# Representations and Deformations of Hom-Lie-Yamaguti Superalgebras

Authors: Shuangjian Guo; Xiaohui Zhang; Shengxiang Wang
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

LetL,α be a Hom-Lie-Yamaguti superalgebra. We first introduce the representation and cohomology theory of Hom-Lie-Yamaguti superalgebras. Also, we introduce the notions of generalized derivations and representations of L,α and present some properties. Finally, we investigate the deformations of L,α by choosing some suitable cohomology.


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## Body

## 1. Introduction

Lie triple systems arose initially in Cartan’s study of Riemannian geometry. Jacobson [1] first introduced Lie triple systems and Jordan triple systems in connection with problems from Jordan theory and quantum mechanics, viewing Lie triple systems as subspaces of Lie algebras that are closed relative to the ternary product. Lie-Yamaguti algebras were introduced by Yamaguti in [2] to give an algebraic interpretation of the characteristic properties of the torsion and curvature of homogeneous spaces with canonical connection in [3]. He called them generalized Lie triple systems at first, which were later called “Lie triple algebras”. Recently, they were renamed as “Lie-Yamaguti algebras” in [4].The theory of Hom-algebra started from Hom-Lie algebras introduced and discussed in [5], motivated by quasi-deformations of Lie algebras of vector fields, in particular q-deformations of Witt and Virasoro algebras. More precisely, Hom-Lie algebras are different from Lie algebras as the Jacobi identity is replaced by a twisted form using morphism. This twisted Jacobi identity is called Hom-Jacobi identity given by
(1)αx,y,z+αy,z,x+αz,x,y=0.So far, many authors have studied Hom-type algebras motivated in part for their applications in physics ([6–10]).In [11], Gaparayi and Issa introduced the concept of Hom-Lie-Yamaguti algebras, which can be viewed as a Hom-type generalization of Lie-Yamaguti algebras. In [12], Ma et al. studied the formal deformations of Hom-Lie-Yamaguti algebras. Recently, in [13], Lin et al. introduced the quasi-derivations of Lie-Yamaguti algebras. In [14], Zhang and Li introduced the representation and cohomology theory of Hom-Lie-Yamaguti algebras and studied deformations and extensions of Hom-Lie-Yamaguti algebras as an application, generalizing the results of [15]. In [16], Zhang et al. introduced the notion of crossed modules for Hom-Lie-Yamaguti algebras and studied their construction of Hom-Lie-Yamaguti algebras.In [17], Gaparayi et al. introduced the concept of Hom-Lie-Yamaguti superalgebras and gave some examples of Hom-Lie-Yamaguti superalgebras. Later, in [18], Gaparayi et al. studied the relation between Hom-Leibniz superalgebras and Hom-Lie-Yamaguti superalgebras.The purpose of this paper is to study the representations and deformations of Hom-Lie-Yamaguti superalgebras. This paper is organized as follows. In Section2, we recall the definitions of Hom-Lie-superalgebras and Hom-Lie supertriple systems. In Section 3, we introduce the representation and cohomology theory of Hom-Lie-Yamaguti superalgebras. In Section 4, we introduce the notions of generalized derivations and representations of a Hom-Lie-Yamaguti superalgebra and present some properties. In Section 5, we consider the theory of deformations of a Hom-Lie-Yamaguti superalgebra by choosing a suitable cohomology.


## 2. Preliminaries

Throughout this paper, we work on an algebraically closed fieldK of characteristic different from 2 and 3, all elements like x,y,z,u,v,w should be homogeneous unless otherwise state. We recall the definitions of Hom-Lie-superalgebras and Hom-Lie supertriple systems from [6, 9].Definition 1.
A Hom-Lie superalgebraL is a ℤ2-graded algebra L=L0¯⊕L1¯, endowed with an even bilinear mapping ·,·:L×L⟶L and a homomorphism α:L⟶L satisfying the following conditions, for all x∈Li¯,y∈Lj¯,z∈Lk¯ and i¯,j¯,k¯∈ℤ2:
(2)x,y⊂Li¯+j¯,x,y=−−1i¯j¯y,x,−1k¯i¯αx,y,z+−1i¯j¯αy,z,x+−1j¯k¯αz,x,y=0.Definition 2.
A Hom-Lie supertriple system is aZ2-graded vector space L, endowed with an even trilinear mapping ·,·,·:L×L×L⟶L and a homomorphism α:L⟶L satisfying the following conditions,
(1)
x,y,z=x+y+zmod2(2)
y,x,z=−−1xyx,y,z(3)
−1xzx,y,z+−1yxy,z,x+−1zyz,x,y=0(4)
αx,αy,z,u,v=x,y,z,αu,αv+−1zx+yαz,x,y,u,αv+−1∣x∣+∣y∣∣z∣+∣u∣αz,αu,x,y,v
for allx,y,z,u,v∈L, and ∣x∣ denotes the degree of the element x∈L.


