# Derivations and Deformations ofδ-Jordan Lie Supertriple Systems

Authors: Shengxiang Wang; Xiaohui Zhang; Shuangjian Guo
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

LetT be a δ-Jordan Lie supertriple system. We first introduce the notions of generalized derivations and representations of T and present some properties. Also, we study the low-dimensional cohomology and the coboundary operator of T, and then we investigate the deformations and Nijenhuis operators of T by choosing some suitable cohomologies.


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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. Lister [2] investigated notions of the radical, semisimplicity and solvability as defined for Lie triple systems and determined all simple Lie triple systems over an algebraically closed field. Later, the representation theory, the central extension, the deformation theory, bilinear forms, and the generalized derivation of Lie triple systems and Jordan triple systems have been developed; see [3–11].In [12], Okubo and Kamiya introduced the notion of δ-Jordan Lie triple system, where δ=±1, which is a generalization of both Lie triple systems (δ=1) and Jordan Lie triple systems (δ=-1). Later, Kamiya and Okubo [13] studied a construction of simple Jordan superalgebras from certain triple systems. Recently, Ma and Chen [14] discussed the cohomology theory, the deformations, Nijenhuis operators, abelian extensions, and T∗-extensions of δ-Jordan Lie triple system.As a natural generalization of Lie triple systems, Okubo [15] introduced the notion of Lie supertriple systems in the study of Yang-Baxter equations. Lie supertriple systems have many applications in high energy physics, and many important results on Lie supertriple systems have been obtained; see [13, 15–17]. In [12], Okubo and Kamiya introduced the notion of δ-Jordan Lie supertriple system (they still call it Jordan Lie triple system); they presented some nontrivial examples and discussed their quasiclassical property. In the present paper, we hope to study generalized derivations, cohomology theories, and deformations of δ-Jordan Lie supertriple systems.This paper is organized as follows. In Section2, we recall the definition of δ-Jordan Lie supertriple systems and construct a kind of δ-Jordan Lie supertriple systems. Also, we study generalized derivation algebra of a δ-Jordan Lie supertriple system. In Section 3, we introduce notions of the representation and low-dimensional cohomology of a δ-Jordan Lie supertriple system. In Section 4, we consider the theory of deformations of a δ-Jordan Lie supertriple system by choosing a suitable cohomology. In Section 5, we study Nijenhuis operators for a δ-Jordan Lie supertriple system to describe trivial deformations.


## 2. Generalized Derivations ofδ-Jordan Lie Supertriple Systems

In this section, we start by recalling the definition ofδ-Jordan Lie supertriple systems and then we study its generalized derivations.Definition 1 ([12]).
Aδ-Jordan Lie supertriple system is a Z2-graded vector space T together with a triple linear product [·,·,·]:V⊗V⊗V→V satisfying(1)1a,b,c=a+b+cmod2;(2)2b,a,c=-δ-1aba,b,c;(3)3-1aca,b,c+-1bab,c,a+-1cbc,a,b=0;(4)4a,b,c,d,e=a,b,c,d,e+-1ca+bc,a,b,d,e+δ-1a+bc+dc,d,a,b,e,for all a,b,c,d,e∈T, where δ=±1 and a denotes the degree of the element a∈T.Remark 2.
Clearly,T0¯ is an ordinary δ-Jordan Lie triple system in [14]. The case of δ=1 defines, especially, a Lie supertriple system while the other case of δ=-1 may be termed an anti-Lie supertriple system as in [18].Example 3 ([12]).
Let(T,[·,·]) be a δ-Jordan Lie superalgebra. Then (T,[·,·,·]) becomes a δ-Jordan Lie supertriple system, where [a,b,c]=[[a,b],c], for all a,b,c∈T.Example 4.
LetT be a δ-Jordan Lie supertriple system and t an indeterminate. Set T′={∑i≥0x⊗ti∣x∈T}; then T′ is a δ-Jordan Lie supertriple system with a triple linear product ·,·,·′ defined by (5)a⊗ti,b⊗tj,c⊗tk′=a,b,c⊗ti+j+k,for all a⊗ti,b⊗tj,c⊗tk∈T′, where a⊗ti=a.Definition 5.
LetT be a δ-Jordn Lie supertriple system and k a nonnegative integer. A homogeneous linear map D:T→T is said to be a homogeneous k-derivation of T if it satisfies(6)δkDa,b,c+δk-1Daa,Db,c+δk-1Da+ba,b,Dc=Da,b,c,for all a,b,c∈T, where D denotes the degree of D.
We denote byDerT=⨁k≥0Derk(T), where Derk(T) is the set of all homogeneous k-derivations of T. Obviously, Der(T) is a subalgebra of End(T) and has a normal Lie superalgebra structure via the bracket product (7)D,D′=DD′--1DD′D′D.Definition 6.
LetT be a δ-Jordan Lie supertriple system and k a nonnegative integer. D∈Ends¯(T) is said to be a homogeneous generalized k-derivation of T, if there exist three endomorphisms D′,D′′,D′′′∈Ends¯(T) such that(8)δkDa,b,c+δk-1saa,D′b,c+δk-1sa+ba,b,D′′c=D′′′a,b,c,for all a,b,c∈T.Definition 7.
LetT be a δ-Jordan Lie supertriple system and k a nonnegative integer. D∈Ends¯(T) is said to be a homogeneous k-quasiderivation of T, if there exists an endomorphism D′∈Ends¯(T) such that(9)δkDa,b,c+δk-1saa,Db,c+δk-1sa+ba,b,Dc=D′a,b,c,for all a,b,c∈T.
LetGDer(T) and QDer(T) be the sets of homogeneous generalized k-derivations and of homogeneous k-quasiderivations, respectively. That is, (10)GDerT=⨁k≥0GDerkT,QDerT=⨁k≥0QDerkT.Definition 8.
LetT be a δ-Jordan Lie supertriple system and k a nonnegative integer. The k-centroid of T is the space of linear transformations on T given by(11)CkT=D∈EndT∣δkDa,b,c=δk-1Daa,Db,c=δk-1Da+ba,b,Dc=Da,b,c.We denote C(T)=⨁k≥0Ck(T) and call it the centroid of T.Definition 9.
