{\prime},\omega+\omega^{\prime})$ .

Proposition 4.1.

With the above notations, $(\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g}),[\ ,\ ]_{LY})$ is a graded Lie algebra, where

(73)

[P,Q]_{LY}=\left\{\begin{aligned} &P\bullet Q+Q\bullet P+P\circ Q+Q\circ P,&p=q=1,\\ &P\circ Q-(-1)^{pq}Q\circ P,&otherwise,\end{aligned}\right.

for all $P\in\mathcal{C}^{p}(\mathfrak{g},\mathfrak{g}),Q\in\mathcal{C}^{q}(\mathfrak{g},\mathfrak{g}).$ Furthermore, $\Pi=(\chi,\omega)\in\mathcal{C}^{1}(\mathfrak{g},\mathfrak{g})$ defines a Lie-Yamaguti algebra structure on $\mathfrak{g}$ if and only if $[\ ,\ ]_{LY}=0$ , that is, $\Pi$ is a Maurer-Cartan element of the graded Lie algebra $(\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g}),[\ ,\ ]_{LY})$ . We write $[P,Q]_{LY}=([P,Q]_{I},[P,Q]_{II}).$

Proof.

Take the same procedure of the proof of Proposition 4.1 [31], we can check that Eq. (67) holds. Clearly, Eq. (67) implies that Eq. (1) holds. For all $\Pi=(\chi,\omega)\in\mathcal{C}^{1}(\mathfrak{g},\mathfrak{g})$ , $[\Pi,\Pi]_{LY}=2\Pi\circ\Pi+2\Pi\bullet\Pi$ and for all $x_{1},x_{2},x_{3},x_{4}\in\mathfrak{g}$ , we have

	$\displaystyle(\Pi\bullet\Pi)_{I}(x_{1},x_{2},x_{3})=$	$\displaystyle\chi(\chi(x_{1},x_{2}),x_{3})+\chi(\chi(x_{2},x_{3}),x_{1})+\chi(\chi(x_{3},x_{1}),x_{2})$
		$\displaystyle+\omega(x_{1},x_{2},x_{3})+\omega(x_{2},x_{3},x_{1})+\omega(x_{3},x_{1},x_{2}),$

and

		$\displaystyle(\Pi\bullet\Pi)_{II}(x_{1},x_{2},x_{3},x_{4})$
	$\displaystyle=$	$\displaystyle\omega(\chi(x_{1},x_{2}),x_{3},x_{4})+\omega(\chi(x_{2},x_{3}),x_{1},x_{4})+\omega(\chi(x_{3},x_{1}),x_{2},x_{4}).$

Combining Theorem 3.1 [31], $\Pi=(\chi,\omega)\in\mathcal{C}^{1}(\mathfrak{g},\mathfrak{g})$ defines a Lie-Yamaguti algebra structure on $\mathfrak{g}$ if and only if $\Pi$ is a Maurer-Cartan element of the graded Lie algebra $(\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g}),[\ ,\ ]_{LY})$ . ∎

By Proposition 4.1, we rewrite Theorem 3.3 [31] as follows:

Theorem 4.2.

Let $(\mathfrak{g},\chi_{\mathfrak{g}},\omega_{\mathfrak{g}})$ be a Lie-Yamaguti algebra. Then $(\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g}),[\ ,\ ]_{LY},d_{\Pi})$ is a differential graded Lie algebra, where $d_{\Pi}$ with $\Pi=(\chi_{\mathfrak{g}},\omega_{\mathfrak{g}})$ is given by

(74)

d_{\Pi}(\nu)=[\Pi,\nu]_{LY},~{}~{}\forall~{}\nu\in\mathcal{C}^{n-1}(\mathfrak{g},\mathfrak{g}).

Moreover, $\Pi+\Pi^{\prime}$ with $\Pi^{\prime}\in\mathcal{C}^{1}(\mathfrak{g},\mathfrak{g})$ defines a Lie-Yamaguti algebra structure on $\mathfrak{g}$ if and only if $\Pi^{\prime}$ is a Maurer-Cartan element of the differential graded Lie algebra $(\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g}),[\ ,\ ]_{LY},d_{\Pi})$ .

Denote

\bar{\mu}(x+a,y+b)=\mu(x)b-\mu(y)a

and

\bar{\theta}(x+a,y+b,z+c)=D(x,y)c+\theta(y,z)a-\theta(x,z)b

for all $x,y,z\in\mathfrak{g}$ and $a,b,c\in V$ .

Proposition 4.3.

With the above notations, $(V,\mu,\theta,D)$ is a representation of Lie-Yamaguti algebra $(\mathfrak{g},\chi_{\mathfrak{g}},\omega_{\mathfrak{g}})$ if and only if $\bar{\Pi}\in\mathcal{L}^{*}(\mathfrak{g}\ltimes V,\mathfrak{g}\ltimes V)$ is a Maurer-Cartan element of the differential graded Lie algebra $(\mathcal{L}^{*}(\mathfrak{g}\ltimes V,\mathfrak{g}\ltimes V),[\ ,\ ]_{LY},d_{\bar{\Pi}})$ , where $\bar{\Pi}=(\bar{\mu},\bar{\theta}).$

Proof.

It follows from Theorem 4.2. ∎

Let $(\mathfrak{g},\chi_{\mathfrak{g}},\omega_{\mathfrak{g}})$ and $(\mathfrak{h},\chi_{\mathfrak{h}},\omega_{\mathfrak{h}})$ be two Lie-Yamaguti algebras. Then $(\mathfrak{g}\oplus\mathfrak{h},\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})$ is a Lie-Yamaguti algebra, where $\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}$ are defined by

\chi_{\mathfrak{g}\oplus\mathfrak{h}}(x+a,y+b)=\chi_{\mathfrak{g}}(x,y)+\chi_{\mathfrak{h}}(a,b),~{}~{}\omega_{\mathfrak{g}\oplus\mathfrak{h}}(x+a,y+b,z+c)=\omega_{\mathfrak{g}}(x,y,z)+\omega_{\mathfrak{h}}(a,b,c)

for all $x,y,z\in\mathfrak{g},a,b,c\in\mathfrak{h}$ .

In view of Theorem 4.2, $(\mathcal{L}^{*}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{g}\oplus\mathfrak{h}),[\ ,\ ]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$ is a differential graded Lie algebra. Define $\mathcal{C}_{>}^{n}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})\subset\mathcal{C}^{n}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h}),~{}\mathbb{C}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})\subset\mathbb{C}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})$ respectively by

\mathcal{C}^{n}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})=\mathcal{C}_{>}^{n}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})\oplus\mathcal{C}^{n}(\mathfrak{h},\mathfrak{h}),~{}\mathbb{C}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})=\mathbb{C}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})\oplus\mathbb{C}(\mathfrak{h},\mathfrak{h}).

Denote by $\mathcal{C}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})=\oplus_{n}\mathcal{C}_{>}^{n}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})$ and $\mathcal{L}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})=\mathcal{C}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})\oplus\mathbb{C}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})$ .

Similar to the case of $3$ -Lie algebras [27], we have

Proposition 4.4.

With the above notations, $(\mathcal{L}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h}),[\ ,\ ]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$ is a differential graded Lie subalgebra of $(\mathcal{L}^{*}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{g}\oplus\mathfrak{h}),[\ ,\ ]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$ .

Proposition 4.5.

The following conditions are equivalent:

$(i)$

$(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\{\ ,\ ,\ \}_{\omega})$ is a Lie-Yamaguti algebra, which is a non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ .

(ii)

$\Pi=(\bar{\chi},\bar{\omega})$ is a Maurer-Cartan element of the differential graded Lie algebra $(C_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h}),[\ ,\ ]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$ , where

\bar{\chi}(x+a,y+b)=\chi(x,y)+\mu(x)b-\mu(y)a,

	$\displaystyle\bar{\omega}(x+a,y+b,z+c)=$	$\displaystyle\omega(x,y,z)+D(x,y)c+\theta(y,z)a-\theta(x,z)b$
		$\displaystyle+T(z)(a,b)+\rho(x)(b,c)-\rho(y)(a,c),$

for all $x,y,z\in\mathfrak{g},a,b,c\in\mathfrak{h}$ .

Proof.

In view of the definition of Maurer-Cartan element, $\Pi=(\bar{\chi},\bar{\omega})$ is a Maurer-Cartan element of the differential graded Lie algebra $(\mathcal{L}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h}),[\ ,\ ]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$ if and only if

d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\Pi+\frac{1}{2}[\Pi,\Pi]_{LY}=0,

that is,

(75)

[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),(\bar{\chi},\bar{\omega})]_{LY}+\frac{1}{2}[(\bar{\chi},\bar{\omega}),(\bar{\chi},\bar{\omega})]_{LY}=0.

On the other hand, by Proposition 4.1, we know that $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\{\ ,\ ,\ \}_{\omega})$ is a Lie-Yamaguti algebra if and only if

(76)

[(\chi_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\chi},\omega_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\omega}),(\chi_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\chi},\omega_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\omega})]_{LY}=0.

