# On Killing Forms and Invariant Forms of Lie-Yamaguti Superalgebras

Authors: Patricia L. Zoungrana; A. Nourou Issa
Journal: International Journal of Mathematics and Mathematical Sciences
 (2017)

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## Abstract

The notions of the Killing form and invariant form in Lie algebras are extended to the ones in Lie-Yamaguti superalgebras and some of their properties are investigated. These notions are alsoZ2-graded generalizations of the ones in Lie-Yamaguti algebras.


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## Body

## 1. Introduction

ALie-Yamaguti algebra is a triple (T,∗,[·,·,·]) consisting of a vector space T, a bilinear map ∗:T×T→T, and a trilinear map [·,·,·]:T×T×T→T such that(LY1)
x∗y=-y∗x,(LY2)
[x,y,z]=-[y,x,z],(LY3)
↺x,y,z{((x∗y)∗z)+[x,y,z]}=0,(LY4)
↺x,y,z[x∗y,z,u]=0,(LY5)
[x,y,u∗v]=[x,y,u]∗v+u∗[x,y,v],(LY6)
[u,v,[x,y,z]]=[[u,v,x],y,z]+[x,[u,v,y],z]+[x,y,[u,v,z]],for all u,v,x,y,z, in T, where ↺x,y,z denotes the sum over cyclic permutation of x, y, z. The bilinear map ∗ sometimes will be denoted by juxtaposition. If x∗y=0, ∀x,y∈T, one gets aLie triple system(T,[·,·,·]), while [x,y,z]=0 in (T,∗,[·,·,·]) induces a Lie algebra (T,∗).Lie-Yamaguti algebras were introduced by Yamaguti [1] (who formerly called them “generalized Lie triple systems”) in an algebraic study of the characteristic properties of the torsion and curvature of a homogeneous space with canonical connection [2]. Later on, these algebraic objects were called “Lie triple algebras” [3] and the terminology of “Lie-Yamaguti algebras” is introduced in [4] for these algebras. For further development of the theory of Lie-Yamaguti algebras one may refer, for example, to [5–8]. From the standard enveloping Lie algebra of a given Lie-Yamaguti algebra, the notions of the Killing-Ricci form and the invariant form of a Lie-Yamaguti algebra are introduced and studied in [9]. Further properties of invariant forms of Lie-Yamaguti algebras were considered in [10].Lie superalgebras as aZ2-graded generalization of Lie algebras are considered in [11, 12] while a Z2-graded generalization of Lie triple systems (called Lie supertriple systems) was first considered in [13]. For an application of Lie supertriple systems in physics, one may refer to [14]. Next,Lie-Yamaguti superalgebras as a Z2-graded generalization of Lie-Yamaguti algebras were first considered in [15].Definition 1 (see [16]).
A Lie-Yamaguti superalgebra is aZ2-graded vector space T=T0⊕T1 with a binary operation denoted by juxtaposition satisfying TiTj⊆Ti+j and a ternary operation [·,·,·] satisfying [Ti,Tj,Tk]⊆Ti+j+k (i,j,k∈Z2) such that(LYS1)
xy=-(-1)x-y-yx,(LYS2)
[x,y,z]=-(-1)x-y-[y,x,z],(LYS3)
↺x,y,z(-1)x-z-{((xy)z)+[x,y,z]}=0,(LYS4)
↺x,y,z(-1)x-z-[xy,z,u]=0,(LYS5)
[x,y,uv]=[x,y,u]v+(-1)u-(x-+y-)u[x,y,v],(LYS6)
[u,v,[x,y,z]]=[[u,v,x],y,z]+(-1)x-(u-+v-)[x,[u,v,y],z]+-1x-+y-u-+v-[x,y,[u,v,z]],for all u,v,x,y,z, in T.Observe thatT0 is a Lie-Yamaguti algebra.As a part of the general theory of superalgebras, the notion of the Killing form of Lie algebras is extended to the one of Lie triple systems (see [17] and references therein), Lie superalgebras [12], and next Lie supertriple systems [18] (see also [19]).In this paper we define and study the Killing form and invariant form of Lie-Yamaguti superalgebras as a generalization of the ones of both of Lie-Yamaguti algebras [9] and Lie supertriple systems [18, 19] including Lie superalgebras [12].The paper is organized as follows. In Section2 we record some useful results on Lie-Yamaguti superalgebras (see [16]). In Section 3 the Killing form of a Lie-Yamaguti superalgebra is defined (see Theorem 10 and Definition 11) and some of its properties are investigated (Proposition 13, Theorem 14, and Corollary 15). In Section 4 the invariant form of a Lie-Yamaguti superalgebra is defined (Definition 16) and, under some conditions, it is shown (Theorem 21) that the Killing form of a Lie-Yamaguti superalgebra T is nondegenerate if and only if the standard enveloping Lie superalgebra of T is semisimple.All vector spaces and algebras are finite-dimensional over a fixed ground fieldK of characteristic 0.