LetT be a δ-Jordan Lie supertriple system. The quasicentroid of T is the space of linear transformations on T given by(12)QCT=D∈EndT∣Da,b,c=Da,b,c,∀a,b,c∈T.Remark 10.
LetT be a δ-Jordan Lie supertriple system. Then QC(T)⊆C(T).
For anyD∈QC(T) and a,b,c∈T, we have(13)Da,b,c=Da,b,c=-1Daa,Db,c=-1Da+ba,b,Dc.In fact, by the definition of the δ-Jordan Lie supertriple system, we have (14)Da,b,c=-δ-1abDb,a,c=-δ-1abDb,a,c=δ2-1ab-1ab+Da,Db,c=-1Daa,Db,c.Similarly, we have (15)-1aD+ca,b,Dc=--1abb,Dc,a--1bD+cDc,a,b=--1ba+DDb,c,a--1bD+cDc,a,b=--1bDD-1bab,c,a+-1cbc,a,b=-1bDD-1aca,b,c.It follows that [a,b,D(c)]=(-1)Da+b[a,b,D(c)].Definition 11.
LetT be a δ-Jordan Lie supertriple system. D∈End(T) is said to be a central derivation of T if(16)Da,b,c=Da,b,c=0,for all a,b,c∈T. Denote the set of all central derivations by ZDer(T).Remark 12.
LetT be a δ-Jordan Lie supertriple system. Then (17)ZDerT⊆DerT⊆QDerT⊆GDerT⊆EndT.Definition 13.
LetT be a δ-Jordan Lie supertriple system. If Z(T)={a∈T∣[a,b,c]=0,∀b,c∈T}, then Z(T) is called the center of T.Proposition 14.
LetT be a δ-Jordan Lie supertriple system; then the following statements hold:(1)
GDer(T),QDer(T), and C(T) are subalgebras of End(T).(2)
ZDer(T) is an ideal of Der(T).Proof.
( 1 ) We only prove that GDer(T) is a subalgebra of End(T), and similarly for cases of QDer(T) and C(T). For any D1∈GDerk(T),D2∈GDerl(T) and a,b,c∈T, we have (18)D1D2a,b,c=δkD1′′′D2a,b,c--1D1D2+aD2a,D1′b,c--1D1D2+a+bD2a,b,D1′′c=δkD1′′′δlD2′′′a,b,c--1D2aa,D2′b,c--1D2a+ba,b,D2′′c--1D1D2+aD2a,D1′b,c--1D1D2+a+bD2a,b,D1′′c=δk+lD1′′′D2′′′a,b,c-δl-1D2aD1′′′a,D2′b,c-δl-1D2a+bD1′′′a,b,D2′′c--1D1D2+aD2a,D1′b,c--1D1D2+a+bD2a,b,D1′′c=δk+lD1′′′D2′′′a,b,c-δk+l-1D2aD1a,D2′b,c-δk+l-1D1+D2aa,D1′D2′b,c-δk+l-1D2a+D1a+b+D2a,D2′b,D1′′c-δk+l-1D2a+bD1a,b,D2′′c-δk+l-1D2a+b+D1aa,D1′b,D2′′c-δk+l-1D1+D2a+ba,b,D1′′D2′′c--1D1D2+aD2a,D1′b,c--1D1D2+a+bD2a,b,D1′′c.Similarly, we have (19)D2D1a,b,c=δk+lD2′′′D1′′′a,b,c-δk+l-1D1aD2a,D1′b,c-δk+l-1D1+D2aa,D2′D1′b,c-δk+l-1D1a+D2a+b+D1a,D1′b,D2′′c-δk+l-1D1a+bD2a,b,D1′′c-δk+l-1D1a+b+D2aa,D2′b,D1′′c-δk+l-1D1+D2a+ba,b,D2′′D1′′c--1D2D1+aD1a,D2′b,c--1D2D1+a+bD1a,b,D2′′c.It follows that (20)D1,D2a,b,c=D1D2a,b,c--1k+lD2D1a,b,c=δk+lD1′′′D2′′′--1D1+D2D2′′′D1′′′a,b,c--1aD1+D2a,D1′D2′--1D1+D2D2′D1′b,c--1a+bD1+D2a,b,D1′′D2′′--1D1+D2D2′′D1′′c=δk+lD1′′′,D2′′′,a,b,c--1aD1+D2a,D1′,D2′b,c--1a+bD1+D2a,b,D1′′,D2′′c.Obviously, D1′,D2′,[D1′′,D2′′] and [D1′′′,D2′′′] are contained in End(T); thus [D1,D2]∈GDerk+l(T)⊆GDer(T); that is, GDer(T) is a subalgebra of End(T).
( 2 ) For any D1∈ZDer(T),D2∈Derk(T), and a,b,c∈T, we have (21)D1,D2a,b,c=D1D2a,b,c--1D1D2D2D1a,b,c=0.Also, we have (22)D1,D2a,b,c=D1D2a,b,c--1D1D2D2D1a,b,c=0--1D1D2D2D1a,b,c=--1D1D2δkD2D1a,b,c--1D2D1+aD1a,D2b,c+-1D1D2-1D2D1+a+bD1a,b,D2c=0.It follows that [D1,D2]∈ZDer(T). That is, ZDer(T) is an ideal of Der(T).Proposition 15.
LetT be a δ-Jordan Lie supertriple system; then the following statements hold:(1)
[Der(T),C(T)]⊆C(T).(2)
[QDer(T),QC(T)]⊆QC(T).(3)
[QC(T),QC(T)]⊆QDer(T).(4)
C(T)⊆QDer(T).Proof.
( 1 ) For any D1∈Derk(T),D2∈Cl(T), and a,b,c∈T, we have (23)D1,D2a,b,c=D1D2a,b,c--1D1D2D2D1a,b,c=δlD1D2a,b,c-δk-1D1D2D2D1a,b,c-δk-1D1D2+aD2a,D1b,c-δk-1D1D2+a+bD2a,b,D1c=δk+lD1D2a,b,c+δk+l-1D1D2+aD2a,D1b,c+δk+l-1D1D2+a+bD2a,b,D1c-δk+l-1D1D2D2D1a,b,c-δk+l-1D1D2+aD2a,D1b,c-δk+l-1D1D2+a+bD2a,b,D1c=δk+lD1,D2a,b,c.Similarly, one can check that (24)D1,D2a,b,c=-1ak+la,D1,D2b,c=-1a+bk+la,b,D1,D2c.It follows that [D1,D2]∈C(T); thus [Der(T),C(T)]⊆C(T).