Since $(\mathfrak{g}\oplus\mathfrak{h},\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})$ is a Lie-Yamaguti algebra, by computations, we have

		$\displaystyle[(\chi_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\chi},\omega_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\omega}),(\chi_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\chi},\omega_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\omega})]_{LY}$
	$\displaystyle=$	$\displaystyle[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})]_{LY}+[(\bar{\chi},\bar{\omega}),(\bar{\chi},\bar{\omega})]_{LY}+2[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),(\bar{\chi},\bar{\omega})]_{LY}$
	$\displaystyle=$	$\displaystyle[(\bar{\chi},\bar{\omega}),(\bar{\chi},\bar{\omega})]_{LY}+2[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),(\bar{\chi},\bar{\omega})]_{LY},$

which yields that Eq. (75) holds if and only if Eq. (76) holds. This completes the proof. ∎

Proposition 4.6.

Two non-abelian extensions $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi_{0}},\{\ ,\ ,\ \}_{\omega_{0}})$ and $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\{\ ,\ ,\ \}_{\omega})$ are equivalent if and only if the Maurer-Cartan elements $\Pi_{0}=(\bar{\chi}_{0},\bar{\omega}_{0})$ and $\Pi=(\bar{\chi},\bar{\omega})$ are gauge equivalent.

Proof.

Let $\Pi_{0},\Pi$ be two Maurer-Cartan elements of the differential graded Lie algebra $(\mathcal{L}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h}),[\ ,\ ]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$ . $\Pi_{0},\Pi$ are gauge equivalent if and only if there is a linear map $\varphi\in\mathrm{Hom}(\mathfrak{g},\mathfrak{h})$ such that

	$\displaystyle\Pi_{0}=$	$\displaystyle e^{ad_{\varphi}}\Pi-\frac{e^{ad_{\varphi}}-1}{ad_{\varphi}}d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi$
	$\displaystyle=$	$\displaystyle(id+ad_{\varphi}+\frac{1}{2!}ad_{\varphi}^{2}+\cdot\cdot\cdot++\frac{1}{n!}ad_{\varphi}^{n}+\cdot\cdot\cdot)\Pi$
		$\displaystyle-(id+\frac{1}{2!}ad_{\varphi}+\frac{1}{3!}ad_{\varphi}^{2}+\cdot\cdot\cdot+\frac{1}{n!}ad_{\varphi}^{n-1}+\cdot\cdot\cdot)d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi.$

In the following, we denote by

ad_{\varphi}\Pi=[\varphi,\Pi]_{LY}=([\varphi,\Pi]_{I},[\varphi,\Pi]_{II}),~{}~{}ad_{\varphi}^{2}\Pi=[\varphi,[\varphi,\Pi]_{LY}]_{LY}=([\varphi,[\varphi,\Pi]_{LY}]_{I},[\varphi,[\varphi,\Pi]_{LY}]_{II}),

d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi=[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{LY}=([(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{I},[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}),

and

[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{LY}=([\varphi,[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{LY}]_{I},[\varphi,[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{LY}]_{II}).

Using Eqs. (68)-(71), for all $w_{i}=x_{i}+a_{i}\in\mathfrak{g}\oplus\mathfrak{h}~{}(i=1,2,3)$ , we get

	$\displaystyle[\varphi,\Pi]_{I}(w_{1},w_{2})=$	$\displaystyle\varphi\bar{\chi}(w_{1},w_{2})-\bar{\chi}(w_{1},\varphi(x_{2}))-\bar{\chi}(\varphi(x_{1}),w_{2})$
	$\displaystyle=$	$\displaystyle-\mu(x_{1})\varphi(x_{2})+\mu(x_{2})\varphi(x_{1}),$

	$\displaystyle[\varphi,\Pi]_{II}(w_{1},w_{2},w_{3})=$	$\displaystyle\varphi\bar{\omega}(w_{1},w_{2},w_{3})-\bar{\omega}(\varphi(x_{1}),w_{2},w_{3})-\bar{\omega}(w_{1},w_{2},\varphi(x_{3}))-\bar{\omega}(w_{1},\varphi(x_{2}),w_{3})$
	$\displaystyle=$	$\displaystyle\theta(x_{1},x_{3})\varphi(x_{2})-T(x_{3})(a_{1},\varphi(x_{2}))-\rho(x_{1})(\varphi(x_{2}),a_{3})$
		$\displaystyle-\theta(x_{2},x_{3})\varphi(x_{1})-T(x_{3})(\varphi(x_{1}),a_{2})+\rho(x_{2})(\varphi(x_{1}),a_{3})$
		$\displaystyle+\rho(x_{2})(a_{1},\varphi(x_{3}))-D(x_{1},x_{2})\varphi(x_{3})-\rho(x_{1})(a_{2},\varphi(x_{3})),$

[\varphi,[\varphi,\Pi]_{LY}]_{I}(w_{1},w_{2})=\varphi[\varphi,\Pi]_{I}(w_{1},w_{2})-[\varphi,\chi]_{I}(w_{1},\varphi(x_{2}))-[\varphi,\Pi]_{I}(\varphi(x_{1}),w_{2})=0,

	$\displaystyle[\varphi,[\varphi,\Pi]_{LY}]_{II}(w_{1},w_{2},w_{3})=$	$\displaystyle\varphi[\varphi,\Pi]_{II}(w_{1},w_{2},w_{3})-[\varphi,\Pi]_{II}(w_{1},\varphi(x_{2}),w_{3})$
		$\displaystyle-[\varphi,\Pi]_{II}(\varphi(x_{1}),w_{2},w_{3})-[\varphi,\Pi]_{II}(w_{1},w_{2},\varphi(x_{3}))$
	$\displaystyle=$	$\displaystyle 2T(x_{3})(\varphi(x_{1}),\varphi(x_{2}))+2\rho(x_{1})(\varphi(x_{2}),\varphi(x_{3}))-2\rho(x_{2})(\varphi(x_{1}),\varphi(x_{3})),$

	$\displaystyle[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{I}(w_{1},w_{2})=$	$\displaystyle\chi_{\mathfrak{g}\oplus\mathfrak{h}}(w_{1},\varphi(x_{2}))+\chi_{\mathfrak{g}\oplus\mathfrak{h}}(\varphi(x_{1}),w_{2})-\varphi\chi_{\mathfrak{g}\oplus\mathfrak{h}}(w_{1},w_{2})$
	$\displaystyle=$	$\displaystyle[a_{1},\varphi(x_{2})]_{\mathfrak{h}}+[\varphi(x_{1}),a_{2}]_{\mathfrak{h}}-\varphi([x_{1},x_{2}]_{\mathfrak{g}}),$

		$\displaystyle[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}(w_{1},w_{2},w_{3})$
	$\displaystyle=$	$\displaystyle\omega_{\mathfrak{g}\oplus\mathfrak{h}}(w_{1},\varphi(x_{2}),w_{3})+\omega_{\mathfrak{g}\oplus\mathfrak{h}}(\varphi(x_{1}),w_{2},w_{3})+\omega_{\mathfrak{g}\oplus\mathfrak{h}}(w_{1},w_{2},\varphi(x_{3}))-\varphi\omega_{\mathfrak{g}\oplus\mathfrak{h}}(w_{1},w_{2},w_{3})$
	$\displaystyle=$	$\displaystyle\{a_{1},\varphi(x_{2}),a_{3}\}_{\mathfrak{h}}+\{\varphi(x_{1}),a_{2},a_{3}\}_{\mathfrak{h}}+\{a_{1},a_{2},\varphi(x_{3})\}_{\mathfrak{h}}-\varphi(\{x_{1},x_{2},x_{3}\}_{\mathfrak{g}}),$
		$\displaystyle[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{I}(w_{1},w_{2})$
	$\displaystyle=$	$\displaystyle\varphi[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{I}(w_{1},w_{2})-[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{I}(w_{1},\varphi(x_{2}))-[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{I}(\varphi(x_{1}),w_{2})$
	$\displaystyle=$	$\displaystyle-2[\varphi(x_{1}),\varphi(x_{2})]_{\mathfrak{h}},$
		$\displaystyle[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{II}(w_{1},w_{2},w_{3})$
	$\displaystyle=$	$\displaystyle\varphi[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}(w_{1},w_{2},w_{3})-[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}(w_{1},\varphi(x_{2}),w_{3})$
		$\displaystyle-[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}(\varphi(x_{1}),w_{2},w_{3})-[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}(w_{1},w_{2},\varphi(x_{3}))$
	$\displaystyle=$	$\displaystyle-\{\varphi(x_{1}),\varphi(x_{2}),a_{3}\}_{\mathfrak{h}}-\{a_{1},\varphi(x_{2}),\varphi(x_{3})\}_{\mathfrak{h}}-\{\varphi(x_{1}),\varphi(x_{2}),a_{3}\}_{\mathfrak{h}}$
		$\displaystyle-\{\varphi(x_{1}),a_{2},\varphi(x_{3})\}_{\mathfrak{h}}-\{a_{1},\varphi(x_{2}),\varphi(x_{3})\}_{\mathfrak{h}}-\{\varphi(x_{1}),a_{2},\varphi(x_{3})\}_{\mathfrak{h}}$
	$\displaystyle=$	$\displaystyle-2\{\varphi(x_{1}),\varphi(x_{2}),a_{3}\}_{\mathfrak{h}}-2\{a_{1},\varphi(x_{2}),\varphi(x_{3})\}_{\mathfrak{h}}-2\{\varphi(x_{1}),a_{2},\varphi(x_{3})\}_{\mathfrak{h}},$
		$\displaystyle[\varphi,[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{LY}]_{I}(w_{1},w_{2})$
	$\displaystyle=$	$\displaystyle\varphi[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{I}(w_{1},w_{2})-[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{I}(x_{1}+a_{1},\varphi(x_{2}))-[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{I}(\varphi(x_{1}),w_{2})$
	$\displaystyle=$	$\displaystyle 0,$
		$\displaystyle[\varphi,[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{LY}]_{II}(w_{1},w_{2},w_{3})$
	$\displaystyle=$	$\displaystyle[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{II}(x_{1}+a_{1},\varphi(x_{2}),w_{3})+[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{II}(\varphi(x_{1}),w_{2},w_{3})$
		$\displaystyle+[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{II}(w_{1},w_{2},\varphi(x_{3}))-\varphi[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{II}(w_{1},w_{2},w_{3})$
	$\displaystyle=$	$\displaystyle 6\{\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3})\}_{\mathfrak{h}},$