## 2. Some Basics on Lie-Yamaguti Superalgebras

We give here some definitions and results which can be found in [11, 12, 16].A superalgebra overK is a Z2-graded algebra A=A0⊕A1, where AiAj⊆Ai+j, (i,j∈Z2). The subspaces A0 and A1 are called the even and the odd parts of the superalgebra and so are called the elements from A0 and from A1, respectively. Below, all the elements are assumed to be homogeneous, that is, either even or odd, and for a homogeneous element x∈Ai, i=0,1, the notation x-=i is used and means the parity of x.LetG be theGrassmann algebra over K generated by the elements 1,e1,…,en such that ei2=0, eiej=-ejei for i≠j. The elements 1, ei1ei2⋯eir, i1<i2<⋯<ir form a basis of G. Denote by G0 (resp., G1) the span of the products of even length (resp., odd length) in the generators. The product of zero ei’s is by convention equal to 1. Then G=G0⊕G1 is an associative and supercommutative superalgebra; that is, g1g2=-1g-1g-2g2g1, where g1,g2∈G0∪G1. Let A=A0⊕A1 be a superalgebra. Consider the graded tensor product G⊗A which becomes a superalgebra with the product given by x⊗g1y⊗g2=-1x-g-2xy⊗g1g2, for homogeneous elements g1,g2∈G, x,y∈A and grading given by G⊗A0=G0⊗A0⊕G1⊗A1, G⊗A1=G0⊗A1⊕G1⊗A0. The subalgebra GA=G⊗A0=G0⊗A0⊕G1⊗A1 is called theGrassmann envelopeof the superalgebra A.Having in mind that ifV is a homogeneous variety of algebras [20], a superalgebra A=A0⊕A1 is called a V-superalgebra, if its Grassmann envelope GA belongs to V, we can state the following proposition.Proposition 2.
A superalgebraT=T0⊕T1 equipped with bilinear and trilinear products verifying TiTj⊆Ti+j and Ti,Tj,Tk⊆Ti+j+k is a Lie-Yamaguti superalgebra if its Grassmann envelope GT=G0⊗T0⊕G1⊗T1 is a Lie-Yamaguti algebra under the following products: (1)x⊗g1y⊗g2=-1x-y-xy⊗g1g2;x⊗g1,y⊗g2,z⊗g3=-1x-y-+y-z-+x-z-x,y,z⊗g1g2g3.Proof.
The proof is straightforward by using the fact that, for any elementx⊗g in GT, we have x-=g-.Example 3.
(1) Lie superalgebras are Lie-Yamaguti superalgebras withx,y,z=0.
(2) Ifxy=0 for any x,y∈T0∪T1 then (LYS2), (LYS3), and (LYS6) define a Lie supertriple system.
(3) LetM=M0⊕M1 be a Malcev superalgebra; that is, for any x,y,z,t in T, (2)xy=--1x-y-yx;--1y-z-xzyt=xyzt+-1x-y-+z-+t-yztx+-1x-+y-z-+t-ztxy+-1t-x-+y-+z-txyz.It is shown in [16] that M becomes a Lie-Yamaguti superalgebra if we set x,y,z=xyz--1x-y-yxz+xyz. Conversely, if on a Malcev superalgebra (M,·) we define a trilinear operation by [x,y,z]=x·yz-(-1)x-y-y·xz+xy·z then M,·,[·,·,·] is a Lie-Yamaguti superalgebra.Definition 4.