( 2 ) Similar to the proof of (1).
( 3 ) For any D1,D2∈QC(T) and a,b,c∈T, we have (25)D1,D2a,b,c+-1aD1+D2a,D1,D2b,c+-1a+bD1+D2a,b,D1,D2c=D1D2a,b,c+-1aD1+D2a,D1D2b,c+-1a+bD1+D2a,b,D1D2c--1D1D2D2D1a,b,c--1aD1+D2+D1D2a,D2D1b,c--1a+bD1+D2+D1D2a,b,D2D1c.Since D1,D2∈QC(T), we have (26)D1D2a,b,c=-1D1a+D2D2a,D1b,c=-1D1a+D2+aD2a,D2D1b,c.Similarly, one may check that (27)a,D1D2b,c=-1bD1+D2+D1D2a,b,D2D1c,a,b,D1D2c=-1a+bD1+D2+D1D2D2D1a,b,c.It follows that (28)D1,D2a,b,c+-1aD1+D2a,D1,D2b,c+-1a+bD1+D2a,b,D1,D2c=0.Therefore, [D1,D2]⊆QDer(T) and [QC(T),QC(T)]⊆QDer(T).
( 4 ) For any D∈Ck(T) and a,b,c∈T, we have (29)Da,b,c=δkDa,b,c=δk-1Daa,Db,c=δk-1Da+ba,b,Dc.Thus (30)δkDa,b,c+δk-1Daa,Db,c+δk-1Da+ba,b,Dc=3Da,b,c,that is, D∈QDerk(T) and QC(T)⊆QDer(T).Theorem 16.
LetT be a δ-Jordan Lie supertriple system; then [C(T),QC(T)]⊆End(T,Z(T)). Moreover, if Z(T)={0}, then [C(T),QC(T)]={0}.Proof.
For anyD1∈Ck(T),D2∈QCl(T), and a,b,c∈T, we have (31)D1,D2a,b,c=D1D2a,b,c--1D1D2D2D1a,b,c=δkD1D2a,b,c--1D1D2-1D1+aD2D1a,D2b,c=δk-1aD2D1a,D2b,c--1aD2D1a,D2b,c=0.So [D1,D2](a)⊆Z(T) and therefore [C(T),QC(T)]⊆End(T,Z(T)). Moreover, if Z(T)={0}, then it is easy to see that [C(T),QC(T)]={0}.Theorem 17.
LetT be a δ-Jordan Lie supertriple system and Chark≠2; then ZDert=C(T)∩Der(T).Proof.
For anyD∈Ck(T)∩Derk(T) and a,b,c∈T, we have (32)Da,b,c=δkDa,b,c+δk-1Daa,Db,c+δk-1Da+ba,b,Dc,Da,b,c=δkDa,b,c=δk-1Daa,Db,c=δk-1Da+ba,b,Dc.It follows that 2D([a,b,c])=0. Thus D([a,b,c])=0 since Chark≠2; that is, D∈ZDert. So C(T)∩Der(T)∈ZDert.
On the other hand, for anyD∈ZDert and a,b,c∈T, we have D([a,b,c])=0. Clearly, (6) and (11) hold; that is, D∈Ck(T)∩Derk(T), and, therefore, ZDert∈C(T)∩Der(T). And this completes the proof.


## 3. The Cohomology ofδ-Jordan Lie Supertriple Systems

In this section, we introduce the notion of the representation ofδ-Jordan Lie supertriple systems and present its low-dimensional cohomologies.Definition 18.
LetT be a δ-Jordan Lie supertriple system and V a Z2-graded vector space. Suppose that there exists a bilinear mapping θ:T⊗T→End(V) satisfying the following axioms:(33)-1a+bc+dθc,dθa,b-δ-1ab+dc+aθb,dθa,c-θa,b,c,d+-1ab+cDb,cθa,d=0,(34)δ-1a+bc+dθc,dDa,b-δDa,bθc,d+θa,b,c,d+δ-1ca+bθc,a,b,d=0,(35)Da,b,c,d+-1ca+bDc,a,b,d-δDa,bDc,d+-1a+bc+dDc,dDa,b,for a,b,c,d∈T, where D(a,b)=(-1)abθ(b,a)-δθ(a,b); then (V,θ) is called the representation of T; V is called a T-module.Example 19.
LetT be a δ-Jordan Lie supertriple system. Define θ:T⊗T→End(T) by θ(a,b)(x)=(-1)xa+b[x,a,b]. It is not hard to check that D(a,b)(x)=δ[a,b,x] and T itself is a T-module. In this case, (T,θ) is said to be the adjoint representation of T.Proposition 20.
LetT be a δ-Jordan Lie supertriple system and (V,θ) the representation. Then T⊕V has a structure of a δ-Jordan Lie supertriple system.Proof.
Define a triple linear product[·,·,·]:(T⊕V)⊗(T⊕V)⊗(T⊕V)→(T⊕V) by (36)a,u,b,v,c,w=a,b,c,-1ab+cθb,cu-δ-1bcθa,cv+δDa,bw,for all (a,u),(b,v),(c,w)∈T⊕V, where a,u=a.
Now we check that the operation[·,·,·] defined above satisfies axioms in Definition 1. It is easy to see that (1) holds since T is a δ-Jordan Lie supertriple system.
For (2), we take any (a,u),(b,v),(c,w)∈T⊕V and compute (37)-δ-1abb,v,a,u,c,w=-δ-1abb,a,c,-1ba+cθa,cv-δ-1acθb,cu+δDb,aw=-δ-1abb,a,c,-δ-1bcθa,cv--1ab+cθb,cu-δ-1abδDb,aw=a,b,c,-1ab+cθb,cu-δ-1bcθa,cv+δDa,bw=a,u,b,v,c,w.The last equality holds since -(-1)ab[b,a,w]=[a,b,w]=δD(a,b)(w).