## 3. Representations of Hom-Lie-Yamaguti Superalgebras

We recall the basic definition of Hom-Lie-Yamaguti superalgebras from [17].Definition 3.
A Hom-Lie-Yamaguti superalgebra (Hom-LY superalgebra for short) is a quadrupleL,·,·,·,·,·,α in which L is K-vector superspace, ·,· a binary superoperation and ·,·,· a ternary superoperation on L, and α:L⟶L an even linear map such that
(SHLY1)αx,y=αx,αy
(SHLY2)αx,y,z=αx,αy,αz
(SHLY3)x,y=−−1xyy,x
(SHLY4)x,y,z=−−1xyy,x,z
(SHLY5)↺x,y,z−1xzx,y,αz+x,y,z=0
SHLY6↺x,y,z−1xzx,y,αz,αu=0
(SHLY7)αx,αy,u,v=x,y,u,α2v+−1ux+yα2u,x,y,v
(SHLY8)α2x,α2y,u,v,w=x,y,u,α2v,α2w+−1ux+yα2u,x,y,v,α2w+−1u+vx+yα2u,α2v,x,y,wfor all x,y,z,u,v,w∈L and where ↺x,y,z denotes the sum over cyclic permutation of x,y,z, and ∣x∣ denotes the degree of the element x∈L. We denote a Hom-LY superalgebra by L,α.Remark 4.
(1)
Ifα=Id, then the Hom-LY superalgebra L,·,·,·,·,·,α reduces to a LY superalgebra L,·,·,·,·,·(see (SLY 1)-(SLY 6)).(2)
Ifx,y=0, for all x,y∈L, then L,·,·,·,·,·,α becomes a Hom-Lie supertriple system L,·,·,·,α2.(3)
Ifx,y,z=0 for all x,y,z∈L, then the Hom-LY superalgebra L,·,·,·,·,·,α becomes a Hom-Lie superalgebra L,·,·,α.
A homomorphism between two Hom-LY superalgebrasL,α and L′,α′ is a linear map φ:L⟶L′ satisfying φ∘α=α′∘φ and
(3)φx,y=φx,φy′,φx,y,z=φx,φy,φz′.Example 5.
Consider the 5-dimensionalℤ2-graded vector space L=L0¯⊕L1¯, over an arbitrary base filed K of characteristic different from 2, with basis u1,u2,u3 of L0¯ and e1,e2 of L1¯, and the nonzero products on these elements are induced by the following relations:
(4)u2∗u1=−u3,u1∗u2=u3,u1∗u3=−2u1,u3∗u1=2u1,u3∗u2=−2u2,u2∗u3=2u2,e1∗u2=e2,e1∗u3=−e1,e2∗u1=e1,e2∗u3=e2.Define the superspace homomorphismsα:L⟶L by
(5)αui=ui,i=1,2,3,αe1=−e1,αe2=−e2.It is not hard to check thatL,∗,α is a Hom-Leibniz superalgebra. By [18], we can define ·,· and ·,·,·, and the nonzero products on these elements are induced by the following relations:
(6)u1,u2=2u3,u1,u3=−4u1,u2,u3=4u2,e1,u2=e2,e1,u3=−e1,e2,u1=e1,e2,u3=e2,u1,u3,u2=2u3,u2,u3,u1=2u3,e1,u2,u3=−12e2,e1,u2,u1=−12e1,e1,u3,u2=12e2,e1,u3,u3=−12e1,e2,u1,u2=−12e2,e2,u1,u3=12e1,e2,u3,u1=−12e1,e2,u3,u3=−12e2.Then,L,·,·,·,·,·,α becomes a Hom-LY superalgebra.Definition 6.
LetL,α be a Hom-LY superalgebra and V,β be a Hom-vector super-space. A representation of L on V consists of an even linear map ρ:L⟶ End(V) and even bilinear maps D,θ:L×L⟶ End(V) such that the following conditions are satisfied:
(SHR1)ραx∘β=β∘ρx
(SHR2)Dαx,αy∘β=β∘Dx,y
(SHR3)θαx,αy∘β=β∘θx,y
(SHR4)Dx,y−−1xyθy,x+θx,y+ρx,y∘β−ραxρy+−1xyραyρx=0
(SHR5)Dx,y,αz+−1xy+zDy,z,αx+−1zx+yDz,x,αy=0
(SHR6)θx,y,αz∘β=−1yzθαx,αzρy−−1xy+zθαy,αzρx
(SHR7)Dαx,αyρz=−1zx+yρα2zDx,y+ρx,y,z∘β2
(SHR8)θαx,y,z∘β=−1xyρα2yθx,z−−1zx+yρα2zθx,y
(SHR9)Dα2x,α2yθu,v=−1u+vx+yθα2u,α2vDx,y+θx,y,u,α2v∘β2+−1ux+yθα2u,x,y,v∘β2
(SHR10)θα2x,y,z,u∘β2=−1z+ux+yθα2z,α2uθx,y−−1yzθα2y,α2uθx,z+−1xy+zDα2y,α2zθx,ufor any x,y,z,u,v∈L. In this case, V is also called an L-module.Proposition 7.