LetT=T0⊕T1 be a Lie-Yamaguti superalgebra. A graded subspace H=H0⊕H1 ofT is a graded Lie-Yamaguti subalgebra of T if HiHj⊆Hi+j and Hi,Hj,Hk⊆Hi+j+k for any i,j,k∈Z2.Definition 5.
A graded subalgebra of a Lie-Yamaguti superalgebraT is an invariant graded subalgebra (resp., an ideal) of T if T,T,H⊆H(resp., TH⊆H and T,H,T⊆H).IfH is an ideal of T, it is an invariant graded subalgebra of T. Obviously the center ZT of a Lie-Yamaguti superalgebra T defined by ZT=x∈T,xy=0andx,y,z=0,∀y,z∈T is an ideal of T.Definition 6.
LetT=T0⊕T1 and T′=T0′⊕T1′ be Lie-Yamaguti superalgebras. A linear map f:T→T′ is said to be of degree r if fTi⊆Tr+i′ for all r,i∈Z2.Definition 7.
LetT=T0⊕T1 and T′=T0′⊕T1′ be Lie-Yamaguti superalgebras. A linear map f:T→T′ is called a homomorphism of Lie-Yamaguti superalgebras if(1)
f preserves the grading, that is, fTi⊆Ti′, i∈Z2;(2)
fxy=fxfy;(3)
fx,y,z=fx,fy,fz for any x,y,z∈T0∪T1.Recall [11] that if V=V0⊕V1 is a Z2-graded vector space then, if we set EndrV=f∈EndV/fVi⊆Vr+i, we obtain an associative superalgebra EndV=End0V⊕End1V; EndrV consists of the linear mappings of V into itself which are homogeneous of degree r. The bracket f,g=fg--1f-g-gf makes EndV into a Lie superalgebra which we denote by lV or lm,n where m=dim⁡V0 and n=dim⁡V1. Let e1,…,em,em+1,…,em+n be a basis of V. In this basis the matrix of a∈lm,n is expressed as αβγδ, α being an m×m-, δ an n×n-, β an m×n-, and γ an n×m-matrix. The matrices of even elements have the form α00δ and those of odd ones 0βγ0. For a=αβγδ, the supertrace of a is defined by stra=trα-trδ and does not depend on the choice of a homogeneous basis. We have stra,b=0 that is strab=-1a-b-strba and str(aba-1)=str(b).Definition 8.
LetT=T0⊕T1 be a Lie-Yamaguti superalgebra; D∈EndrT is a superderivation of T if, for any x,y,z∈T0∪T1,(3)Dxy=Dxy+-1rx-xDy;Dx,y,z=Dx,y,z+-1rx-x,Dy,z+-1rx-+y-x,y,Dz.LetDrT consist of all the superderivations of degree r and DT=D0T⊕D1T. It is easy to check that DT is a graded subalgebra of EndT called the Lie superalgebra of superderivations of T.LetT=T0⊕T1 be a Lie-Yamaguti superalgebra. For any x,y∈T0∪T1, denote by Dx,y the endomorphism of T defined by Dx,yz=x,y,z for any z∈T. We have, for any x,y∈T0∪T1, r∈Z2, Dx,yTr⊆Tr+x-+y-; that is, Dx,y is a linear map of degree x-+y-. Moreover, it comes from (LYS5) and (LYS6) that (4)Dx,yzw=Dx,yzw+-1z-x-+y-zDx,yw;(5)Dx,yz,v,w=Dx,yz,v,w+-1z-x-+y-z,Dx,yv,w+-1z-+v-x-+y-z,v,Dx,ywfor any x,y,z,v,w∈T0∪T1. It follows that Dx,y is a superderivation of T called aninner superderivation of T.LetDT,T be the vector space spanned by all Dx,yx,y∈T.We can define naturally aZ2-gradation by setting DT,T=D0T,T⊕D1T,T, where DrT,T consists of the superderivation Dx,y of degree r. From (5) we also have that, for any x,y,z,v,w∈T0∪T1, (6)Dx,y,Dz,v=Dx,y,z,v+-1z-x-+y-Dz,x,y,v.It is clear from (6) that DT,T is a Z2-graded Lie subalgebra of DT called theLie superalgebra of all inner superderivations of T.Now, let(T,·,[·,·,·]) be a Lie-Yamaguti superalgebra.SetLiT=Ti⊕Di(T,T), i=0,1, and define a new bracket operation in LT=L0T⊕L1T=T⊕D(T,T) as follows: for any x,y∈T0∪T1, D1,D2∈D0T,T∪D1T,T,(7)x,y=xy+Dx,y;D,x=--1x-D-x,D=Dx;D1,D2=D1D2--1D-1D-2D2D1.Theorem 9.