For (3), we have (38)-1aca,u,b,v,c,w+-1bab,v,c,w,a,u+-1cbc,w,a,u,b,v=-1aca,b,c+-1bab,c,a+-1cbc,a,b,Ω=0,Ω,where (39)Ω=-1abθb,cu-δ-1a+bcθa,cv+δ-1acDa,bw+-1bcθc,av-δ-1b+caθb,aw+δ-1baDb,cu+-1cbθa,bw-δ-1c+abθc,bu+δ-1cbDc,av=-1abθb,cu+-1bcθc,av+-1cbθa,bw--1acθa,bw+-1baθb,cu--1bcθc,av=0.The second equality holds since D(a,b)=(-1)abθ(b,a)-δθ(a,b). Then we have (40)-1aca,u,b,v,c,w+-1bab,v,c,w,a,u+-1cbc,w,a,u,b,v=0,0,as desired.
For (4), we take any (a,u),(b,v),(c,w),(d,m),(e,n)∈T⊕V. First, we calculate the following expression: (41)a,u,b,v,c,w,d,m,e,n=a,b,c,-1ab+cθb,cu-δ-1bcθa,cv+δDa,bw,d,m,e,n=a,b,c,d,e,Ω1,where (42)Ω1=-1a+b+cd+e-1ab+cθd,eθb,cu-δ-1a+b+cd+e-1bcθd,eθa,cv+δ-1a+b+cd+eθd,eDa,bw-δ-1deθa,b,c,em+δDa,b,c,dn.Second, we compute the expression (-1)ca+b[(c,w),[(a,u),(b,v),(d,m)],(e,n)]: (43)-1ca+bc,w,a,u,b,v,d,m,e,n=-1ca+bc,w,a,b,d,-1ab+dθb,du-δ-1bdθa,dv+δDa,bm,e,n=-1ca+bc,a,b,d,e,Ω2,where (44)Ω2=-δ-1a+b+de-1ca+b+ab+dθc,eθb,du+-1a+b+de-1ca+b+bdθc,eθa,dv+-1a+b+d+ec-1ca+bθa,b,d,ew--1a+b+de-1ca+bθc,eDa,bm+δ-1ca+bDc,a,b,dn.Third, we compute the expression δ(-1)a+bc+d[(c,w),(d,m),[(a,u),(b,v),(e,n)]]: (45)δ-1a+bc+dc,w,d,m,a,u,b,v,e,n=δ-1a+bc+dc,w,d,m,a,b,e,-1ab+eθb,eu-δ-1beθa,ev+δDa,bn=δ-1a+bc+dc,d,a,b,e,Ω3,where (46)Ω3=-1a+bc+d-1ab+eDc,dθb,eu-δ-1a+bc+d-1beDc,dθa,ev+δ-1a+bc+d-1a+b+d+ecθd,a,b,ew--1a+bc+d-1a+b+edθc,a,b,em+δ-1a+bc+dDc,dDa,bn.Fourth, we compute the expression [(a,u),(b,v),[(c,w),(d,m),(e,n)]]: (47)a,u,b,v,c,w,d,m,e,n=a,u,b,v,c,d,e,-1cd+eθd,ew-δ-1deθc,em+δDc,dn=a,b,c,d,e,Ω4,where (48)Ω4=-1ab+c+d+eθb,c,d,eu-δ-1bc+d+eθa,c,d,ev+δ-1cd+eDa,bθd,ew--1deDa,bθc,em+Da,bDc,dn.Finally, by (33), (34), and (35), we have (49)a,u,b,v,c,w,d,m,e,n+-1ca+bc,w,a,u,b,v,d,m,e,n+δ-1a+bc+dc,w,d,m,a,u,b,v,e,n=a,b,c,d,e+-1ca+bc,a,b,d,e+δ-1a+bc+dc,d,a,b,e,Ω1+Ω2+Ω3=a,b,c,d,e,Ω4=a,u,b,v,c,w,d,m,e,n,as desired, and this finishes the proof.Corollary 21.
Anyδ-Jordan Lie supertriple system can be considered as a subspace of a δ-Jordan Lie superalgebra.Proof.
It is straightforward from Example19 and Proposition 20.Definition 22.
LetT be a δ-Jordan Lie supertriple system and V a T-module by a bilinear map θ. If an n-linear map f:T×T×⋯×T→T satisfies the following axioms(1)
f(x1,x2,…,x,y,…,xn)=-δ(-1)xyf(x1,x2,…,x,y,…,xn);(2)
(-1)xzf(x1,x2,…,xn-3,x,y,z)+(-1)yxf(x1,x2,…,xn-3,y,z,x)+(-1)zyf(x1,x2,…,xn-3,z,x,y)=0,
thenf is called an n-cochain on T. Denote by Cδn(T,V) the set of all n-cochains, for n≥1.Definition 23.
LetT be a δ-Jordan Lie supertriple system and V a T-module by a bilinear map θ. For n=1,2, the coboundary operator dn:Cδn(T,V)→Cδn+2(T,V) is defined as follows:(i)
Iff∈Cδ1(T,V), then (50)d1fx1,x2,x3=-1f+x1x2+x3θx2,x3fx1-fx1,x2,x3+δ-1fx1+x2Dx1,x2fx3-δ-1x2x3+fx1+x3θx1,x3fx2.(ii)
Iff∈Cδ2(T,V), then (51)d2fy,x1,x2,x3=-1f+y+x1x2+x3θx2,x3fy,x1-fy,x1,x2,x3-δ-1x2x3+f+y1x1+x3θx1,x3fy,x2+δ-1f+yx1+x2Dx1,x2fy,x3.(iii)
Iff∈Cδ3(T,V), then (52)d3fx1,x2,x3,x4,x5=-1f+x1+x2+x3x4+x5θx4,x5fx1,x2,x3-δ-1f+x1+x2x3+x5+x4x5θx3,x5fx1,x2,x4-δ-1fx1+x2Dx1,x2fx3,x4,x5+-1f+x1+x2x3+x4Dx3,x4fx1,x2,x5+fx1,x2,x3,x4,x5-fx1,x2,x3,x4,x5+-1x3x1+x2fx3,x1,x2,x4,x5+δ-1x1+x2x3+x4fx3,x4,x1,x2,x5.(iv)
Iff∈Cδ4(T,V), then (53)d4fy,x1,x2,x3,x4,x5=-1f+y+x1+x2+x3x4+x5θx4,x5fy,x1,x2,x3-δ-1f+y+x1+x2x3+x5+x4x5θx3,x5fy,x1,x2,x4-δ-1f+yx1+x2Dx1,x2fy,x3,x4,x5+-1f+y+x1+x2x3+x4Dx3,x4fy,x1,x2,x5+fy,x1,x2,x3,x4,x5-fy,x1,x2,x3,x4,x5+-1x3x1+x2fy,x3,x1,x2,x4,x5+δ-1x1+x2x3+x4fy,x3,x4,x1,x2,x5.Theorem 24.