LetL,α be a Hom-LY superalgebra and V be a Hom-vector superspace. Assume we have a map ρ from L to End(V) and maps D,θ:L×L⟶ End(V) satisfying (SHR1)–(SHR10). Then, ρ,D,θ is a representation of L,α on V,β if and only if L⊕V is a Hom-LY superalgebra under the following maps:
(7)α+βx+u≔αx+βu,x+u,y+v≔x,y+ρxv−−1xyρyu,x+u,y+v,z+w≔x,y,z+Dx,yw−−1yzθx,zv+−1xy+zθy,zu,for any x,y,z∈L and u,v,w∈V.Proof.
It is easy to check that the conditions (SHLY1)–(SHLY4) hold, we only verify that conditions (SHLY5)–(SHLY8) hold for maps defined onL⊕V.
For (SHLY5), we have(8)x+u,y+v,z+w+c.p.=x,y,z+Dx,yw−−1yzθx,zv+−1xy+zθy,zu+c.p.
and(9)x+u,y+v,αz+βw+c.p.=x,y+ρxv−−1xyρyu,αz+βw+c.p.=x,y,αz+ρx,yβw−−1xzραzρxv+−1x+zyραzρyu+c.p.
Thus by (SHR4), the condition (SHLY5) holds.
For (SHLY6), we have(10)−1xzx+u,y+v,αz+βw,αp+βt+c.p.=−1xzx,y,αz,αp+Dx,y,αzβt−θx,y,αpβw+−1z+p+yxθαz,αpρxv−−1z+p+yxθαz,αpρyu+c.p.=0.
Thus by (SHR5), the condition (SHLY6) holds.
For (SHLY7), we have(11)αx+βu,αy+βv,z+w,p+t=αx,αy,z,p+Dαx,αyρzt−−1zpDαx,αyρpw−−1z+pyθx,z,pv+−1z+p+yxθy,z,pu,x+u,y+v,z+w,α2p+β2t+−1zx+yα2z+β2w,x+u,y+v,p+t=x,y,z,α2p+ρx,y,zt−−1x+y+zpρpDx,yw−−1x+y+zp+yzρpθx,zv+−1z+yp+x+pxρpθy,zu+−1zx+yα2z,x,y,p+−1zx+yρzDx,yt−−1xz+x+pyρzθx,pv+−1zp+xz+y+pρzθy,pu−−1pzρx,y,pw.
Thus by (SHR6), the condition (SHLY7) holds.
Now it suffices to verify (SHLY8). By the definition of the Hom-LY superalgebra, we have(12)α2x1+β2u1,α2x2+β2u2,y1+v1,y2+v2,y3+v3=α2x1,α2x2,y1,y2,y3−−1x2y1+y2+y3θα2x1,y1,y2,y3β2u2+−1x1x2+y1+y2+y3θα2x2,y1,y2,y3β2u1+Dα2x1,α2x2Dy1,y2v3−−1y2y3Dα2x1,α2x2θy1,y3v2+−1y1y2+y3Dα2x1,α2x2θy2,y3v1,x1+u1,x2+u2,y1+v1,α2y2+β2v2,α2y3+β2v3=x1,x2,y1,α2y2+α2y3+Dx1,x2,y1,α2y2β2v3+−1y2y3θx1,x2,y1,α2y3β2u1+−1y2+y3x1+x2+y1θα2y2,α2y3Dx1,x2v1−−1y1+y2+y3x2+x1+y1y2+y3θα2y2,α2y3θx1,y1u2+−1y2+y3x2+y3+x1y2+y3+x2+y3θα2y2,α2y3θx2,y1u1,−1y1x1+x2α2y1+β2v1,x1+u1,x2+u2,y2+v2,α2y3+β2v3=−1y1x1+x2α2y1,x1,x2,y2,α2y3+−1y1x1+x2Dα2y1,x1,x2,y2β2v3+−1y1y2+y3θx1,x2,y2,α2y3β2v1−−1y2y3θα2y1,α2y3Dx1,x2v3+−1y1+y3x1+y2+x2y1+y2+y3+y1y2θα2y1,α2y3θx1,y2u2−θα2y1,α2y3θx2,y2u1−1y1+y2x1+x2α2y1+β2v1,α2y2+β2v2,x1+u1,x2+u2,y3+v3=−1y1+y2x1+x2α2y1,α2y2,x1,x2,y3−−1y1x1+x2+y2y3−θα2y1,x1,x2,y3β2v2+−1∣y2∣∣x1∣+∣x2∣+∣y1∣y2+y3θα2y2,x1,x2,y3β2v1+−1y1+y2x1+x2Dα2y1,α2y2Dx1,x2v3−−1y1+y2x1+x3+x2y3Dα2y1,α2y2θx1,y3u2+−1y1+y2x2+y3+x1x2+y1+y2+y3Dα2y1,α2y2θx2,y3u1.