and

ad_{\varphi}^{n}\Pi=0,~{}~{}ad_{\varphi}^{n}(d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\Pi)=0,~{}~{}\forall~{}~{}n\geq 3.

Thus, $\Pi$ and $\Pi_{0}$ are gauge equivalent Maurer-Cartan elements if and only if

(77)

\chi_{0}=\chi+[\varphi,\Pi]_{I}-d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi-\frac{1}{2!}[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi]_{I},

	$\displaystyle\omega_{0}=$	$\displaystyle\omega+[\varphi,\Pi]_{II}+\frac{1}{2!}[\varphi,[\varphi,\Pi]_{LY}]_{II}-d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi$
(78)			$\displaystyle-\frac{1}{2!}[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi]_{II}-\frac{1}{3!}[\varphi,[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi]_{LY}]_{II}.$

Therefore, Eqs. (77) and (78) hold if and only if Eqs. (53)-(58) hold. This finishes the proof.

∎

5. Extensibility of a pair of Lie-Yamaguti algebra automorphisms

In this section, we study extensibility of pairs of Lie-Yamaguti algebra automorphisms and characterize them by equivalent conditions.

Definition 5.1.

A pair of automorphisms $(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$ is said to be extensible with respect to a non-abelian extension

\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0

if there is an automorphism $\gamma\in\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})$ such that $i\beta=\gamma i,~{}p\gamma=\alpha p$ , that is, the following commutative diagram holds:

It is natural to ask: when is a pair of automorphisms $(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$ extensible? We discuss this problem in the following.

Theorem 5.2.

Let $0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$ and $(\chi,\omega,\mu,\theta,D,\rho,T)$ the corresponding non-abelian (2,3)-cocycle induced by $s$ . A pair $(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$ is extensible if and only if there is a linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ satisfying the following conditions:

		$\displaystyle\beta\omega(x,y,z)-\omega(\alpha(x),\alpha(y),\alpha(z))=T(\alpha(z))(\varphi(x),\varphi(y))-\rho(\alpha(y))(\varphi(x),\varphi(z))-\theta(\alpha(y),\alpha(z))\varphi(x)$
(79)		$\displaystyle+$	$\displaystyle\rho(\alpha(x))(\varphi(y),\varphi(z))+\theta(\alpha(x),\alpha(z))\varphi(y)-D(\alpha(x),\alpha(y))\varphi(z)+\varphi(\{x,y,z\}_{\mathfrak{h}})-\{\varphi(x),\varphi(y),\varphi(z)\}_{\mathfrak{h}},$

(80)

\beta\chi(x,y)-\chi(\alpha(x),\alpha(y))=[\varphi(x),\varphi(y)]_{\mathfrak{h}}+\varphi([x,y]_{\mathfrak{g}})-\mu(\alpha(x))\varphi(y)+\mu(\alpha(y))\varphi(x),

(81)

\beta(\theta(x,y)a)-\theta(\alpha(x),\alpha(y))\beta(a)=\{\beta(a),\varphi(x),\varphi(y)\}_{\mathfrak{h}}-T(\alpha(y))(\beta(a),\varphi(x))+\rho(\alpha(x))(\beta(a),\varphi(y)),

(82)

\beta D(x,y)a-D(\alpha(x),\alpha(y))\beta(a)=\{\varphi(x),\varphi(y),\beta(a)\}_{\mathfrak{h}}-\rho(\alpha(x))(\varphi(y),\beta(a))+\rho(\alpha(y))(\varphi(x),\beta(a)),

(83)

\beta(\rho(x)(a,b))-\rho(\alpha(x))(\beta(a),\beta(b))=\{\beta(a),\varphi(x),\beta(b)\}_{\mathfrak{h}},

(84)

\beta T(x)(a,b)-T(\alpha(x))(\beta(a),\beta(b))=\{\beta(b),\beta(a),\varphi(x)\}_{\mathfrak{h}},

(85)

\beta\mu(x)a-\mu(\alpha(x))\beta(a)=[\beta(a),\varphi(x)]_{\mathfrak{h}},

for all $x,y,z\in\mathfrak{g}$ and $a\in\mathfrak{h}$ .

Proof.

Assume that $(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$ is extensible, that is, there is an automorphism $\gamma\in\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})$ such that $\gamma i=i\beta$ and $p\gamma=\alpha p$ . Due to $s$ being a section of $p$ , for all $x\in\mathfrak{g}$ ,

p(s\alpha-\gamma s)(x)=\alpha(x)-\alpha(x)=0,

which implies that $(s\alpha-\gamma s)(x)\in\mathrm{ker}p=\mathfrak{h}$ . So we can define a linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ by

\varphi(x)=(s\alpha-\gamma s)(x),~{}~{}\forall~{}x\in\mathfrak{g}.

Using Eqs. (63) and (64), for $x,y\in\mathfrak{g},a\in\mathfrak{h}$ , we get

			$\displaystyle\beta(\theta(x,y)a)-\theta(\alpha(x),\alpha(y))\beta(a)$
		$\displaystyle=$	$\displaystyle\beta\{a,s(x),s(y)\}_{\hat{\mathfrak{g}}}-\{\beta(a),s\alpha(x),s\alpha(y)\}_{\hat{\mathfrak{g}}}$
		$\displaystyle=$	$\displaystyle\{\beta(a),\beta s(x),\beta s(y)\}_{\hat{\mathfrak{g}}}-\{\beta(a),s\alpha(x),s\alpha(y)\}_{\hat{\mathfrak{g}}}$
		$\displaystyle=$	$\displaystyle\{\beta(a),\beta s(x)-s\alpha(x),\beta s(y)\}_{\hat{\mathfrak{g}}}+\{\beta(a),s\alpha(x),\beta s(y)\}_{\hat{\mathfrak{g}}}-\{\beta(a),s\alpha(x),s\alpha(y)\}_{\hat{\mathfrak{g}}}$
		$\displaystyle=$	$\displaystyle\{\beta(a),-\varphi(x),\beta s(y)\}_{\hat{\mathfrak{g}}}+\{\beta(a),s\alpha(x),-\varphi(y)\}_{\hat{\mathfrak{g}}}$
		$\displaystyle=$	$\displaystyle-\{\beta(a),\varphi(x),\beta s(y)-s\alpha(y)\}_{\hat{\mathfrak{g}}}-\{\beta(a),\varphi(x),s\alpha(y)\}_{\hat{\mathfrak{g}}}-\{\beta(a),s\alpha(x),\varphi(y)\}_{\hat{\mathfrak{g}}}$
		$\displaystyle=$	$\displaystyle\{\beta(a),\varphi(x),\varphi(y)\}_{\hat{\mathfrak{g}}}-\{\beta(a),\varphi(x),s\alpha(y)\}_{\hat{\mathfrak{g}}}+\{s\alpha(x),\beta(a),\varphi(y)\}_{\hat{\mathfrak{g}}}$
		$\displaystyle=$	$\displaystyle\{\beta(a),\varphi(x),\varphi(y)\}_{\mathfrak{h}}-T(\alpha(y))(\beta(a),\varphi(x))+\rho(\alpha(x))(\beta(a),\varphi(y)),$

which indicates that Eq. (81) holds. Take the same procedure, we can prove that Eqs. (5.2), (80) and (82)-(85) hold.

Conversely, suppose that $(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$ and there is a linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ satisfying Eqs. (5.2)-(85). Since $s$ is a section of $p$ , all $\hat{w}\in\hat{\mathfrak{g}}$ can be written as $\hat{w}=a+s(x)$ for some $a\in\mathfrak{h},x\in\mathfrak{g}.$ Define a linear map $\gamma:\hat{\mathfrak{g}}\longrightarrow\hat{\mathfrak{g}}$ by

\gamma(\hat{w})=\gamma(a+s(x))=\beta(a)-\varphi(x)+s\alpha(x).