LetT=T0⊕T1 be a Lie-Yamaguti superalgebra. Then (1)
LT is a Lie superalgebra called the standard enveloping Lie superalgebra of T and D(T,T) becomes a graded subalgebra of LT.(2)
IfH is an ideal of T then H⊕D(T,H) is an ideal of T⊕D(T,T).Proof.
The bracket·,· is bilinear by definition and XY=--1X-Y-YX for any X,Y∈LT by (LYS1) and (LYS2). Jacobi’s superidentity follows from (LTS3–6).
(2) is obvious.


## 3. Killing Forms of Lie-Yamaguti Superalgebras

The definition of the Killing form given here for Lie-Yamaguti superalgebras stems from [9] in the case of Lie-Yamaguti algebras and extends the one given in [18] for Lie supertriple systems. Let T=T0⊕T1 be an n-dimensional Lie-Yamaguti superalgebra. Denote by α the Killing form of the standard enveloping Lie superalgebra LT=T0⊕D0(T,T)⊕T1⊕D1(T,T). Consider the bilinear form β of T obtained by restricting α to T×T. For any x,y,z in T, define the endomorphisms Lx and Rx,y of the vector space T by Lxy=xy and Rx,yz=-1z-x-+y-z,x,y=-1z-x-+y-Dz,xy. It is clear that Rx,y is of degree x-+y- and Dz,t,Rx,y=Rz,t,x,y+-1z-+t-x-Rx,z,t,y.Theorem 10.
Forx,y∈T, we have (8)βx,y=strLxLy+strRx,y+-1x-y-Ry,x.Proof.
Letai, bi, ui, vi be bases of T0, T1, D0T,T, D1T,T, respectively. For these bases, we express the operations of T and DT,T as follows: (9)aiaj=∑lSijlal;aiaj∈T0,aibj=∑lTijlbl;aibj∈T1,bibj=∑lRijlal;bibj∈T0,Dai,aj=∑αDijαuα;Dai,aj∈D0T,T,Dai,bj=∑αCijαvα;Dai,bj∈D1T,T,Dbi,bj=∑αXijαuα;Dbi,bj∈D0T,T,uα,ai=uαai=∑jKαijaj;vα,ai=vαai=∑jLαijbj;uα,bi=uαbi=∑jHαijbj;vα,bi=vαbi=∑jQαijaj.To prove the theorem, it suffices to show that βai,aj=αai,aj, βai,bj=αai,bj and βbi,bj=αbi,bj. Since LaiLbjT0⊆T1 and LaiLbjT1⊆T0, we have strLaiLbj=0. Also, Rai,bjT1⊆T0 and Rai,bjT0⊆T1 give strRai,bj+Rbj,ai=0 and then βai,bj=0=αai,bj because of the consistency property of α (ai∈T0⊕D0(T,T),bj∈T1⊕D1(T,T)). Hence, it remains to show that βai,aj=αai,aj and βbi,bj=αbi,bj. The operations in T and the identities (7) imply the