Thus by (SHR7), the condition (SHLY8) holds. Therefore, we obtain thatL⊕V is a Hom-LY superalgebra.LetV be a representation of Hom-LY superalgebra L. Let us define the cohomology group of L with coefficients in V. Let f:L×L×⋯×L⟶V be an n-linear map such that the following conditions are satisfied:
(13)fαx1,⋯,αxn=βfx1,⋯,xn,fx1,⋯,x2i−1,x2i,⋯,xn=0,ifx2i−1=x2i.The vector space spanned by such linear maps is called ann-cochain of L, which is denoted by CnL,V for n≥1.Definition 8.
For anyf,g∈C2nL,V×C2n+1L,V, the coboundary operator δ:f,g⟶δIf,δIIg is a mapping from C2nL,V×C2n+1L,V into C2n+2L,V×C2n+3L,V defined as follows:
(14)δIfx1,x2,⋯,x2n+2=−1x2n+1x1+x2+⋯+x2nρα2nx2n+1gx1,x2,⋯,x2n,x2n+2−−1x2n+2x1+x2+⋯+x2n+1ρα2nx2n+2gx1,x2,⋯,x2n,x2n+1−gαx1,αx2,⋯,αx2n,x2n+1,x2n+2+∑k=1n−1n+k+1+x2k−1+x2kx1+x2+⋯+x2k−2Dα2n−1x2k−1,α2n−1x2kfx1,⋯,x^2k−1,x^2k,⋯,x2n+2+∑k=1n∑j=2k+12n+2−1n+k+x2k−1+x2kx2k+1+x2k+2+⋯+xj−1fα2x1,⋯,x^2k−1,x^2k,⋯,x2k−1,x2k,xj,⋯,α2x2n+2,=−1x2n+2+x2n+3g+x1+x2+⋯+x2n+1θα2nx2n+2,α2nx2n+3gx1,⋯,x2n+1−−1x2n+1+x2n+3g+x1+x2+⋯+x2n+x2n+2x2n+3θα2nx2n+1,α2nx2n+3gx1,⋯,x2n,x2n+1+∑k=1n+1−1n+k+1+x2k−1+x2kg+x1+x2+⋯+x2k−2Dα2nx2k−1,α2nx2kgx1,⋯,x^2k−1,x^2k,⋯,x2n+3+∑k=1n+1∑j=2k+12n+3−1n+k+x2k−1+x2kx2k+1+x2k+2+⋯+xj−1gα2x1,⋯,x^2k−1,x^2k,⋯,x2k−1,x2k,xj,⋯,α2x2n+3.Proposition 9.
The coboundary operator defined above satisfiesδ∘δ=0, that is δI∘δI=0 and δII∘δII=0.Proof.
Similar to [14].The subspaceZ2nL,V×Z2n+1L,V of C2nL,V×C2n+1L,V spanned by all the f,g's such that δf,g=0 is called the space of cocycles while the space B2nL,V×B2n+1L,V≕δC2n−2L,V×C2n−1L,V is called the space of coboundaries.Definition 10.
For the casen≥2, the 2n,2n+1-cohomology group of a Hom-LY superalgebra L with coefficients in V is defined to be the quotient space:
(15)H2nL,V×H2n+1L,V≔Z2nL,V×Z2n+1L,V/B2nL,V×B2n+1L,V.In conclusion, we obtain a cochain complex whose cohomology group is called the cohomology group of a Hom-LY superalgebraL with coefficients in V.


## 4.αk-Derivations of Hom-Lie-Yamaguti Superalgebras

In this section, we give the definition ofαk-derivations of Hom-LY superalgebras, then, we study its generalized derivations.Definition 11.
A linear mapD:L⟶L is called an αk-derivation of L if it satisfies
(16)Dx,y=−1Dxαkx,Dy+Dx,αky,Dx,y,z=Dx,αky,αkz+−1Dxαkx,Dy,αkz+−1Dx+yαkx,αky,Dz,for all x,y,z∈L, where ∣D∣ denotes the degree of D.We denote byDerL=⊕k≥0DerαkL, where DerαkL is the set of all homogeneous αk-derivations of L. Obviously, DerL is a subalgebra of EndL.Theorem 12.
DerL is a Lie superalgebra, where the bracket product is defined as follows:
(17)D,D′=DD′−−1DD′D′D.Proof.
It is sufficient to proveDerαkL,DerαsL⊆Derαk+sL. Note that
(18)D,D′x,y=Dαsx,D′y+−1D′xD′x,αsy−−1DD′D′αkx,Dy+−1DxDx,αky=−1D′xDαsx,αkD′y+−1Dx+D′xαk+sx,DD′y+DD′x,αk+sy+−1DD′xαkD′x,Dαsy−−1Dx+D′xD′αkx,αsDy−−1D′x+Dx+DD′αk+sx,D′Dy−−1DD′D′Dx,αk+sy−−1D′x+D+DD′αsDx,D′αky=D,D′x,αk+sy+−1D′+Dxαk+sx,D,D′y.