It is easy to check that $i\beta=\gamma i,~{}p\gamma=\alpha p$ and $\gamma(\mathfrak{h})=\mathfrak{h}$ . In the sequel, firstly, we prove that $\gamma$ is bijective. Indeed if $\gamma(\hat{w})=0,$ we have $s\alpha(x)=0$ and $\beta(a)-\varphi(x)=0$ . In view of $s$ and $\alpha$ being injective, we get $x=0$ , which follows that $a=0$ . Thus, $\hat{w}=a+s(x)=0$ , that is $\gamma$ is injective. For any $\hat{w}=a+s(x)\in\hat{\mathfrak{g}}$ ,

\gamma(\beta^{-1}(a)+\beta^{-1}\varphi\alpha^{-1}(x)+s\alpha^{-1}(x))=a+s(x)=\hat{w},

which yields that $\gamma$ is surjective. In all, $\gamma$ is bijective.

Secondly, we show that $\gamma$ is a homomorphism of the Lie-Yamaguti algebra $\hat{\mathfrak{g}}$ . In fact, for all $\hat{w}_{i}=a_{i}+s(x_{i})\in\hat{\mathfrak{g}}~{}(i=1,2,3)$ ,

			$\displaystyle\{\gamma(\hat{w}_{1}),\gamma(\hat{w}_{2}),\gamma(\hat{w}_{3})\}_{\hat{\mathfrak{g}}}$
		$\displaystyle=$	$\displaystyle\{\beta(a_{1})-\varphi(x_{1})+s\alpha(x_{1}),\beta(a_{2})-\varphi(x_{2})+s\alpha(x_{2}),\beta(a_{3})-\varphi(x_{3})+s\alpha(x_{3})\}_{\hat{\mathfrak{g}}}$
		$\displaystyle=$	$\displaystyle\{\beta(a_{1}),\beta(a_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}-\{\beta(a_{1}),\beta(a_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}+\{\beta(a_{1}),\beta(a_{2}),s\alpha(x_{3})\}_{\hat{\mathfrak{g}}}$
			$\displaystyle-\{\beta(a_{1}),\varphi(x_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}+\{\beta(a_{1}),\varphi(x_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}-\{\beta(a_{1}),\varphi(x_{2}),s\alpha(x_{3})\}_{\hat{\mathfrak{g}}}$
			$\displaystyle+\{\beta(a_{1}),s\alpha(x_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}-\{\beta(a_{1}),s\alpha(x_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}+\{\beta(a_{1}),s\alpha(x_{2}),s\alpha(x_{3})\}_{\hat{\mathfrak{g}}}$
			$\displaystyle-\{\varphi(x_{1}),\beta(a_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}+\{\varphi(x_{1}),\beta(a_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}-\{\varphi(x_{1}),\beta(a_{2}),s\alpha(x_{3})\}_{\hat{\mathfrak{g}}}$
			$\displaystyle+\{\varphi(x_{1}),\varphi(x_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}-\{\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}+\{\varphi(x_{1}),\varphi(x_{2}),s\alpha(x_{3})\}_{\hat{\mathfrak{g}}}$
			$\displaystyle-\{\varphi(x_{1}),s\alpha(x_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}+\{\varphi(x_{1}),s\alpha(x_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}-\{\varphi(x_{1}),s\alpha(x_{2}),s\alpha(x_{3})\}_{\hat{\mathfrak{g}}}$
			$\displaystyle+\{s\alpha(x_{1}),\beta(a_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}-\{s\alpha(x_{1}),\beta(a_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}+\{s\alpha(x_{1}),\beta(a_{2}),s\alpha(x_{3})\}_{\hat{\mathfrak{g}}}$
			$\displaystyle-\{s\alpha(x_{1}),\varphi(x_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}+\{s\alpha(x_{1}),\varphi(x_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}-\{s\alpha(x_{1}),\varphi(x_{2}),s\alpha(x_{3})\}_{\hat{\mathfrak{g}}}$
			$\displaystyle+\{s\alpha(x_{1}),s\alpha(x_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}-\{s\alpha(x_{1}),s\alpha(x_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}+\{s\alpha(x_{1}),s\alpha(x_{2}),s\alpha(x_{3})\}_{\hat{\mathfrak{g}}}$
		$\displaystyle=$	$\displaystyle\{\beta(a_{1}),\beta(a_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}-\{\beta(a_{1}),\beta(a_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}+T(\alpha(x_{3}))(\beta(a_{1}),\beta(a_{2}))$
			$\displaystyle-\{\beta(a_{1}),\varphi(x_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}+\{\beta(a_{1}),\varphi(x_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}-T(\alpha(x_{3}))(\beta(a_{1}),\varphi(x_{2}))$
			$\displaystyle-\rho(\alpha(x_{2}))(\beta(a_{1}),\beta(a_{3}))+\rho(\alpha(x_{2}))(\beta(a_{1}),\varphi(x_{3}))+\theta(\alpha(x_{2}),\alpha(x_{3}))\beta(a_{1})$
			$\displaystyle-\{\varphi(x_{1}),\beta(a_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}+\{\varphi(x_{1}),\beta(a_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}-T(\alpha(x_{3}))(\varphi(x_{1}),\beta(a_{2}))$
			$\displaystyle+\{\varphi(x_{1}),\varphi(x_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}-\{\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3})\}_{\hat{\mathfrak{g}}}+T(\alpha(x_{3}))(\varphi(x_{1}),\varphi(x_{2}))$
			$\displaystyle+\rho(\alpha(x_{2}))(\varphi(x_{1}),\beta(a_{3}))-\rho(\alpha(x_{2}))(\varphi(x_{1}),\varphi(x_{3}))-\theta(\alpha(x_{2}),\alpha(x_{3}))\varphi(x_{1})$
			$\displaystyle+\rho(\alpha(x_{1}))(\beta(a_{2}),\beta(a_{3}))-\rho(\alpha(x_{1}))(\beta(a_{2}),\varphi(x_{3}))-\theta(\alpha(x_{1}),\alpha(x_{3}))\beta(a_{2})$
			$\displaystyle-\rho(\alpha(x_{1}))(\varphi(x_{2}),\beta(a_{3}))+\rho(\alpha(x_{1}))(\varphi(x_{2}),\varphi(x_{3}))+\theta(\alpha(x_{1}),\alpha(x_{3}))\varphi(x_{2})$
			$\displaystyle+D(\alpha(x_{1}),\alpha(x_{2}))\beta(a_{3})-D(\alpha(x_{1}),\alpha(x_{2}))\varphi(x_{3})+\omega(\alpha(x_{1}),\alpha(x_{2}),\alpha(x_{3}))+s\alpha\{x_{1},x_{2},x_{3}\}_{\mathfrak{g}}$

and

			$\displaystyle\gamma(\{\hat{w}_{1},\hat{w}_{2},\hat{w}_{3}\}_{\hat{\mathfrak{g}}})$
		$\displaystyle=$	$\displaystyle\gamma(\{a_{1},a_{2},a_{3}\}_{\hat{\mathfrak{g}}}+\{a_{1},a_{2},s(x_{3})\}_{\hat{\mathfrak{g}}}+\{a_{1},s(x_{2}),a_{3}\}_{\hat{\mathfrak{g}}}+\{a_{1},s(x_{2}),s(x_{3})\}_{\hat{\mathfrak{g}}}$
			$\displaystyle+\{s(x_{1}),a_{2},s(x_{3})\}_{\hat{\mathfrak{g}}}+\{s(x_{1}),s(x_{2}),a_{3}\}_{\hat{\mathfrak{g}}}+[s(x_{1}),a_{2},a_{3}]_{\hat{\mathfrak{g}}}+\omega(x_{1},x_{2},x_{3})+s\{x_{1},x_{2},x_{3}\}_{\mathfrak{g}})$
		$\displaystyle=$	$\displaystyle\{\beta(a_{1}),\beta(a_{2}),\beta(a_{3})\}_{\hat{\mathfrak{g}}}+\beta(T(x_{3})(a_{1},a_{2})-\rho(x_{2})(a_{1},a_{3})+\theta(x_{2},x_{3})a_{1}-\theta(x_{1},x_{3})a_{2}$
			$\displaystyle+D(x_{1},x_{2})a_{3}+\rho(x_{1})(a_{2},a_{3}))+\beta(\{s(x_{1}),s(x_{2}),s(x_{3})\}_{\hat{\mathfrak{g}}})+\beta\omega(x_{1},x_{2},x_{3})$
			$\displaystyle+s\alpha\{x_{1},x_{2},x_{3}\}_{\mathfrak{g}}-\varphi(\{x_{1},x_{2},x_{3}\}_{\mathfrak{g}}).$

Thanks to Eqs. (5.2)-(83), we have

\gamma(\{\hat{w}_{1},\hat{w}_{2},\hat{w}_{3}\}_{\hat{\mathfrak{g}}})=\{\gamma(\hat{w}_{1}),\gamma(\hat{w}_{2}),\gamma(\hat{w}_{3})\}_{\hat{\mathfrak{g}}}.