following:(10)ai,aj,ak=ai,ajak+Daj,ak=ai,∑mSjkmam+∑αDjkαuα=∑mSjkmaiam+Dai,am-∑αDjkα∑lKαilal=∑mSjkm∑lSimlal+∑αDimαuα-∑α,lDjkαKαilal=∑m,lSjkmSimlal+∑m,αSjkmDimαuα-∑α,lDjkαKαilal.In a similar way, we get(11)ai,aj,bk=∑m,lTjkmTimlbl+∑m,αTjkmCimαvα-∑α,lCjkαLαilbl,ai,aj,uα=-ai,uαaj=-ai,∑mKαjmam=-∑m,lKαjmSimlal-∑m,βKαjmDimβuβ,ai,aj,vα=-ai,vαaj=-ai,∑mLαjmbm=-∑m,lLαjmTimlbl-∑m,βLαjmCimβvβ.Also,(12)LaiLajak=aiajak=ai∑pSjkpap=∑pSjkpaiap=∑p,lSjkpSiplal,LaiLajbk=aiajbk=ai∑pTjkpbp=∑pTjkpaibp=∑p,lTjkpTiplbl,Rai,ajak=ak,ai,aj=Dak,aiaj=∑αDkiαuαaj=∑αDkiα∑mKαjmam=∑α,mDkiαKαjmam;Rai,ajbk=bk,ai,aj=Dbk,aiaj=-∑αCikαvαaj=-∑αCikα∑mLαjmbm=-∑α,mCikαLαjmbm.By interchanging i and j, we have(13)Raj,aiak=∑α,mDkjαKαimam;Raj,aibk=-∑α,mCjkαLαimbm.Therefore,(14)βai,aj=αai,aj=∑m,kSjkmSimk-∑α,kDjkαKαikak-∑m,kTjkmTimk+∑α,kCjkαLαik-∑m,αKαjmDimα+∑m,αLαjmCimα,strLaiLaj+strRaj,ai+Raj,ai=∑p,kSjkpSipk-∑p,kTjkpTipk+∑α,kDkiαKαjk+∑α,kDkjαKαik+∑α,kCikαLαjk+∑α,kCjkαLαik=βai,aj.It remains to show that βbi,bj=strLbiLbj+strRbj,bi+Rbj,bi,(15)bi,bj,bk=bi,bjbk+Dbj,bk=bi,∑mRjkmam+∑αXjkαuα=-∑m,lRjkmTmilbl-∑m,αRjkmimαvα-∑α,lXjkαHαilbl.Likewise, we have(16)bi,bj,ak=-∑m,lTkjmRimlal-∑m,αTkjmXimαuα-∑α,lCkjαQαilal,bi,bj,uα=-bi,uαbj=-bi,∑lHαjlbl=-∑lHαjlRilmam-∑l,βHαjlXilβuβ,bi,bj,vα=bi,vαbj=bi,∑lQαjlal=-∑l,mQαjlTlimbm-∑l,βQαjlCliβvβ.Therefore,(17)βai,aj=αai,aj=-∑m,kTkjmRimk-∑α,kCkjαQαik+∑m,kRjkmTmik+∑α,kXjkαHαik-∑l,αHαjlXilα+∑l,αQαjlCliα.Now,(18)LbiLbjak=bibjak=-bi∑mTjkmbm=-∑mTjkmbibm=-∑m,lTjkmRimlal;LbiLbjbk=bibjbk=bi∑mRjkmam=-∑mRjkmambi=-∑m,lRjkmTmilbl;Rbi,bjak=ak,bi,bj=Dak,bibj=∑αCkiαvαbj=∑αCkiα∑lQαjlal=∑α,lCkiαQαjlal;Rbi,bjbk=bk,bi,bj=Dbk,bibj=∑αXkiαuαbj=∑αXkiα∑lHαjlbl=∑α,lXkiαHαjlbl.By interchanging i and j, we have(19)Rbj,biak=∑α,lCkjαQαilal;Rbj,bibk=∑α,lXkjαHαilbl,strLbiLbj+strRbj,bi+Rbj,bi=-∑m,kTjkmRimk+∑m,kRjkmTmik+∑α,kCkiαQαjk-∑α,kXkiαHαjk-∑α,kCkjαQαik+∑α,kXkjαHαikbl=βbi,bj.Hence the theorem is proved.Definition 11.
The bilinear formβ defined on the Lie-Yamaguti superalgebra T=T0⊕T1 by (20)βx,y=strLxLy+strRx,y+-1x-y-Ry,xfor x,y∈T is called the Killing form of T.Remark 12.