LetT be a δ-Jordan Lie supertriple system and V a T-module by a bilinear map θ. The coboundary operator dn defined above satisfies dn+2dn=0, n=1,2.Proof.
From the definition of the coboundary operator, it follows immediately thatd3d1=0 implies d4d2=0. Then, we only need to prove d3d1=0:(54)d3d1fx1,x2,x3,x4,x5(55)=-1f+x1+x2+x3x4+x5θx4,x5d1fx1,x2,x3(56)-δ-1f+x1+x2x3+x5+x4x5θx3,x5d1fx1,x2,x4(57)-δ-1fx1+x2Dx1,x2d1fx3,x4,x5(58)+-1f+x1+x2x3+x4Dx3,x4d1fx1,x2,x5(59)+d1fx1,x2,x3,x4,x5-d1fx1,x2,x3,x4,x5(60)+-1x3x1+x2d1fx3,x1,x2,x4,x5(61)+δ-1x1+x2x3+x4d1fx3,x4,x1,x2,x5.By Definition 23, we have(62)55=-1f+x1+x2+x3x4+x5θx4,x5d1fx1,x2,x3=-1f+x1+x2+x3x4+x5θx4,x5-f(x1,x2,x3+-1f+x1x2+x3θx2,x3fx1+δ-1fx1+x2Dx1,x2fx3-δ-1x2x3+fx1+x3θx1,x3fx2,(63)56=-δ-1f+x1+x2x3+x5+x4x5θx3,x5d1fx1,x2,x4=-δ-1f+x1+x2x3+x5+x4x5θx3,x5-fx1,x2,x4+-1f+x1x2+x4θx2,x4fx1+δ-1fx1+x2Dx1,x2fx4-δ-1x2x4+fx1+x4θx1,x4fx2,(64)57=-δ-1fx1+x2Dx1,x2d1fx3,x4,x5=-δ-1fx1+x2Dx1,x2-fx3,x4,x5+-1f+x3x4+x5θx4,x5fx3+δ-1fx3+x4Dx3,x4fx5-δ-1x4x5+fx3+x5θx3,x5fx4,(65)58=-1f+x1+x2x3+x4Dx3,x4d1fx1,x2,x5=-1f+x1+x2x3+x4Dx3,x4-fx1,x2,x5+-1f+x1x2+x5θx2,x5fx1+δ-1fx1+x2Dx1,x2fx5-δ-1x2x5+fx1+x5θx1,x5fx2,(66)59=d1fx1,x2,x3,x4,x5-d1fx1,x2,x3,x4,x5=-1f+x1+x2+x3x4+x5θx4,x5fx1,x2,x3-fx1,x2,x3,x4,x5+δ-1fx1+x2+x3+x4Dx1,x2,x3,x4fx5-δ-1x4x5+fx1+x2+x3+x5θx1,x2,x3,x5fx4--1f+x1x2+x3+x4+x5θx2,x3,x4,x5fx1+fx1,x2,x3,x4,x5-δ-1fx1+x2Dx1,x2fx3,x4,x5+δ-1fx1+x3+x4+x5+x2x3+x4+x5θx1,x3,x4,x5,fx2,(67)60=-1x3x1+x2d1fx3,x1,x2,x4,x5=-1x3x1+x2+f+x3x1+x2+x4+x5θx1,x2,x4,x5fx3--1x3x1+x2fx3,x1,x2,x4,x5+δ-1x3x1+x2+f(x1+x2+x3+x4Dx3,x1,x2,x4fx5-δ-1f+x1+x2x3+x5+x4x5θx3,x5fx1,x2,x4,(68)61=δ-1x1+x2x3+x4d1fx3,x4,x1,x2,x5=δ-1x1+x2x3+x4-1f+x3x1+x2+x4+x5θx4,x1,x2,x5fx3-fx3,x4,x1,x2,x5+δ-1fx3+x4Dx3,x4)fx1,x2,x5-δ-1x4x1+x2+x5+fx1+x2+x3+x5θx3,x1,x2,x5fx4.By (62)–(68), we have (69)d3d1fx1,x2,x3,x4,x5=-fx1,x2,x3,x4,x5--1x3x1+x2fx3,x1,x2,x4,x5-δ-1x1+x2x3+x4fx3,x4,x1,x2,x5+fx1,x2,x3,x4,x5+-1f+x1x2+x3+x4+x5Λ1fx1-δ-1f+x2x3+x4+x5+fx1Λ2fx2+-1fx1+x2+x4+x5+x3x4+x5Λ3fx3--1fx1+x2+x3+x5+x4x5Λ4fx4+δ-1fx1+x2+x3+x4Λ5fx5=-1f+x1x2+x3+x4+x5Λ1fx1-δ-1f+x2x3+x4+x5+fx1Λ2fx2+-1fx1+x2+x4+x5+x3x4+x5Λ3fx3--1fx1+x2+x3+x5+x4x5Λ4fx4+δ-1fx1+x2+x3+x4Λ5fx5,where (70)Λ1=-1x2+x3x4+x5θx4,x5θx2,x3-δ-1x2x3+x2+x4x5θx3,x5θx2,x4-θx2,x3,x4,x5+δ-1x2x3+x4Dx3,x4θx2,x5=0,byEq.33Λ2=-1x1+x3x4+x5θx4,x5θx1,x3-δ-1x1x3+x5x1+x4θx3,x5θx1,x4-θx1,x3,x4,x5+δ-1x1x3+x4Dx3,x4θx1,x5=0,byEq.33Λ3=-1x1+x2x4+x5θx4,x5Dx1,x2-δDx1,x2θx4,x5+θx1,x2,x4,x5+δ-1x4x1+x2θx4,x1,x2,x5=0,byEq.34Λ4=-1x1+x2x3+x5θx3,x5Dx1,x2-δDx1,x2θx3,x5+θx1,x2,x3,x5+δ-1x3x1+x2θx3,x1,x2,x5=0,byEq.34Λ5=Dx1,x2,x3,x4+-1x3x1+x2Dx3,x1,x2,x4-δDx1,x2Dx3,x4+-1x1+x2x3+x4Dx3,x4Dx1,x2=0.byEq.35It follows that d3(d1f)(x1,x2,x3,x4,x5)=0, as desired. And this finishes the proof.Forn=1,2, the map f∈Cδn(T,V) is called an n-cocycle if dnf=0. We denote by Zδn(T,V) the subspace spanned by n-cocycles and Bδn(T,V)=dn-2Cδn-2(T,V). By Theorem 24, Bδn(T,V) is a subspace of Zδn(T,V). Therefore, we can define a cohomology space Hδn(T,V) of the δ-Jordn Lie supertriple system T as the factor space Zδn(T,V)/Bδn(T,V).