By the same token, $\gamma([\hat{w}_{1},\hat{w}_{2}]_{\hat{\mathfrak{g}}})=[\gamma(\hat{w}_{1}),\gamma(\hat{w}_{2})]_{\hat{\mathfrak{g}}}.$ Hence, $\gamma\in\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})$ . This completes the proof. ∎

Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$ and $(\chi,\omega,\mu,\theta,D,\rho,T)$ be the corresponding non-abelian (2,3)-cocycle induced by $s$ . For all $(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$ , define maps $\chi_{(\alpha,\beta)}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h},~{}\omega_{(\alpha,\beta)}:\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h},~{}\mu_{(\alpha,\beta)}:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h}),~{}\theta_{(\alpha,\beta)},D_{(\alpha,\beta)}:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h}),~{}\rho_{(\alpha,\beta)},T_{(\alpha,\beta)}:\mathfrak{g}\longrightarrow\mathrm{Hom}(\mathfrak{h}\wedge\mathfrak{h},\mathfrak{h})$ respectively by

(86)

\omega_{(\alpha,\beta)}(x,y,z)=\beta\omega(\alpha^{-1}(x),\alpha^{-1}(y),\alpha^{-1}(z)),~{}~{}\chi_{(\alpha,\beta)}(x,y)=\beta\chi(\alpha^{-1}(x),\alpha^{-1}(y)),

(87)

\theta_{(\alpha,\beta)}(x,y)a=\beta(\theta(\alpha^{-1}(x),\alpha^{-1}(y))\beta^{-1}(a)),~{}~{}D_{(\alpha,\beta)}(x,y)a=\beta D(\alpha^{-1}(x),\alpha^{-1}(y))\beta^{-1}(a),

(88)

\rho_{(\alpha,\beta)}(x)(a,b)=\beta\rho(\alpha^{-1}(x))(\beta^{-1}(a),\beta^{-1}(b)),~{}~{}T_{(\alpha,\beta)}(x)(a,b)=\beta T(\alpha^{-1}(x))(\beta^{-1}(a),\beta^{-1}(b)),

(89)

\mu_{(\alpha,\beta)}(x)a=\beta\mu(\alpha^{-1}(x))\beta^{-1}(a),

for all $x,y,z\in\mathfrak{g},a,b\in\mathfrak{h}.$

Proposition 5.3.

With the above notations, $(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})$ is a non-abelian (2,3)-cocycle.

Proof.

By Eqs. (• ‣ 3.3), (86) and (87), for all $x_{1},x_{2},y_{1},y_{2},y_{3}\in\mathfrak{g}$ , we get

			$\displaystyle D_{(\alpha,\beta)}(x_{1},x_{2})\omega_{(\alpha,\beta)}(y_{1},y_{2},y_{3})+\omega_{(\alpha,\beta)}(x_{1},x_{2},\{y_{1},y_{2},y_{3}\}_{\mathfrak{g}})$
		$\displaystyle=$	$\displaystyle\beta D(\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}))\beta^{-1}\beta\omega_{(\alpha,\beta)}(\alpha^{-1}(y_{1}),\alpha^{-1}(y_{2}),\alpha^{-1}(y_{3}))$
			$\displaystyle-\beta\omega_{(\alpha,\beta)}(\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(\{y_{1},y_{2},y_{3}\}_{\mathfrak{g}}))$
		$\displaystyle=$	$\displaystyle\beta\omega(\{\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{1})\}_{\mathfrak{g}},\alpha^{-1}(y_{2}),\alpha^{-1}(y_{3}))$
			$\displaystyle+\beta\theta(\alpha^{-1}(y_{2}),\alpha^{-1}(y_{3}))\omega(\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{1}))$
			$\displaystyle+\beta\omega(\alpha^{-1}(y_{1}),\{\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{2})\}_{\mathfrak{g}},\alpha^{-1}(y_{3}))$
			$\displaystyle-\beta\theta(\alpha^{-1}(y_{1}),\alpha^{-1}(y_{3}))\omega(\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{2}))$
			$\displaystyle+\beta\omega(\alpha^{-1}(y_{1}),\alpha^{-1}(y_{2}),\{\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{3})\}_{\mathfrak{g}})$
			$\displaystyle+\beta D(\alpha^{-1}(y_{1}),\alpha^{-1}(y_{2}))\omega(\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{3}))$
		$\displaystyle=$	$\displaystyle\omega_{(\alpha,\beta)}(\{x_{1},x_{2},y_{1}\}_{\mathfrak{g}},y_{2},y_{3})+\theta_{(\alpha,\beta)}(y_{2},y_{3})\omega_{(\alpha,\beta)}(x_{1},x_{2},y_{1})+\omega_{(\alpha,\beta)}(y_{1},\{x_{1},x_{2},y_{2}\}_{\mathfrak{g}},y_{3})$
			$\displaystyle-\theta_{(\alpha,\beta)}(y_{1},y_{3})\omega_{(\alpha,\beta)}(x_{1},x_{2},y_{2})+\omega_{(\alpha,\beta)}(y_{1},y_{2},\{x_{1},x_{2},y_{3}\}_{\mathfrak{g}})+D_{(\alpha,\beta)}(y_{1},y_{2})\omega_{(\alpha,\beta)}(x_{1},x_{2},y_{3}),$

which implies that Eq. (• ‣ 3.3) holds for $(\omega_{(\alpha,\beta)},\theta_{(\alpha,\beta)})$ . Similarly, we can check that Eqs. (• ‣ 3.3)-(52) hold. This completes the proof. ∎

Theorem 5.4.

Let $0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian extension of a Lie-Yamaguti algebra $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$ and $(\chi,\omega,\mu,\theta,D,\rho,T)$ be the corresponding non-abelian (2,3)-cocycle induced by $s$ . A pair $(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$ is extensible if and only if the non-abelian (2,3)-cocycles $(\chi,\omega,\mu,\theta,D,\rho,T)$ and $(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})$ are equivalent.

Proof.

Suppose $(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$ is extensible, by Theorem 5.2, there is a linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ satisfying Eqs. (5.2)-(85). For all $x,y\in\mathfrak{g},a\in\mathfrak{h}$ , there exist $x_{0},y_{0}\in\mathfrak{g},a_{0}\in\mathfrak{h}$ such that $x=\alpha(x_{0}),y=\alpha(y_{0}),a=\beta(a_{0})$ . Thus, by Eqs. (81), (87) and (88), we have

			$\displaystyle\theta_{(\alpha,\beta)}(x,y)a-\theta(x,y)a$
		$\displaystyle=$	$\displaystyle\beta(\theta(\alpha^{-1}(x),\alpha^{-1}(y))\beta^{-1}(a))-\theta(x,y)a$
		$\displaystyle=$	$\displaystyle\beta(\theta(x_{0},y_{0})a_{0})-\theta(\alpha(x_{0}),\alpha(y_{0}))\beta(a_{0})$
		$\displaystyle=$	$\displaystyle\{\beta(a_{0}),\varphi(x_{0}),\varphi(y_{0})\}_{\mathfrak{h}}-T(\alpha(y_{0}))(\beta(a_{0}),\varphi(x_{0}))+\rho(\alpha(x_{0}))(\beta(a_{0}),\varphi(y_{0}))$
		$\displaystyle=$	$\displaystyle\{a,\varphi\alpha^{-1}(x),\varphi\alpha^{-1}(y)\}_{\mathfrak{h}}-T(y)(a,\varphi\alpha^{-1}(x))+\rho(x)(a,\varphi\alpha^{-1}(y)),$

which indicates that Eq. (56) holds. Analogously, Eqs. (53)-(55) and (57)-(58) hold. Thus, $(\chi,\omega,\mu,\theta,D,\rho,T)$ and $(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})$ are equivalent via the linear map $\varphi\alpha^{-1}:\mathfrak{g}\longrightarrow\mathfrak{h}$ .

The converse part can be obtained analogously.

∎

6. Wells exact sequences for Lie-Yamaguti algebras

In this section, we consider the Wells map associated with non-abelian extensions of Lie-Yamaguti algebras. Then we interpret the results gained in Section 5 in terms of the Wells map.

(90)

it+sp=I_{\hat{\mathfrak{g}}}.

Assume that $(\chi,\omega,\mu,\theta,D,\rho,T)$ is the corresponding non-abelian (2,3)-cocycle induced by $s$ . Define a linear map $W:\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})\longrightarrow H^{(2,3)}_{nab}(\mathfrak{g},\mathfrak{h})$ by

(91)

W(\alpha,\beta)=[(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})-(\chi,\omega,\mu,\theta,D,\rho,T)].

The map $W$ is called the Wells map.

Proposition 6.1.

The Wells map $W$ does not depend on the choice of sections.

Proof.