Recall that ifT is a Lie superalgebra, then the Killing form β on T is defined as β(x,y)=str(LxLy), x,y∈T. Likewise, if T is a Lie supertriple system (resp., a Lie-Yamaguti algebra), the Killing form on T is defined as β(x,y)=str(Rx,y+(-1)x-y-Ry,x) (resp., β(x,y)=tr(LxLy)+tr(Rx,y+Ry,x)) with Lu and Ru,v defined according to the considered structure on T. So if a Lie-Yamaguti superalgebra T is reduced to a Lie superalgebra (resp., a Lie supertriple system, a Lie-Yamaguti algebra), then β as defined in Definition 11 is the Killing form of the Lie superalgebra (resp., the Lie supertriple system, the Lie-Yamaguti algebra) T.Proposition 13.
LetT=T0⊕T1 be a Lie-Yamaguti superalgebra with a Killing form denoted by β. Then, (1)
βT0,T1=0 (consistence),(2)
βx,y=-1x-y-βy,x (supersymmetry),(3)
βAx,Ay=βx,y, A∈AutT.Proof.
AsLT0LT1T1⊆T0, LT0LT1T0⊆T1, RT0,T1T0⊆T1 and RT0,T1T1⊆T0 we can state that βT0,T1=0.
(2) comes from the definition ofβ.
Now, for anyA in AutT, x in T, Ax¯=x-, and (21)βAx,Ay=strLAxLAy+strRAx,Ay+-1x-y-RAy,Ax.
AsAxy=AxAy then ALxy=LAxAy; that is, ALx=LAxA and ALxA-1=LAx.
Hence,strLAxLAy=strALxA-1ALyA-1=strALxLyA-1=strLxLy. Also, Ax,y,z=Ax,Ay,Az gives ARy,zx=RAy,AzAx; that is, ARy,zA-1=RAy,Az. Then,(22)βAx,Ay=strLAxLAy+strRAx,Ay+-1x-y-RAy,Ax=strLxLy+strARx,yA-1+-1x-y-ARy,xA-1=strLxLy+strARx,y+-1x-y-Ry,xA-1=strLxLy+strRx,y+-1x-y-Ry,x=βx,y.Now, letγ be a trilinear form in T given by γx,y,z=strDx,yLz for any x,y,z∈T. We can easily see that, for any x,y,z∈T, γx,y,z=--1x-y-γy,x,z and that γ vanishes identically if T is reduced to Lie superalgebra or Lie supertriple system.Theorem 14.
LetT=T0⊕T1 be a Lie-Yamaguti superalgebra with a Killing form denoted by β. Then, β satisfies the identities (23)βxy,z+-1x-y-βy,xz=-1x-y-γy,x,z+-1x-+y-z-γz,x,y;(24)βx,y,z,w+-1w-y-+z-βx,w,y,z=-1w-y-+z-γx,w,yz--1x-y-+z-γy,z,xwfor all x,y,z∈T.Proof.
The Killing formα of L=T0⊕D0(T,T)⊕T1⊕D1(T,T) satisfies αy,x,z+-1x-y-αx,y,z=0; that is, αy,x,z=αy,x,z. But, using (7), we have (25)αx,y,z=αxy+Dx,y,z=αxy,z+αDx,y,z=βxy,z+strDx,yLz=βxy,z+γx,y,z,αy,x,z=αy,xz+Dx,z=αy,xz+strLyDx,z=βy,xz+-1y-x-+z-strDx,zLy=βy,xz+-1y-x-+z-γx,z,y.Then the identity αy,x,z+-1x-y-αx,y,z=0 gives βxy,z+γx,y,z+-1x-y-βy,xz+-1y-z-γx,z,y=0 that is βxy,z+-1x-y-βy,xz=-γx,y,z--1y-z-γx,z,y=-1x-y-γy,x,z+-1x-+y-z-γz,x,y and (23) is obtained.