Similarly, we can check that(19)D,D′x,y,z=D,D′x,αk+sy,αk+sz+−1D′+Dxαk+sx,D,D′y,αk+sz+−1D′+Dx+yαk+sx,αk+sy,D,D′z.
It follows thatD,D′∈Derαk+sL.Definition 13.
LetL,α be a Hom-LY superalgebra. D∈EndsL is said to be a homogeneous generalized αk-derivation of L, if there exist three endomorphisms D′,D″,D′″∈EndsL such that
(20)Dx,αky+−1sxαkx,D′y=D″x,y,Dx,αky,αkz+−1sxαkx,D′y,αkz+−1sx+yαkx,αky,D′′z=D′″x,y,z,for all x,y,z∈L.Definition 14.
LetL,α be a Hom-LY superalgebra. D∈EndsL is said to be a homogeneous αk-quasiderivation of L, if there exist endomorphisms D′,D′′∈EndsL such that
(21)Dx,αky+−1sxαkx,Dy=D′x,y,Dx,αky,αkz+−1sxαkx,Dy,αkz+−1sx+yαkx,αky,Dz=D″x,y,z,for all x,y,z∈L.LetGDerL and QDerL be the sets of homogeneous generalized αk-derivations and of homogeneous αk-quasiderivations, respectively. That is,
(22)GDerL≔⨁k≥0GDerαkL,QDerL≔⨁k≥0QDerαkL.Definition 15.
LetL,α be a Hom-LY superalgebra. The αk-centroid of L is the space of linear transformations on L given by
(23)CαkL=D∈EndLDx,αky=−1Dxαkx,Dy=Dx,yandDx,αky,αkz=−1Dxαkx,Dy,αkz=−1Dx+yαkx,αky,Dc=Dx,y,z.
We denoteCL=⨁k≥0CαkL and call it the centroid of L.Definition 16.
LetL,α be a Hom-LY superalgebra. The quasicentroid of L is the space of linear transformations on L given by
(24)QCαkL=D∈EndL∣Dx,αky=−1Dxαkx,DyandDx,αky,αkz=−1Dxαkx,Dy,αkz=−1Dx+yαkx,αky,Dz,for all x,y,z∈L. We denote QCL=⊕k≥0QCαkL and call it the quasicentroid of L.Remark 17.
LetL,α be a Hom-LY superalgebra. Then CL⊆QCL.Definition 18.
LetL,α be a Hom-LY superalgebra. D∈EndL is said to be a central αk-derivation of L if
(25)Dx,αky=Dx,y=0,Dx,y,z=Dx,αky,αkz=0,for all x,y,z∈L. Denote the set of all central αk-derivations by ZDerL.Remark 19.
LetL,α be a Hom-LY superalgebra. Then
(26)ZDerL⊆DerL⊆QDerL⊆GDerL⊆EndL.Definition 20.
LetL,α be a Hom-LY superalgebra. If ZL=x∈L∣x,y,z=0,∀x,y,z∈L, then ZL is called the center of L.Proposition 21.
LetL,α be a Hom-LY superalgebra, then the following statements hold:
(1)
GDerL,QDerL, and CL are subalgebras of EndL(2)
ZDerL is an ideal of DerL.Proof.
(1)
We only prove thatGDerL is a subalgebra of EndL, and similarly for cases of QDerL and CL. For any D1∈GDerαkL,D2∈GDerαsL and x,y,z∈L, we have
(27)D1D2x,αk+sy,αk+sz=D″′1D2x,αsy,αsz−−1∣D1∣∣D2∣+∣x∣αkD2x,D′1αsy,αk+sz−−1∣D1∣∣D2∣+∣x∣+∣y∣αkD2x,αk+sy,D″1αsz=D″′1D′″2x,y,z−−1D2xαsx,D′2y,αsz−−1∣D2∣∣x∣+∣y∣αsx,αsy,D″2z−−1∣D1∣∣D2∣+∣x∣αkD2x,D′1αsy,αk+sz−−1∣D1∣∣D2∣+∣x∣+∣y∣αkD2x,αk+sy,D″1αsz=D′″1D′″2x,y,z−−1D2xD′″1x,D′2y,z−−1∣D2∣∣x∣+∣y∣D′′′1x,y,D′′2z−−1∣D1∣∣D2∣+∣x∣D2x,D′1y,z−−1∣D1∣∣D2∣+∣x∣+∣y∣D2x,y,D″1z=D′″1D″′2x,y,z−−1D2xD1αsx,αkD′2y,αk+sz−−1∣D1∣+∣D2∣∣x∣αk+sx,D′1D′2y,αk+sz−−1D2x+∣D1∣∣x∣+∣y∣+∣D2∣αk+sx,αkD′2y,D″1αsz−−1∣D2∣∣x∣+∣y∣D1αsx,αk+sy,αkD″2z−−1∣D2∣∣x∣+∣y∣+D1xαk+sx,D′1αsy,αkD″2z−−1∣D1∣+∣D2∣∣x∣+∣y∣αk+sx,αk+sy,D″1D″2z−−1∣D1∣∣D2∣+∣x∣αkD2x,D′1αsy,αk+sz−−1∣D1∣∣D2∣+∣x∣+∣y∣αkD2x,αk+sy,D″1αsz.