For all $x,y,z\in\mathfrak{g}$ , there are elements $x_{0},y_{0},z_{0}\in\mathfrak{g}$ such that $x=\alpha(x_{0}),y=\alpha(y_{0}),z=\alpha(z_{0})$ . Assume that $(\chi^{\prime},\omega^{\prime},\mu^{\prime},\theta^{\prime},D^{{}^{\prime}},\rho^{{}^{\prime}},T^{{}^{\prime}})$ is another non-abelian (2,3)-cocycle corresponding to the non-abelian extension $\mathcal{E}$ . By Lemma 3.7, we know that $(\chi^{\prime},\omega^{\prime},\mu^{\prime},\theta^{\prime},D^{{}^{\prime}},\rho^{{}^{\prime}},T^{{}^{\prime}})$ and $(\chi,\omega,\mu,\theta,D,\rho,T)$ are equivalent non-abelian (2,3)-cocycles via a linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ According to Eqs. (3.4) and (86)-(88), denote $\psi=\beta\varphi\alpha^{-1}$ , which follows that

		$\displaystyle\omega^{{}^{\prime}}_{(\alpha,\beta)}(x,y,z)-\omega_{(\alpha,\beta)}(x,y,z)$
	$\displaystyle=$	$\displaystyle\beta\omega^{{}^{\prime}}(\alpha^{-1}(x),\alpha^{-1}(y),\alpha^{-1}(z))-\beta\omega(\alpha^{-1}(x),\alpha^{-1}(y),\alpha^{-1}(z))$
	$\displaystyle=$	$\displaystyle\beta\omega^{{}^{\prime}}(x_{0},y_{0},z_{0})-\beta\omega(x_{0},y_{0},z_{0})$
	$\displaystyle=$	$\displaystyle\beta\Big{(}\theta(x_{0},z_{0})\varphi(y_{0})-D(x_{0},y_{0})\varphi(z_{0})+\rho(x_{0})(\varphi(y_{0}),\varphi(z_{0}))-\theta(y_{0},z_{0})\varphi(x_{0})$
		$\displaystyle+T(z_{0})(\varphi(x_{0}),\varphi(y_{0}))-\rho(y_{0})(\varphi(x_{0}),\varphi(z_{0}))-\{\varphi(x_{0}),\varphi(y_{0}),\varphi(z_{0})\}_{\mathfrak{h}}+\varphi\{x_{0},y_{0},z_{0}\}_{\mathfrak{g}}\Big{)}$
	$\displaystyle=$	$\displaystyle\beta\Big{(}\theta(\alpha^{-1}(x),\alpha^{-1}(z))\beta^{-1}\psi(y)-D(\alpha^{-1}(x),\alpha^{-1}(y))\beta^{-1}\psi(z)+\rho(\alpha^{-1}(x))(\beta^{-1}\psi(y),\beta^{-1}\psi(z))$
		$\displaystyle-\theta(\alpha^{-1}(y),\alpha^{-1}(z))\beta^{-1}\psi(x)+T(\alpha^{-1}(z))(\beta^{-1}\psi(x),\beta^{-1}\psi(y))-\rho(\alpha^{-1}(y))(\beta^{-1}\psi(x),\beta^{-1}\psi(z))\Big{)}$
		$\displaystyle-\{\psi(x),\psi(y),\psi(z)\}_{\mathfrak{h}}+\psi(\{x,y,z\}_{\mathfrak{g}})$
	$\displaystyle=$	$\displaystyle\theta_{(\alpha,\beta)}(x,z)\psi(y)-D_{(\alpha,\beta)}(x,y)\psi(z)+\rho_{(\alpha,\beta)}(x)(\psi(y),\psi(z))-\theta_{(\alpha,\beta)}(y,z)\psi(x)$
		$\displaystyle+T_{(\alpha,\beta)}(z)(\psi(x),\psi(y))-\rho_{(\alpha,\beta)}(y)(\psi(x),\psi(z))-\{\psi(x),\psi(y),\psi(z)\}_{\mathfrak{h}}+\psi(\{x,y,z\}_{\mathfrak{g}}).$

By the same token,

	$\displaystyle\chi^{{}^{\prime}}_{(\alpha,\beta)}(x,y)-\chi_{(\alpha,\beta)}(x,y)=[\psi(x),\psi(y)]_{\mathfrak{h}}+\psi([x,y]_{\mathfrak{g}})-\mu_{(\alpha,\beta)}(x)\psi(y)+\mu_{(\alpha,\beta)}(y)\psi(x),$
	$\displaystyle\theta^{{}^{\prime}}_{(\alpha,\beta)}(x,y)a-\theta_{(\alpha,\beta)}(x,y)a=\rho_{(\alpha,\beta)}(x)(a,\psi(y))-T(y)(a,\psi(x))+\{a,\psi(x),\psi(y)\}_{\mathfrak{h}},$
	$\displaystyle D^{{}^{\prime}}_{(\alpha,\beta)}(x,y)a-D_{(\alpha,\beta)}(x,y)a=\rho_{(\alpha,\beta)}(y)(\psi(x),a)-\rho_{(\alpha,\beta)}(x)(\psi(y),a)+\{\psi(x),\psi(y),a\}_{\mathfrak{h}},$
	$\displaystyle\mu^{{}^{\prime}}_{(\alpha,\beta)}(x)a-\mu_{(\alpha,\beta)}(x)a=[a,\psi(x)]_{\mathfrak{h}},~{}~{}~{}~{}\rho^{{}^{\prime}}_{(\alpha,\beta)}(x)(a,b)-\rho_{(\alpha,\beta)}(x)(a,b)=\{a,\psi(x),b\}_{\mathfrak{h}},$
	$\displaystyle T^{{}^{\prime}}_{(\alpha,\beta)}(x)(a,b)-T_{(\alpha,\beta)}(x)(a,b)=\{b,a,\psi(x)\}_{\mathfrak{h}}.$

So, $(\chi^{\prime}_{(\alpha,\beta)},\omega^{\prime}_{(\alpha,\beta)},\mu^{\prime}_{(\alpha,\beta)},\theta^{\prime}_{(\alpha,\beta)},D^{\prime}_{(\alpha,\beta)},\rho^{\prime}_{(\alpha,\beta)},T^{\prime}_{(\alpha,\beta)})$ and $(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})$ are equivalent non-abelian (2,3)-cocycles via the linear map $\psi=\beta\varphi\alpha^{-1}$ . Combining Lemma 3.7, we know that

\displaystyle(\chi^{\prime}_{(\alpha,\beta)},\omega^{\prime}_{(\alpha,\beta)},\mu^{\prime}_{(\alpha,\beta)},\theta^{\prime}_{(\alpha,\beta)},D^{\prime}_{(\alpha,\beta)},\rho^{\prime}_{(\alpha,\beta)},T^{\prime}_{(\alpha,\beta)})-(\chi^{\prime},\omega^{\prime},\mu^{\prime},\theta^{\prime},D^{{}^{\prime}},\rho^{{}^{\prime}},T^{{}^{\prime}})

and

\displaystyle(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})-(\chi,\omega,\mu,\theta,D,\rho,T)

are equivalent via the linear map $\beta\varphi\alpha^{-1}-\varphi$ . ∎

Proposition 6.2.

(92)

K:\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})\longrightarrow\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{g}),~{}~{}K(\gamma)=(p\gamma s,\gamma|_{\mathfrak{h}}),~{}\forall~{}\gamma\in\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}}).

Then $K$ is a homomorphism of groups.

Proof.

One can take the same procedure of Lie algebras, see [1]. ∎

Theorem 6.3.

Assume that $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ is a non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $\hat{\mathfrak{g}}$ . Then there is an exact sequence:

1\longrightarrow\mathrm{Aut}_{\mathfrak{h}}^{\mathfrak{g}}(\hat{\mathfrak{g}})\stackrel{{\scriptstyle H}}{{\longrightarrow}}\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})\stackrel{{\scriptstyle K}}{{\longrightarrow}}\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})\stackrel{{\scriptstyle W}}{{\longrightarrow}}H^{(2,3)}_{nab}(\mathfrak{g},\mathfrak{h}),

where $\mathrm{Aut}_{\mathfrak{h}}^{\mathfrak{g}}(\hat{\mathfrak{g}})=\{\gamma\in\mathrm{Aut}(\hat{\mathfrak{g}})|K(\gamma)=(I_{\mathfrak{g}},I_{\mathfrak{h}})\}$ .

Proof.