Fromαy,x,z=αy,x,z we deduce αx,w,y,z=αx,w,y,z=αx,w,y,z that is --1w-y-+z-αx,y,z,w=αx,w,y,z and (26)αx,y,z,w+-1w-y-+z-αx,w,y,z=0.Then, using (7) again and developing (26), we have αx,yz+Dy,z,w+-1w-y-+z-αxw+Dx,w,y,z=0 and we get (27)αx,yzw+Dyz,w+y,z,w+-1w-y-+z-αxwy+Dxw,y+x,w,y,z=0.This gives (28)βx,yzw+αx,Dyz,w+βx,y,z,w+-1w-y-+z-βxwy,z+-1w-y-+z-αDxw,y,z+-1w-y-+z-βx,w,y,z=0,βx,yzw+strLxLDyz,w+βx,y,z,w+-1w-y-+z-βxwy,z+-1w-y-+z-strLDxw,yLz+-1w-y-+z-βx,w,y,z=0.Thus,(29)βx,yzw+-1x-y-+z-+w-strLDyz,wLx+βx,y,z,w+-1w-y-+z-βxwy,z+-1w-y-+z-strLDxw,yLz+-1w-y-+z-βx,w,y,z=0;that is,(30)βx,yzw+-1x-y-+z-+w-γyz,w,x+βx,y,z,w+-1w-y-+z-βxwy,z+-1w-y-+z-γxw,y,z+-1w-y-+z-βx,w,y,z=0.This implies(31)βx,y,z,w+-1w-y-+z-βx,w,y,z=-βx,yzw--1w-y-+z-βxwy,z--1x-y-+z-+w-γyz,w,x--1w-y-+z-γxw,y,z.But (23) gives (32)-1w-x-βx,wyz+βwx,yz=-1w-x-γx,w,yz+-1w-+x-y-+z-γyz,w,x,-1w-x-βxwy,z+βwx,yz=γwx,y,z+-1w-+x-y-+z-+y-z-γz,y,wx.Then, (33)-βx,wyz--1w-y-+z-βxwy,z=-1w-y-+z-γx,w,yz+-1x-w-+y-+z-γyz,w,x--1w-x-+y-+z-γwx,y,z--1x-w-+y-+z-+y-z-γz,y,wx.Hence, βx,y,z,w+-1w-y-+z-βx,w,y,z=-1w-y-+z-γx,w,yz--1x-y-+z-γz,y,wx and (24) is proved.Corollary 15.
LetT=T0⊕T1 be a Lie-Yamaguti superalgebra with a Killing form denoted by β. Then, β satisfies the following for x,y,z∈T: (34)y,z,x,w+-1x-y-+z-βx,y,z,w=0;(35)βRw,yx,z--1w-+y-x-+w-y-βx,Ry,wz=-1x-w-+y-γx,w,yz+-1w-y-+z-+x-z-γy,z,xw.Proof.
Using (24) we have (36)βy,z,x,w=-1w-+x-y-+z-βx,w,y,z--1w-+x-y-+z-γx,w,yz--1w-x-γy,z,wx;(37)-1x-y-+z-βx,y,z,w=--1w-+x-y-+z-βx,w,y,z+-1w-+x-y-+z-γx,w,yz+-1w-x-γy,z,wx.By adding memberwise (36) and (37) we obtain the identity (34).
Also, the identity (37) is equivalent to --1y-z-βx,z,y,w+-1w-y-+z-βx,w,y,z=-1w-y-+z-γx,w,yz--1x-y-+z-γy,z,xw. Then, we obtain -1w-y-+x-w-+y-βRw,yx,z,z-βx,Ry,wz=-1w-y-γw,x,yz--1x-y-+z-+w-z-γy,z,xw that is βRw,yx,z--1w-+y-x-+w-y-βx,Ry,wz=-1x-w-+y-γx,w,yz+-1w-y-+z-+x-z-γy,z,xw and the remaining assertion is proved.


## 4. Invariant Forms of Lie-Yamaguti Superalgebras

In this section we introduced the concept of invariant forms of Lie-Yamaguti superalgebras as generalizations of those of Lie superalgebras and Lie supertriple systems.Definition 16.
An invariant formb of a Lie-Yamaguti superalgebra T=T0⊕T1 is a supersymmetric bilinear form on T satisfying the identities (38)bxy,z+-1x-y-by,xz=0;(39)bx,w,y,z+-1w-y-+z-bx,y,z,w=0for all x,y,z,w in T.Remark 17.
(1) Ifγ=0, the Killing form of T is an invariant form of T.
(2) IfT is reduced to a Lie supertriple system (resp., a Lie superalgebra, a Lie-Yamaguti algebra), then b is reduced to an invariant form of a Lie supertriple system [19] (resp., a Lie superalgebra [12], a Lie-Yamaguti algebra [10]).Definition 18.