Similarly, we have(28)D2D1x,αk+sy,αk+sz=D′″2D′″1x,y,z−−1D1xD2αkx,αsD1′y,αk+sz−−1D1+D2xαk+sx,D2′D1′y,αk+sz−−1D1x+D2x+y+D1αk+sx,αsD1′y,D″2αkz−−1D1x+yD2αkx,αk+sy,αsD″1z−−1D1x+y+D2xαk+sx,D′2αky,αsD″1z−−1D1+D2x+yαk+sx,αk+sy,D″2D″1z−−1D2D1+xαsD1x,D′2αky,αk+sz−−1D2D1+x+yαsD1x,αk+sy,D″2αkz.
It follows that(29)D1,D2x,αk+sy,αk+sz=D1D2x,y,z−−1D1D2D2D1x,y,z=D″′1D″′2−−1∣D1∣+∣D2∣D″′2D′″1x,y,z−−1xD1+D2αk+sx,D′1D′2−−1D1+D2D′2D′1y,αk+sz−−1x+yD1+D2αk+sx,αk+sy,D″1D″2−−1D1+D2D′′2D′′1z=D″′1,D″′2x,y,z−−1xD1+D2αk+sx,D′1,D′2y,αk+sz−−1x+yD1+D2αk+sx,αk+sy,D″1,D″2z,and it is easy to check that
(30)D1,D2x,αk+sy=D″1,D″2x,y−−1xD1+D2αk+sx,D′1,D′2y.
Obviously,D′1,D′2,D″1,D″2 and D′″1,D″′2 are contained in EndL, thus D1,D2∈GDerαk+lL⊆GDerL, that is, GDerL is a subalgebra of EndL.
(2)
For anyD1∈ZDerαkL,D2∈DerαsL and x,y,z∈L, we have(31)D1,D2x,y,z=D1D2x,y,z−−1D1D2D2D1x,y,z=0.
Also, we have(32)D1,D2x,αk+sy,αk+sz=D1D2x,αk+sy,αk+sz−−1D1D2D2D1x,αk+sy,αk+sz=0−−1D1D2D2D1x,αk+sy,αk+sz=−−1D1D2D2D1x,αky,αkz−−1D2D1+xαsD1x,D2αky,αk+sz+−1D1D2−1D2D1+x+yαsD1x,αk+sy,D2αkz=0,and it is easy to check that
(33)D1,D2x,αk+sy=0.
It follows thatD1,D2∈ZDerαk+sL. That is, ZDerL is an ideal of DerL.Lemma 22.
LetL,α be a Hom-LY superalgebra, then the following statements hold:
(34)DerL,CL⊆CL.QDerL,QCL⊆QCL.QCL,QCL⊆QDerL.CL⊆QDerL.QDerL+QCL⊆GDerL.Proof.
(1)–(4) are easy to prove and we omit them, we only check (5). In fact, letD1∈QDerαkL,D2∈QCαkL. Then, there exist D′1,D′′1∈EndsL, for any x,y,z∈L, we have
(35)D1x,αky+−1sxαkx,D1y=D1′x,y,D1x,αky,αkz+−1sxαkx,D1y,αkz+−1sx+yαkx,αky,D1z=D1″x,y,z.
Thus, for anyx,y,z∈L, we have
(36)D1+D2x,αky=D1x,αky+D2x,αky=D1′x,y−−1sxαkx,D1y+−1sxαkx,D2y=D1′x,y−−1sxαkx,D1−D2y,D1+D2x,αky,αkz=D1x,αky,αkz+D2x,αky,αkz=D1″x,y,z−−1sxαkx,D1y,αkz−−1sx+yαkx,αky,D1z+−1sxαkx,D2y,αkz=D1″x,y,z−−1sxαkx,D1−D2y,αkz−−1sx+yαkx,αky,D1z.
Therefore,D1+D2∈GDerαkL.Proposition 23.
LetL,α be a Hom-LY superalgebra, then QCL+QCL,QCL is a subalgebra of GDerL.Proof.
By Lemma22, (3) and (5), we have
(37)QCL+QCL,QCL⊆GDerL,and it follows that
(38)QCL+QCL,QCL,QCL+QCL,QCL⊆QCL+GDerL,QCL+QCL,QCL⊆QCL,QCL+QCL,QCL,QCL+QDerL,QCLQDerL,QCL,QCL.