Obviously, $\mathrm{Ker}K=\mathrm{Im}H$ and $H$ is injective. We only need to prove that $\mathrm{Ker}W=\mathrm{Im}K$ . By Theorem 5.4, for all $(\alpha,\beta)\in\mathrm{Ker}W$ , we get that $(\alpha,\beta)$ is extensible with respect to the non-abelian extension $\mathcal{E}$ , that is, there is a $\gamma\in\mathrm{Aut}_{\mathfrak{h}}^{\mathfrak{g}}(\hat{\mathfrak{g}})$ , such that $i\beta=\gamma i,~{}p\gamma=\alpha p$ , which follows that $\alpha=\alpha ps=p\gamma s,~{}\beta=\gamma|_{\mathfrak{h}}.$ Thus, $(\alpha,\beta)\in\mathrm{Im}K$ . On the other hand, for any $(\alpha,\beta)\in\mathrm{Im}K$ , there is an isomorphism $\gamma\in\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})$ , such that Eq. (92) holds. Combining Eq. (90) and $\mathrm{Im}i=\mathrm{Ker}p$ , we obtain $\alpha p=p\gamma sp=p\gamma(I_{\hat{\mathfrak{g}}}-it)=p\gamma$ and $i\beta=\gamma i$ . Hence, $(\alpha,\beta)$ is extensible with respect to the non-abelian extension $\mathcal{E}$ . According to Theorem 5.4, $(\alpha,\beta)\in\mathrm{Ker}W$ . In all, $\mathrm{Ker}W=\mathrm{Im}K$ . ∎

Denote

(93)

\displaystyle Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})=

\displaystyle\left\{\varphi:\mathfrak{g}\rightarrow\mathfrak{h}\left|\begin{aligned} &[a,\varphi(x)]_{\mathfrak{h}}=\{a,b,\varphi(x)\}_{\mathfrak{h}}=\{\varphi(x),a,b\}_{\mathfrak{h}}=0,\\ &\{\varphi(x),\varphi(y),a\}_{\mathfrak{h}}-\rho(x)(\varphi(y),a)+\rho(y)(\varphi(x),a)=0,\\ &\{a,\varphi(x),\varphi(y)\}_{\mathfrak{h}}-T(y)(a,\varphi(x))+\rho(x)(a,\varphi(y))=0,\\ &\mu(x)\varphi(y)-\mu(y)\varphi(x)=\varphi([x,y]_{\mathfrak{g}})+[\varphi(x),\varphi(y)]_{\mathfrak{h}},~{}\\ &T(z)(\varphi(x),\varphi(y))-\rho(y)(\varphi(x),\varphi(z))-\theta(y,z)\varphi(x)\\ &+\rho(x)(\varphi(y),\varphi(z))+\theta(x,z)\varphi(y)-D(x,y)\varphi(z)\\ &=\{\varphi(x),\varphi(y),\varphi(z)\}_{\mathfrak{h}}-\varphi(\{x,y,z\}_{\mathfrak{h}}),~{}\forall~{}x,y,z\in{\mathfrak{g}},a,b\in{\mathfrak{h}}\end{aligned}\right.\right\}.

It is easy to check that $Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$ is an abelian group, which is called a non-abelian 1-cocycle on $\mathfrak{g}$ with values in $\mathfrak{h}$ .

Proposition 6.4.

With the above notations, we have

$(i)$

The linear map $S:\mathrm{Ker}K\longrightarrow Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$ defined by

(94) $S(\gamma)(x)=\varphi_{\gamma}(x)=s(x)-\gamma s(x),~{}\forall~{}~{}\gamma\in\mathrm{Ker}K,~{}x\in\mathfrak{g}$

is a homomorphism of groups.
$(ii)$

$S$ is an isomorphism, that is, $\mathrm{KerK}\simeq Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$ .

Proof.

(i)

By Eqs. (62)-(64), (92) and (94), for all $x,y,z\in\mathfrak{g}$ , we have,

		$\displaystyle\{\varphi_{\gamma}(x),\varphi_{\gamma}(y),\varphi_{\gamma}(z)\}_{\mathfrak{h}}-T(z)(\varphi_{\gamma}(x),\varphi_{\gamma}(y))+\rho(y)(\varphi_{\gamma}(x),\varphi_{\gamma}(z))+\theta(y,z)\varphi_{\gamma}(x)$
		$\displaystyle-\rho(x)(\varphi_{\gamma}(y),\varphi_{\gamma}(z))-\theta(x,z)\varphi_{\gamma}(y)+D(x,y)\varphi_{\gamma}(z)-\varphi_{\gamma}(\{x,y,z\}_{\mathfrak{g}})$
	$\displaystyle=$	$\displaystyle\{s(x)-\gamma s(x),s(y)-\gamma s(y),s(z)-\gamma s(z)\}_{\hat{\mathfrak{g}}}-\{s(x)-\gamma s(x),s(y)-\gamma s(y),s(z)\}_{\hat{\mathfrak{g}}}$
		$\displaystyle+\{s(y),s(x)-\gamma s(x),s(z)-\gamma s(z)\}_{\hat{\mathfrak{g}}}+\{s(x)-\gamma s(x),s(y),s(z)\}_{\hat{\mathfrak{g}}}-\{s(x),s(y)-\gamma s(y),s(z)-\gamma s(z)\}_{\hat{\mathfrak{g}}}$
		$\displaystyle-\{s(y)-\gamma s(y),s(x),s(z)\}_{\hat{\mathfrak{g}}}+\{s(x),s(y),s(z)-\gamma s(z)\}_{\hat{\mathfrak{g}}}+\gamma s(\{x,y,z\}_{\mathfrak{g}})-s(\{x,y,z\}_{\mathfrak{g}})$
	$\displaystyle=$	$\displaystyle\gamma s(\{x,y,z\}_{\mathfrak{g}})-\{\gamma s(x),\gamma s(y),\gamma s(z)\}_{\hat{\mathfrak{g}}}+\{s(x),s(y),s(z)\}_{\hat{\mathfrak{g}}}-s(\{x,y,z\}_{\mathfrak{g}})$
	$\displaystyle=$	$\displaystyle\omega(x,y,z)-\gamma\omega(x,y,z)$
	$\displaystyle=$	$\displaystyle 0.$

Analogously, we can check that $\varphi_{\gamma}$ satisfies the other identities in $Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$ . Thus, $S$ is well-defined. For any $\gamma_{1},\gamma_{2}\in\mathrm{Ker}K$ and $x\in\mathfrak{g}$ , suppose $S(\gamma_{1})=\varphi_{\gamma_{1}}$ and $S(\gamma_{2})=\varphi_{\gamma_{2}}$ . By Eqs. (92) and (94), we have

	$\displaystyle S(\gamma_{1}\gamma_{2})(x)$	$\displaystyle=s(x)-\gamma_{1}\gamma_{2}s(x)$
		$\displaystyle=s(x)-\gamma_{1}(s(x)-\varphi_{\gamma_{2}}(x))$
		$\displaystyle=s(x)-\gamma_{1}s(x)+\gamma_{1}\varphi_{\gamma_{2}}(x)$
		$\displaystyle=\varphi_{\gamma_{1}}(x)+\varphi_{\gamma_{2}}(x)$

which means that $S(\gamma_{1}\gamma_{2})=S(\gamma_{1})+S(\gamma_{2})$ is a homomorphism of groups.

$(ii)$

For all $\gamma\in\mathrm{Ker}K$ , we obtain that $K(\gamma)=(p\gamma s,\gamma|_{\mathfrak{h}})=(I_{\mathfrak{g}},I_{\mathfrak{h}})$ . If $S(\gamma)=\varphi_{\gamma}=0$ , we can get $\varphi_{\gamma}(x)=s(x)-\gamma s(x)=0$ , that is, $\gamma=I_{\hat{\mathfrak{g}}}$ , which indicates that $S$ is injective. Secondly, we prove that $S$ is surjective. Since $s$ is a section of $p$ , all $\hat{x}\in\hat{\mathfrak{g}}$ can be written as $a+s(x)$ for some $a\in\mathfrak{h},x\in\mathfrak{g}$ . For any $\varphi\in Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$ , define a linear map $\gamma:\hat{\mathfrak{g}}\rightarrow\hat{\mathfrak{g}}$ by

(95) $\gamma(\hat{x})=\gamma(a+s(x))=s(x)-\varphi(x)+a,~{}\forall~{}\hat{x}\in\hat{\mathfrak{g}}.$

It is obviously that $(p\gamma s,\gamma|_{\mathfrak{h}})=(I_{\mathfrak{g}},I_{\mathfrak{h}})$ . We need to verify that $\gamma$ is an automorphism of Lie-Yamaguti algebra $\hat{\mathfrak{g}}$ . One can take the same procedure as the proof of the converse part of Theorem 5.2. It follows that $\gamma\in\mathrm{Ker}K$ . Thus, $S$ is surjective. In all, $S$ is bijective. So $\mathrm{Ker}K\simeq Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$ .

∎

Combining Theorem 6.3 and Proposition 6.4, we have

Theorem 6.5.

0\longrightarrow Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})\stackrel{{\scriptstyle i}}{{\longrightarrow}}\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})\stackrel{{\scriptstyle K}}{{\longrightarrow}}\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})\stackrel{{\scriptstyle W}}{{\longrightarrow}}H^{(2,3)}_{nab}(\mathfrak{g},\mathfrak{h}).

7. Particular case: abelian extensions of Lie-Yamaguti algebras

In this section, we discuss the results of previous section in particular case.

We fix the abelian extension $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$ . Assume that $(\chi,\omega)$ is a (2,3)-cocycle corresponding to $\mathcal{E}$ . In the case of abelian extensions $\mathcal{E}$ , the maps $\rho,T$ defined by (64) become to zero. Then the quadruple $(\mathfrak{h},\mu,\theta,D)$ given by Eq. (63) is a representation of $\mathfrak{g}$ [35]. Moreover,

Theorem 7.1 ([35]).