## 4.1-Parameter Formal Deformations of δ-Jordan Lie Supertriple Systems

LetT be a δ-Jordan Lie supertriple system and k[[t]] the power series ring in one variable t with coefficients in k. Assume that T[[t]] is the set of formal series whose coefficients are elements of the vector space T.Definition 25.
LetT be a δ-Jordan Lie supertriple system. A 1-parameter formal deformation of T is a formal power series ft:T[[t]]×T[[t]]×T[[t]]→T[[t]] given by (71)ft=∑i≥0fix1,x2,x3ti=f0x1,x2,x3+f1x1,x2,x3t+f2x1,x2,x3t2+⋯,where each fi is a k-trilinear map fi:T×T×T→T (extended to be k[[t]]-trilinear) and f0(x1,x2,x3)=[x1,x2,x3], satisfying the following axioms:(72)ftx1,x2,x3=x1+x2+x3;(73)ftx2,x1,x3=-δ-1x1x2ftx1,x2,x3;(74)-1x1x3ftx1,x2,x3+-1x2x1ftx2,x3,x1+-1x3x2ftx3,x1,x2=0;(75)ftx1,x2,ftx3,x4,x5=-1x3x1+x2ftx3,ftx1,x2,x4,x5+ftftx1,x2,x3,x4,x5+δ-1x1+x2x3+x4ftx3,x4,ftx1,x2,x5.Remark 26.
Equations (72)–(75) are equivalent to (n=0,1,2,⋯)(76)fix2,x1,x3=x1+x2+x3;(77)fix2,x1,x3=-δ-1x1x2fix1,x2,x3;(78)-1x1x3fix1,x2,x3+-1x2x1fix2,x3,x1+-1x3x2fix3,x1,x2=0;(79)∑i+j=nfix1,x2,fjx3,x4,x5=∑i+j=n-1x3x1+x2fix3,fjx1,x2,x4,x5+fifjx1,x2,x3,x4,x5+δ-1x1+x2x3+x4fix3,x4,fjx1,x2,x5.
Furthermore, we can rewrite the deformation Equation (79) by the equality ∑i+j=nfifj=0, where (80)fifjx1,x2,x3,x4,x5=-fix1,x2,fjx3,x4,x5+-1x3x1+x2fix3,fjx1,x2,x4,x5+fifjx1,x2,x3,x4,x5+δ-1x1+x2x3+x4fix3,x4,fjx1,x2,x5.
Whenn=1, (79) is equivalent to f0f1+f1f0=0. When n≥2, (79) is equivalent to -(f0fn+fnf0)=f1fn-1+f2fn-2+⋯+fn-1f1.By Example19, (T,θ) is the adjoint representation of T itself, where θ(a,b)(x)=(-1)xa+b[x,a,b] and D(a,b)(x)=δ[a,b,x]. It is easy to see that fi∈Cδ3(T,V) and therefore fifj∈Cδ5(T,V). Since f0(x1,x2,x3)=[x1,x2,x3], we have (81)f0f1x1,x2,x3,x4,x5=-f0x1,x2,f1x3,x4,x5+-1x3x1+x2f0x3,f1x1,x2,x4,x5+f0f1x1,x2,x3,x4,x5+δ-1x1+x2x3+x4f0x3,x4,f1x1,x2,x5=-x1,x2,f1x3,x4,x5+-1x3x1+x2x3,f1x1,x2,x4,x5+f1x1,x2,x3,x4,x5+δ-1x1+x2x3+x4x3,x4,f1x1,x2,x5=-δDx1,x2fx3,x4,x5-δ-1x1+x2x3+x5+x4x5Dx3,x4fx1,x2,x5+-1x1+x2+x3x4+x5θx4,x5fx1,x2,x3+-1x1+x2x3+x4Dx3,x4fx1,x2,x5Similarly, we have (82)f1f0x1,x2,x3,x4,x5=-f1x1,x2,f0x3,x4,x5+-1x3x1+x2f1x3,f0x1,x2,x4,x5+f1f0x1,x2,x3,x4,x5+δ-1x1+x2x3+x4f1x3,x4,f0x1,x2,x5=-f1x1,x2,x3,x4,x5+-1x3x1+x2f1x3,x1,x2,x4,x5+f1x1,x2,x3,x4,x5+δ-1x1+x2x3+x4f1x3,x4,x1,x2,x5.It follows that (83)f0f1+f1f0x1,x2,x3,x4,x5=d3f1x1,x2,x3,x4,x5.Therefore, we deduce d3f1=0 since f0f1+f1f0=0. Also we can obtain -d3fn=f1fn-1+f2fn-2+⋯+fn-1f1. And f1 is called the infinitesimal deformation of ft.Definition 27.
LetT be a δ-Jordan Lie supertriple system. Two 1-parameter formal deformations ft and ft′ of T are said to be equivalent, denoted by ft~ft′, if there exists a formal isomorphism of k[[t]]-modules (84)ϕtx=∑i≥0ϕixti:Tt,ft,δ→Tt,ft′,δ,where ϕ0=idT, ϕi:T→T is an k-linear map (extended to be k[[t]]-linear) such that (85)ϕtftx1,x2,x3=ft′ϕtx1,ϕtx2,ϕtx3,∀x1,x2,x3∈T.In particular, iff1=f2=⋯=0, then f1=f0 is called the null deformation. If ft~f0, then ft is called the trivial deformation. If every 1-parameter formal deformation ft is trivial, then T is called an analytically rigid δ-Jordan Lie supertriple system.Theorem 28.