It is easy to verify thatQDerL,QCL,QCL⊆QCL,QCL by the Jacobi identity of Hom-Lie algebras. Thus,
(39)QCL+QCL,QCL,QCL+QCL,QCL⊆QCL+QCL,QCL⊆GDerL.
The proof is finished.Theorem 24.
LetL,α be a Hom-Lie-Yamaguti superalgebra, where α is surjective, then CL,QCL⊆EndL,ZL. Moreover, if ZL=0, then CL,QCL=0.Proof.
For anyD1∈CαkL,D2∈QCαsL and x,y,z∈L, since α is surjective, there exist y′,z′∈L such that y=αk+sy′,z=αk+sz′, then we have
(40)D1,D2x,y=D1,D2x,αk+sy′=D1D2x,αk+sy′−−1D1D2D2D1x,αk+sy′=D1D2x,αsy′−−1D1D2αsD1x,D2αky′=D1D2x,αsy′−D1D2x,αsy′=0,D1,D2x,y,z=D1D2x,αk+sy′,αk+sz′−−1D1D2D2D1x,αk+sy′,αk+sz′=D1D2x,αsy′,αsz′−−1D1D2−1D1+xD2αsD1x,D2αky′,αk+sz′=−1xD2D1αsx,D2y′,αsz′−−1aD2D1αsx,D2y′,αsz′=0.
SoD1,D2x⊆ZL and therefore, CL,QCL⊆EndL,ZL. Moreover, if ZL=0, it is easy to see that CL,QCL=0.


## 5. 1-Parameter Formal Deformations of Hom-Lie-Yamaguti Superalgebras

LetL,α be a Hom-LY superalgebra over K and Kt the power series ring in one variable t with coefficients in K. Assume that Lt is the set of formal series whose coefficients are elements of the vector space L.Definition 25.
LetL,α be a Hom-LY superalgebra. A 1-parameter formal deformations of L is a pair of formal power series ft,gt of the form
(41)ft=·,·+∑i≥1fiti,gt=·,·,·+∑i≥1giti,where each fi is a K-bilinear map fi:L×L⟶L (extended to be Kt-bilinear) and each gi is a K-trilinear map gi:L×L×L⟶L (extended to be Kt-trilinear) such that Lt,ft,gt,α is a Hom-Lie-Yamaguti superalgebra over Kt. Set f0=·,· and g0=·,·,·, then ft and gt can be written as ft=∑i≥0fiti,gt=∑i≥0giti, respectively.
SinceLt,ft,gt,α is a Hom-LY superalgebra. Then, it satisfies the following axioms:
(42)α∘ftx,y=ftαx,αy,(43)α∘gtx,y,z=gtαx,αy,αz,(44)ftx,y=−−1xyfty,x,(45)gtx,y,z=−−1xygty,x,z,(46)Óx,y,z−1xzftftx,y,αz+gtx,y,z=0,(47)Óx,y,z−1xzgtftx,y,αz,αu=0,(48)gtαx,αy,ftu,v=ftgtx,y,u,α2v+−1ux+yftα2u,gtx,y,v,(49)gtα2x,α2y,gtu,v,w=gtgtx,y,u,α2v,α2w+−1ux+ygtα2u,gtx,y,v,α2w+−1u+vx+ygtα2u,α2v,gtx,y,w,for all x,y,z,u,v,w∈L.Remark 26.
Equations (42)–(49) are equivalent to (n=0,1,2,⋯)
(50)α∘fnx,y=fnαx,αy,(51)α∘gnx,y,z=gnαx,αy,αz,(52)fnx,y=−−1xyfny,x,(53)gnx,y,z=−−1xygny,x,z,(54)Óx,y,z−1xz∑i+j=nfifjx,y,αz+gnx,y,z=0,(55)Óx,y,z−1xz∑i+j=ngifjx,y,αz,αu=0,(56)∑i+j=ngiαx,αy,fju,v=∑i+j=nfigjx,y,u,α2v+−1ux+yfiα2u,gjx,y,v,(57)∑i+j=ngiα2x,α2y,gju,v,w=∑i+j=ngigjx,y,u,α2v,α2w+−1ux+ygiα2u,gjx,y,v,α2w+−1u+vx+ygiα2u,α2v,gjx,y,w,for all x,y,z,u,v,w∈L. These equations are called the deformation equations of a Hom-LY superalgebra. Equations (50)–(53) imply f,g∈C2L,L×C3L,L.Letn=1 in Equations (50)–(57). Then,
(58)Óx,y,z−1xzf1x,y,αz+f1x,y,αz+g1x,y,z=0,Óx,y,z−1xzf1x,y,αz,αu+g1x,y,αz,αu=0,αx,αy,f1u,v+g1αx,αy,u,v−g1x,y,u,α2v−f1x,y,u,α2v−−1ux+yα2z,g1x,y,u−−1ux+yf1α2z,x,y,u=0,α2x,α2y,g1u,v,w+g1α2x,α2y,u,v,w−g1x,y,u,α2v,α2w−g1x,y,u,α2v,α2w−−1ux+yα2u,g1x,y,v,α2w−−1ux+yg1α2u,x,y,v,α2w−−1ux+yα2u,α2v,g1x,y,w−−1u+vx+yg1α2u,α2v,x,y,w−−1u+vx+yα2u,α2v,g1x,y,w=0,which imply δI2,δII2f1,g1=0,0, i.e.,
(59)f1,g1∈Z2L,L×Z3L,L.The pairf1,g1 is called the infinitesimal deformation of ft,gt.Definition 27.