Letb be an invariant form of a Lie-Yamaguti superalgebra T and S a subset of T. The orthogonal S⊥ of S with respect to b is defined by S⊥=x∈T,bx,y=0,∀y∈S. The invariant form b is nondegenerate if T⊥=0.Lemma 19.
Letb be an invariant form of a Lie-Yamaguti superalgebra T. Then, for any x, y, z, w in T, we have(40)bx,y,w,z+-1x-+y-z-bw,x,y,z=0.Proof.
By interchangingy and w in (39) we have(41)bx,y,w,z+-1y-w-+z-bx,w,z,y=0that is -1x-+y-z-bz,x,y,w+-1y-w-+z-+w-z-bx,w,z,y=0 by supersymmetry. Also by switching z and w in (41), we obtain bx,y,z,w~+-1y-w-+z-bx,z,w,y=0 that is (42)bx,y,z,w--1y-w-+z-+w-z-bx,w,z,y=0.Thus adding (41) and (42) we get (40) whence the lemma.Lemma 20.
Letb be an invariant form of a Lie-Yamaguti superalgebra T. Then,(1)
T+T,T,T⊥=ZT if b is nondegenerate;(2)
ifH is an ideal of T then H⊥ is an ideal of T. In particular, T⊥ is an ideal of T.Proof.
Considerx in T+T,T,T⊥. Then, for any u,v,w∈T, we have bx,uv=0 and bx,u,v,w=0. This implies, by (38) and (39), that -1u-x-bux,v=0 and -1w-u-+v-bx,w,u,v=0 that is bux,v=0 and bx,w,u,v=0. As b is nondegenerate, we get ux=0 and x,w,u=0 for any u,v,w∈T. This gives x∈ZT.
Conversely, ifx∈ZT, we have, for any u,v,u′,v′,w′∈T, bx,uv+u′,v′,w′=bx,uv+bx,u′,v′,w′=0 and x∈T+T,T,T⊥ whence T+T,T,T⊥=ZT.
Now, suppose thatH is an ideal of T that is TH⊆H and T,H,T⊆H; then for any x,y∈T, u∈H⊥, and h∈H, we have bxu,h=--1x-u-bu,xh=0 and bx,u,y,h=--1x-u-bu,x,y,h=-1x-y-+h-+x-u-bu,y,h,x=0. Then TH⊥⊆H⊥ and T,H⊥,T⊆H⊥ which proves (2).We are now ready to prove the following theorem.Theorem 21.
LetT=T0⊕T1 be a Lie-Yamaguti superalgebra with γ=0. Then the Killing form β is nondegenerate if and only if the standard enveloping Lie superalgebra LT=T⊕DT,T is a semisimple Lie superalgebra.Proof.
Letα be the Killing form of the Lie superalgebra LT. If γ=0, we have, for any x,y,z∈T, γx,y,z=strDx,yLz=0 and (43)αDx,y,z=0.Then, using the invariance of α and (43), we have, for any x,y,z,w∈T:αx,y,Dz,w=αx,y,Dz,w; that is, by (7),
α x y + D x , y , D z , w = - - 1 y - z - + w - α x , z , w , y and αDx,y,Dz,w=--1y-z-+w-αx,z,w,y. This gives(44)αDx,y,Dz,w=--1y-z-+w-βx,z,w,y.Thus, if β is nondegenerate, the restriction of α on DT,T×DT,T is nondegenerate and α is nondegenerate.
Now, suppose thatβ is degenerate. Then by the lemma above, T⊥ is an ideal of T so T⊥⊕DT,T⊥ is a nonzero ideal of T.
Using the identities (43) and (44) we get (45)αT⊥⊕DT,T⊥,T⊕DT,T=αT⊥,T+αT⊥,DT,T+αDT,T⊥,T+αDT,T⊥,DT,T=αT⊥,T+βT,T,T⊥,T=0.It comes that α is degenerate and T⊕DT,T is not semisimple which proves the theorem.The results of this paper could be used for a study of the structure of a pair consisting of a semisimple Lie superalgebra and its semisimple graded subalgebra.



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