Letft=∑i≥0fi(x1,x2,x3)ti and ft′=∑i≥0fi′(x1,x2,x3)ti be two equivalent 1-parameter formal deformations of T. Then the infinitesimal deformations f1 and f1′ belong to the same cohomology class in Hδ3(T,T).Proof.
By the assumption thatf1 and f1′ are equivalent, there exists a formal isomorphism ϕt(x)=∑i≥0ϕi(x)ti of k[[t]]-modules satisfying (86)∑i≥0ϕi∑j≥0fjx1,x2,x3tj=∑i≥0fi′∑k≥0ϕkx1tk,∑l≥0ϕlx2tl,∑m≥0ϕmx3tmti,for any x1,x2,x3∈T. Compared with the coefficients of t1 for two sides of the above equation, we have (87)f1x1,x2,x3+ϕ1x1,x2,x3=f1′x1,x2,x3+ϕ1x1,x2,x3+x1,ϕ1x2,x3+x1,x2,ϕ1x3.It follows that (88)f1x1,x2,x3-f1′x1,x2,x3=ϕ1x1,x2,x3+x1,ϕ1x2,x3+x1,x2,ϕ1x3-ϕ1x1,x2,x3=-1x1x2+x3θx2,x3ϕ1x1-ϕ1x1,x2,x3+δDx1,x2ϕ1x3-δ-1x2x3θx1,x3ϕ1x2=d1ϕ1x1,x2,x3.So f1-f1′=d1ϕ1∈Bδ3(T,T), as desired. The proof is completed.Theorem 29.
LetT be a δ-Jordan Lie supertriple system with Hδ3(T,T)=0; then T is analytically rigid.Proof.
Letft=∑i≥0fiti be a 1-parameter formal deformation of T. Then d3fn=f1fn-1+f2fn-2+⋯+fn-1f1=0. By the assumption Hδ3(T,T)=0, we have fn∈Zδ3(T,T)=Bδ3(T,T); that is, there exits gn∈Cδ1(T,T) such that fn=d1gn.
Setϕt=idT-gntn; then (89)ϕtidT+gntn+gn2t2n+gn3t3n+⋯=idT-gntnidT+gntn+gn2t2n+gn3t3n+⋯=idT+gntn+gn2t2n+gn3t3n+⋯-gntn+gn2t2n+gn3t3n+⋯=idT.Similarly, one may check that (idT+gntn+gn2t2n+gn3t3n+⋯)ϕt=idT. So ϕt:(T[[t]],ft,δ)→(T[[t]],ft′,δ) is a linear isomorphism. Thus we can define another 1-parameter formal deformation by ϕt-1 in the form of (90)ft′x1,x2,x3=ϕt-1fiϕtx1,ϕtx2,ϕtx3.Obviously, ft~ft′. Set ft′=∑i≥0fi′ti; then we have (91)∑i≥0fi′x1,x2,x3ti-∑i≥0gnfix1,x2,x3ti+n=f0x1,x2,x3-f0gnx1,x2,x3+f0x1,gnx2,x3+f0x1,x2,gnx3tn+f0gnx1,gnx2,x3+f0x1,gnx2,gnx3+f0x1,gnx2,gnx3t2n+f0gnx1,gnx2,gnx3t3n+∑i≥nfix1,x2,x3ti-∑i≥nfignx1,x2,x3+fix1,gnx2,x3+fix1,x2,gnx3ti+n+∑i≥nfignx1,gnx2,x3+fix1,gnx2,gnx3+fignx1,x2,gnx3ti+2n-∑i≥nfignx1,gnx2,gnx3ti+3n.By the above equation, it follows that (92)f0′x1,x2,x3=f0x1,x2,x3=x1,x2,x3,f1′x1,x2,x3=f1′x1,x2,x3=⋯=fn-1′x1,x2,x3=0,fn′x1,x2,x3-gnx1,x2,x3=fnx1,x2,x3-gnx1,x2,x3-x1,gnx2,x3-x1,x2,gnx3=fnx1,x2,x3--1x1x2+x3θx2,x3gnx1+δ-1x2x3θx1,x3gnx2-δDx1,x2gnx3.Therefore, we deduce (93)fn′x1,x2,x3=fnx1,x2,x3-d1gnx1,x2,x3=0.It follows that ft′=f0+∑i≥n+1fi′ti. By induction, we have ft~f0; that is, T is analytically rigid. The proof is finished.


## 5. Nijenhuis Operators ofδ-Jordan Lie Supertriple Systems

In this section, we introduce the notion of Nijenhuis operators forδ-Jordan Lie supertriple systems. Also, we give trivial deformations of this kind of operators.LetT be a δ-Jordan Lie supertriple system and let ψ:T×T×T→T be an even trilinear map. Consider a λ-parametrized family of linear operations: (94)x1,x2,x3λ=x1,x2,x3+λψx1,x2,x3,for any x1,x2,x3∈T, where λ is a formal variable.If[·,·,·]λ endow T with a δ-Jordan Lie supertriple system structure which is denoted by Tλ, then we call that ψ generates a λ-parameter infinitesimal deformation of the δ-Jordan Lie supertriple system T.Theorem 30.
LetT be a δ-Jordan Lie supertriple system. Then Tλ is a δ-Jordan Lie supertriple system if and only if(i)
ψ itself defines a δ-Jordan Lie supertriple system structure on T;(ii)
ψ is a 3-cocycle of T.Proof.
Assume thatTλ is a δ-Jordan Lie supertriple system. For any x1,x2,x3∈T, we have (95)x2,x1,x3λ=x2,x1,x3+λψx2,x1,x3=-δ-1x1x2x1,x2,x3+λψx2,x1,x3,x2,x1,x3λ=-δ-1x1x2x1,x2,x3λ=-δ-1x1x2x1,x2,x3-δ-1x1x2λψx1,x2,x3.It follows that(96)ψx2,x1,x3=-δ-1x1x2ψx1,x2,x3.
For (3), we have (97)λ-1x1x3ψx1,x2,x3+-1x2x1ψx2,x3,x1+-1x3x2ψx3,x1,x2=-1x1x3x1,x2,x3λ+-1x2x1x2,x3,x1λ+-1x3x2x3,x1,x2λ--1x1x3x1,x2,x3--1x2x1x2,x3,x1--1x3x2x3,x1,x2=0,as desired. The last equality holds since T and Tλ are both δ-Jordan Lie supertriple systems.