LetL,α be a Hom-LY superalgebra. Two 1-parameter formal deformations ft,gt and f′t,g′t of L are said to be equivalent, denoted by ft,gt~f′t,g′t, if there exists a formal isomorphism of Kt-modules
(60)ϕtx=∑i≥0ϕixti:Lt,ft,gt,α⟶Lt,f′t,g′t,α,where ϕi:L⟶L is a K-linear map (extended to be Kt-linear) such that
(61)ϕ0=idL,ϕt∘α=α∘ϕt,ϕt∘ftx,y=f′tϕtx,ϕty,ϕt∘gtx,y,z=g′tϕtx,ϕty,ϕtz.In particular, iff1,g1=f2,g2=⋯=0,0, then ft,gt=f0,g0 is called the null deformation. If ft,gt~f0,g0, then ft,gt is called the trivial deformation. If every 1-parameter formal deformation ft,gt is trivial, then L is called an analytically rigid Hom-LY superalgebra.Theorem 28.
Letft,gt and f′t,g′t be the two equivalent 1-parameter formal deformations of L. Then, the infinitesimal deformations f1,g1 and f′1,g′1 belong to the same cohomology class in H2L,L×H3L,L.Proof.
By the assumption thatf1,g1 and f′1,g1′ are equivalent, there exists a formal isomorphism ϕtx=∑i≥0ϕixti of Kt-modules satisfying
(62)∑i≥0ϕi∑j≥0fjx1,x2tjti=∑i≥0f′i∑k≥0ϕkx1tk,∑l≥0ϕlx2tlti,∑i≥0ϕi∑j≥0gjx1,x2,x3tjti=∑i≥0g′i∑k≥0ϕkx1tk,∑l≥0ϕlx2tl,∑m≥0ϕmx3tmti,for any x1,x2,x3∈L. Comparing the coefficients of t1 in both sides in each of the equations above, we have
(63)f1x1,x2+ϕ1x1,x2=f′1x1,x2+ϕ1x1,x2+x1,ϕ1x2,g1x1,x2,x3+ϕ1x1,x2,x3=g′1x1,x2,x3+ϕ1x1,x2,x3+x1,ϕ1x2,x3+x1,x2,ϕ1x3.
It follows thatf1−f′1,g1−g′1=δI1,δII1ϕ1,ϕ1∈B2L,L×B3L,L, as desired. The proof is completed.Theorem 29.
LetL,α be a Hom-LY superalgebra with H2L,L×H3L,L=0, then L is analytically rigid.Proof.
Letft,gt be a 1-parameter formal deformation of L. Suppose ft=f0+∑i≥rfiti, gt=g0+∑i≥rgiti. Set n=r in Equations (50)–(53), it follows that
(64)fr,gr∈Z2L,L×Z3L,L=B2L,L×B3L,L.
Then, there existshr∈C1L,L such that fr,gr=δI1hr,δI1hr.
Considerϕt=idL−hrtr, then ϕt:L⟶L is a linear isomorphism and ϕt∘α=α∘ϕt. Thus, we can define another 1-parameter formal deformation by ϕt−1 in the form of
(65)f′tx,y=ϕt−1ftϕtx,ϕty,g′tx,y,z=ϕt−1gtϕtx,ϕty,ϕtz.
Setf′t=∑i≥0f′iti and use the fact ϕtf′tx,y=ftϕtx,ϕty, then we have
(66)idL−hrtr∑i≥0f′ix,yti=f0+∑i≥0fitix−hrxtr,y−hrytr,that is
(67)∑i≥0f′ix,yti−∑i≥0hr∘f′ix,yti+r=f0x,y−f0hrx,ytr−f0x,hrytr+f0hrx,hryt2r+∑i≥rfix,yti−∑i≥rfihrx,y−fix,hryti+r+∑i≥rfihrx,hryti+2r.
By the above equation, it follows that(68)f′0x,y=f0x,y=x,y,f′1x,y=f′1x,y=⋯=f′r−1x,y=0,f′rx,y−hrx,y=frx,y−hrx,y−x,hry.
Therefore, we deduce(69)f′rx,y=frx,y−δI1hrx,y=0.
Sof′t=·,·+∑i≥r+1f′iti. Similarly, we have g′t=·,·,·+∑i≥r+1g′iti. By induction, we have ft,gt~f0,g0, that is, L is analytically rigid. The proof is finished.



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