$(i)$

The triple $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\{\ ,\ ,\ \}_{\omega})$ is a Lie-Yamaguti algebra if and only if $(\chi,\omega)$ is a (2,3)-cocycle of $\mathfrak{g}$ with coefficients in the representation $(\mathfrak{h},\mu,\theta,D)$ .
$(ii)$

Abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ are classified by the cohomology group $H^{(2,3)}(\mathfrak{g},\mathfrak{h})$ of $\mathfrak{g}$ with coefficients in $(\mathfrak{h},\mu,\theta,D)$ .

Theorem 7.2.

	$\displaystyle\beta\omega(x,y,z)-\omega(\alpha(x),\alpha(y),\alpha(z))=$	$\displaystyle\theta(\alpha(x),\alpha(z))\varphi(y)-\theta(\alpha(y),\alpha(z))\varphi(x)$
(96)			$\displaystyle-D(\alpha(x),\alpha(y))\varphi(z)+\varphi(\{x,y,z\}_{\mathfrak{h}}),$

(97)

\beta\chi(x,y)-\chi(\alpha(x),\alpha(y))=\mu(\alpha(y))\varphi(x)-\mu(\alpha(x))\varphi(y)+\varphi([x,y]_{\mathfrak{g}}),

(98)

\beta(\theta(x,y)a)=\theta(\alpha(x),\alpha(y))\beta(a),~{}~{}\beta\mu(x)a=\mu(\alpha(x))\beta(a).

Proof.

It can be obtained directly from Theorem 5.2. ∎

By Eqs. (10) and (98), we obtain

\beta D(x,y)a=D(\alpha(x),\alpha(y))\beta(a).

For all $(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$ , $(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)})$ may not be a (2,3)-cocycle. Indeed, $(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)})$ is a (2,3)-cocycle if Eq. (98) holds. Thus, it is natural to introduce the space of compatible pairs of automorphisms:

\displaystyle C_{(\mu,\theta)}=

\displaystyle\left\{(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})\left|\begin{aligned} &\beta(\theta(x,y)a)=\theta(\alpha(x),\alpha(y))\beta(a),\\ &\beta\mu(x)a=\mu(\alpha(x))\beta(a),~{}\forall~{}x,y\in{\mathfrak{g}},a\in{\mathfrak{h}}\end{aligned}\right.\right\}.

More detail on the space of compatible pairs of automorphisms can be found in [13].

Analogous to Theorem 5.4, we get

Theorem 7.3 ([13]).

In the case of abelian extensions, $Z^{1}_{nab}(\mathfrak{g},\mathfrak{h})$ defined by (93) becomes to ${H}^{1}(\mathfrak{g},\mathfrak{h})$ given in Section 2. In the light of Theorem 6.5 and Theorem 7.3, we have the following exact sequence:

Theorem 7.4 ([13]).

0\longrightarrow H^{1}(\mathfrak{g},\mathfrak{h})\stackrel{{\scriptstyle i}}{{\longrightarrow}}\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})\stackrel{{\scriptstyle K}}{{\longrightarrow}}C_{(\mu,\theta)}\stackrel{{\scriptstyle W}}{{\longrightarrow}}H^{(2,3)}(\mathfrak{g},\mathfrak{h}).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11871421); Natural Science Foundation of Zhejiang Province of China (LY19A010001); and Science and Technology Planning Project of Zhejiang Province (2022C01118).

Statements and Declarations

All datasets underlying the conclusions of the paper are available to readers. No conflict of interest exits in the submission of this manuscript.

References

[1] V. G. Bardakov, M. Singh, Extensions and automorphisms of Lie algebras, J. Algebra Appl. 16 (2017), 1750162.
[2] P. Benito, A. Elduque, F. M. Herce, Irreducible Lie-Yamaguti algebras, J. Pure Appl. Algebra 213 (2009),795-808.
[3] P. Benito, A. Elduque, F. M. Herce, Irreducible Lie-Yamaguti algebras of generic type, J. Pure Appl. Algebra 215 (2011), 108-130.
[4] J. M. Casas, E. Khmaladze, M. Ladra, Low-dimensional non-abelian Leibniz cohomology, Forum Math. 25 (3) (2013), 443–469.
[5] S. Chen, Y. Sheng, Z. Zheng, Non-abelian extensions of Lie 2-algebras, Sci. China Math. 55 (8) (2012), 1655–1668.
[6] A. Das, A. Mandal, Extensions, deformations and categorifications of AssDer pairs, arXiv: 2002.11415.
[7] A. Das, N. Ratheeb, Extensions and automorphisms of Rota-Baxter groups, J. Algebra 636 (2023), 626–665.
[8] L. Du, Y. Tan, Wells sequences for abelian extensions of Lie coalgebras, J. Algebra Appl. 20 (8) (2021), 2150149.
[9] L. Du, Y. Tan, Coderivations, abelian extensions and cohomology of Lie coalgebras, Commun. Algebra 49 (10) (2021), 4519-4542.
[10] S. Eilenberg, S. Maclane, Cohomology theory in abstract groups. II. Group extensions with non-abelian kernel, Ann. Math. 48 (1947), 326-341.
[11] Y. Frégier, Non-abelian cohomology of extensions of Lie algebras as Deligne groupoid, J. Algebra 398 (2014), 243–257.
[12] J. M. W. Goldman, The defoemation theory of representations of fundamental groups of compact Kahler manifolds, Publ. Math. IHES 67 (1988), 43–96.
[13] S. Goswamia, S. K. Mishraa, G. Mukherjee, Automorphisms of extensions of Lie-Yamaguti algebras and inducibility problem, J. Algebra 641 (2024), 268–306.
[14] Y. Guo, B. Hou, Crossed modules and non-abelian extensions of Rota-Baxter Leibniz algebras, J. Geom. Phys. 191 (2023), 104906.
[15] S. K. Hazra, A. Habib, Wells exact sequence and automorphisms of extensions of Lie superalgebras, J. Lie Theory 30 (2020), 179–199.
[16] B. Hou, J. Zhao, Crossed modules, non-abelian extensions of associative conformal algebras and Wells exact sequences, arXiv:2211.10842v1.
[17] N. Inassaridze, E. Khmaladze, M. Ladra, Non-abelian cohomology and extensions of Lie algebras, J. Lie Theory 18 (2008), 413–432.
[18] P. Jin, H. Liu, The Wells exact sequence for the automorphism group of a group extension, J. Algebra 324 (2010), 1219–1228.
[19] M. K. Kinyon, A. Weinstein, Leibniz algebras, Courant algebroids and multiplications on reductive homogeneous spaces, Amer. J. Math. 123 (2001), 525-550.
[20] J. Lin, L. Chen, Y. Ma, On the deformation of Lie-Yamaguti algebras, Acta Math. Sin. (Engl. Ser.) 31 (6) (2015), 938-946.
[21] J. Liu, Y. Sheng, Q. Wang, On non-abelian extensions of Leibniz algebras, Commun. Algebra 46 (2) (2018), 574–587.
[22] S. K. Mishra, A. Das, S. K. Hazra, Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms, Linear. Algebra. Appl. 669 (2023), 147-174.
[23] K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65.
[24] I. B. S. Passi, M. Singh, M. K. Yadav, Automorphisms of abelian group extensions, J. Algebra 324 (2010), 820–830.
[25] Y. Sheng, J. Zhao, Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras, Commun. Algebra 50 (9) (2022), 4056-4073.
[26] Y. Sheng, J. Zhao, Y. Zhou, Nijnhuis operators, product structures and complex structures on Lie-Yamaguti algebras, J. Algebra Appl. (2021), 2150146.
[27] L. Song, A. Makhlouf, R. Tang, On non-abelian extensions of 3-Lie algebras, Commun. Theor. Phys. 69 (2018), 347-356.
[28] N. Takahashi, Modules over quadratic spaces and representations of Lie Yamaguti algebras, J. Lie Theory 31 (4) (2021), 897-932.
[29] C. Wells, Automorphisms of group extensions, Trans. Amer. Math. Soc. 155 (1971), 189-194.
[30] R. Tang, Y. Frégier, Y. Sheng, Cohomologies of a Lie algebra with a derivation and applications, J. Algebra. 534 (2) (2019), 65-99.
[31] J. Zhao, Y. Qiao, Maurer-Cartan characterizations, $L_{\infty}$ -algebras, and cohomology of relative Rota-Baxter operators on Lie Yamaguti algebras, arXiv:2310.05360v1.
[32] J. Zhao, Y. Qiao, Cohomology and deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras, arXiv:2204.04872.
[33] K. Yamaguti, On the cohomology groups of general Lie triple system, Kumamoto J. Sci. Ser. A 8 (1969), 135-146.
[34] K. Yamaguti, On the Lie triple system and its generalization, J. Sci. Hiroshima Univ. Ser. A 21 (1957/58), 155-160.
[35] T. Zhang, J. Li, Deformations and extensions of Lie-Yamaguti algebras, Linear Multilinear Algebra 63 (11) (2015), 2212-2231.