For (3), we take x1,x2,x3,x4,x5∈T and calculate (98)x1,x2,x3,x4,x5λλ=x1,x2,x3,x4,x5+λψx3,x4,x5λ=x1,x2,x3,x4,x5+λψx3,x4,x5+λψx1,x2,x3,x4,x5+λψx3,x4,x5=x1,x2,x3,x4,x5+λx1,x2,ψx3,x4,x5+ψx1,x2,x3,x4,x5+λ2ψx1,x2,ψx3,x4,x5.By similar calculation, we have (99)x1,x2,x3λ,x4,x5λ=x1,x2,x3,x4,x5+λψx1,x2,x3,x4,x5+ψx1,x2,x3,x4,x5+λ2ψψx1,x2,x3,x4,x5,x3,x1,x2,x4λ,x5λ=x3,x1,x2,x4,x5+λx3,ψx1,x2,x4,x5+ψx3,x1,x2,x4,x5+λ2ψx3,ψx1,x2,x4,x5,x3,x4,x1,x2,x5λλ=x3,x4,x1,x2,x5+λx3,x4,ψx1,x2,x5+ψx3,x4,x1,x2,x5+λ2ψx3,x4,ψx1,x2,x5.It follows that (100)x1,x2,x3λ,x4,x5λ+-1x3x1+x2x3,x1,x2,x4λ,x5λ+δ-1x1+x2x3+x4x3,x4,x1,x2,x5λλ=x1,x2,x3,x4,x5+-1x3x1+x2x3,x1,x2,x4,x5+δ-1x1+x2x3+x4x3,x4,x1,x2,x5+λψx1,x2,x3,x4,x5+ψx1,x2,x3,x4,x5+-1x3x1+x2x3,ψx1,x2,x4,x5+-1x3x1+x2ψx3,x1,x2,x4,x5+δ-1x1+x2x3+x4x3,x4,ψx1,x2,x5+δ-1x1+x2x3+x4dψx3,x4,x1,x2,x5+λ2ψψx1,x2,x3,x4,x5+-1x3x1+x2ψx3,ψx1,x2,x4,x5+δ-1x1+x2x3+x4ψx3,x4,ψx1,x2,x5.Therefore, we have(101)x1,x2,ψx3,x4,x5+ψx1,x2,x3,x4,x5=ψx1,x2,x3,x4,x5+ψx1,x2,x3,x4,x5+-1x3x1+x2x3,ψx1,x2,x4,x5+-1x3x1+x2ψx3,x1,x2,x4,x5+δ-1x1+x2x3+x4x3,x4,ψx1,x2,x5+δ-1x1+x2x3+x4ψx3,x4,x1,x2,x5,(102)ψx1,x2,ψx3,x4,x5=ψψx1,x2,x3,x4,x5+-1x3x1+x2ψx3,ψx1,x2,x4,x5+δ-1x1+x2x3+x4ψx3,x4,ψx1,x2,x5.By (102), ψ satisfies (3). So ψ defines a δ-Jordan Lie supertriple system structure on T.
Sinceθ(x1,x2)(x)=(-1)xx1+x2[x,x1,x2] and D(x1,x2)(x)=δ[x1,x2,x], we can rewrite (97) as follows:(103)0=-δDx1,x2ψx3,x4,x5-ψx1,x2,x3,x4,x5+-1x1+x2+x3x4+x5θx4,x5ψx1,x2,x3+ψx1,x2,x3,x4,x5-δ-1x3x1+x2θx3,eψx1,x2,x4+-1x3x1+x2ψx3,x1,x2,x4,x5+δ-1x1+x2x3+x4Dx3,x4ψx1,x2,x5+δ-1x1+x2x3+x4ψx3,x4,x1,x2,x5=d3ψx1,x2,x3,x4,x5.The last equality holds since ψ is an even trilinear map. So d3ψ=0, as required.
Conversely, ifψ satisfies conditions (i) and (ii), it is easy to see that Tλ is a δ-Jordan Lie supertriple system from (96), (97), (101), and (102).Definition 31.
A deformation is said to be trivial if there exists a linear mapN:T→T such that, for all λ, φλ=id+λN:Tλ→T satisfies(104)φλx1,x2,x3λ=φλx1,φλx2,φλx3,for any x1,x2,x3∈T.The left-hand side of (104) is equal to (105)x1,x2,x3+λψx1,x2,x3+Nx1,x2,x3+λ2Nψx1,x2,x3.The right-hand side of (104) is equal to (106)x1,x2,x3+λNx1,x2,x3+x1,Nx2,x3+x1,x2,Nx3+λ2Nx1,Nx2,x3+x1,Nx2,Nx3+Nx1,x2,Nx3)+λ3Nx1,Nx2,Nx3.Therefore, by (104), we have(107)0=Nx1,Nx2,Nx3,(108)Nψx1,x2,x3=Nx1,Nx2,x3+x1,Nx2,Nx3+Nx1,x2,Nx3),(109)ψx1,x2,x3=Nx1,x2,x3+x1,Nx2,x3+x1,x2,Nx3-Nx1,x2,x3=-1x1x2+x3θx2,x3Nx1-δ-1x2x3θx1,x3Nx2+δDx1,x2Nx3-Nx1,x2,x3=d1Nx1,x2,x3.By (108) and (109), we can deduce that(110)N2x1,x2,x3=NNx1,x2,x3+Nx1,Nx2,x3+Nx1,x2,Nx3-Nx1,Nx2,x3-x1,Nx2,Nx3-Nx1,x2,Nx3).Definition 32.
A linear operatorN:T→T is called a Nijenhuis operator if and only if (107) and (110) hold.Theorem 33.
LetN be a Nijenhuis operator for T. Then, a deformation of T can be obtained by putting (111)ψx1,x2,x3=-1x1x2+x3θx2,x3Nx1-δ-1x2x3θx1,x3Nx2+δDx1,x2Nx3-Nx1,x2,x3.Moreover, this deformation is trivial.Proof.
The proof is similar to one in the setting ofδ-Jordan Lie triple system in [14].



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*Source: 3295462-2019-07-11.xml*