# Central Extensions and Nijenhuis Operators of Hom-δ-Jordan Lie Triple Systems

Authors: Qiang Li; Lili Ma
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this paper, the equivalence of central extensions andHα,αV3T,V is proven in the study in Hom-δ-Jordan Lie triple systems. The concepts of Nijenhuis operators of Hom-δ-Jordan Lie triple systems are given. Moreover, a trivial deformation is got.


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## Body

## 1. Introduction

It is well known that Lie triple systems are closely related to geometry. In a symmetric space, its tangent algebra is a Lie triple system. The definitions of the semisimplicity, radicality, and solvability for Lie triple systems are discussed, and the simple Lie triple system is determined by Lister [1]. Cohomologies of Lie triple systems were obtained [2]. Kubo and Taniguchi showed that in Lie triple systems, this kind of cohomology plays an important role in the study of deformations and extensions in 2004 [3]. A generalization of Lie triple systems and δ-Jordan Lie triple systems was defined in [4] by Okubo and Kamiya, where δ=±1. The case of δ=1 yields the Lie triple system, and they call the other case of δ=−1 a Jordan Lie triple system. Then, they obtained a method to construct simple Jordan superalgebras by certain triple systems and studied the F-type Jordan superalgebra of a Jordan Lie triple system [5]. Recently, the cohomologies, Nijenhuis operators, representations, abelian extensions, and T∗-extensions of δ-Jordan Lie triple systems were developed by Ma and Chen [6].The theory of Hom-type algebras has been studied (see [7–16]). In 2012, Yau showed the concept of Hom-Lie triple systems [17]. Later, generalized derivations of Hom-Lie triple systems were determined [18]. The cohomologies, 1-parameter formal deformations, and central extensions of Hom-Lie triple systems were discussed [19]. In 2019, the generalization of δ-Jordan Lie triple systems and Hom-Lie triple systems, 1-parameter formal deformations, and cohomologies of Hom-δ-Jordan Lie triple systems were studied [20]. We pay our main attention to consider central extensions and Nijenhuis operators of Hom-δ-Jordan Lie triple systems.The paper is organized as follows. In Section2, we summarize basic concepts and construct a structure of multiplicative Hom-δ-Jordan Lie triple systems. In Section 3, the equivalence of the third cohomology group and the central extensions of a Hom-δ-Jordan Lie triple system is proven. We discuss Nijenhuis operators of Hom-δ-Jordan Lie triple systems and obtain a trivial deformation using a Nijenhuis operator in Section 4.In this paper, the capital letterF denotes an arbitrary field.


## 2. Preliminaries

We start by recalling the definition of Hom-Lie triple systems.Definition 1 [17].
A Hom-Lie triple systemT,·,·,·,α=α1,α2 consists of an F-vector space T, a trilinear map ·,·,·:T3⟶T, and linear maps αi:T⟶T for i=1,2, called twisted maps, such that for all t1,t2,t3,t4,t5∈T,
(1)t1,t1,t3=0,t1,t2,t3+t2,t3,t1+t3,t1,t2=0,α1t4,α2t5,t1,t2,t3=t4,t5,t1,α1t2,α2t3+α1t1,t4,t5,t2,α2t3+α1t1,α2t2,t4,t5,t3.Definition 2 [20].
A Hom-δ-Jordan Lie triple system T,·,·,·,δ,α=α1,α2 consists of an F-vector space T, a trilinear map ·,·,·:T3⟶T, and linear maps αi:T⟶T for i=1,2, called twisted maps, such that for all t1,t2,t3,t4,t5∈T,
(2)t1,t2,t3=−δt2,t1,t3,(3)t1,t2,t3+t2,t3,t1+t3,t1,t2=0,(4)α1t4,α2t5,t1,t2,t3=t4,t5,t1,α1t2,α2t3+α1t1,t4,t5,t2,α2t3+δα1t1,α2t2,t4,t5,t3.Remark 3.
Whenδ=1, a Hom-δ-Jordan Lie triple system is a Hom-Lie triple system. A Hom-δ-Jordan Lie triple system is a δ-Jordan Lie triple system if the twisted maps αi are both equal to the identity map. So Hom-Lie triple systems and δ-Jordan Lie triple systems are special examples of Hom-δ-Jordan Lie triple systems.A Hom-δ-Jordan Lie triple system is said to be multiplicative if α1=α2=α and αt1t2t3=αt1αt2αt3 and denoted by T,·,·,·,α.A morphismf:T,·,·,·,α=α1,α2⟶T′,·,·,·′,α′=α1′,α2′ of the Hom-δ-Jordan Lie triple system is a linear map satisfying ft1t2t3=ft1ft2ft3′ and f∘αi=αi′∘f for i=1,2. An isomorphism is a bijective morphism.Definition 4 [20].
LetT,·,·,·,δ,α be multiplicative Hom-δ-Jordan Lie triple systems, V be an F-vector space, and A∈EndV. V is said to be a T,·,·,·,δ,α-module with respect to A if there exists a bilinear map θ:T2⟶EndV, t1,t2↦θt1,t2 such that for all t1,t2,t3,t4∈T,
(5)θαt1,αt2∘A=A∘θt1,t2,(6)θαt3,αt4θt1,t2−δθαt2,αt4θt1,t3−θαt1,t2,t3,t4∘A+Dαt2,αt3θt1,t4=0,(7)δθαt3,αt4Dt1,t2−δDαt1,αt2θt3,t4+θt1,t2,t3,αt4∘A+δθαt3,t1,t2,t4∘A=0,(8)δDαt3,αt4Dt1,t2−Dαt1,αt2Dt3,t4+δDt1,t2,t3,αt4∘A+δDαt3,t1,t2,t4∘A=0,where Dt1,t2=θt2,t1−δθt1,t2.Then,θ is said to be the representation of T,·,·,·,δ,α on V with respect to A. In the case θ=0, V is said to be the trivial T,·,·,·,δ,α-module with respect to A.In the case thatδ=1, i.e., Hom-δ-Jordan Lie triple systems are Hom-Lie triple systems, we can get (8) from (7) by a direct calculation. But it is not true in the other case δ=−1.Particularly,Dt1,t2t3=δt1,t2,t3 and (5), (6), and (7) hold, if V=T, A=α, and θt1,t2t3=t3,t1,t2. In this case, T is called the adjoint T,·,·,·,α-module and θ is called the adjoint representation of T,·,·,·,α on itself with respect to α.In the following, the semidirect product of multiplicative Hom-δ-Jordan Lie triple systems and its module for general algebras were introduced.Proposition 5.
Assume thatT,·,·,·,δ,α is a multiplicative Hom-δ-Jordan Lie triple system on V with respect to A and θ is a representation of T,·,·,·,δ,α. Then, TC=T⊕V has a structure of a multiplicative Hom-δ-Jordan Lie triple system.Proof.
We define the operation·,·,·V:TC3⟶TC by t1,a,t2,b,t3,cV=t1,t2,t3,θt2,t3a−δθt1,t3b+δDt1,t2c and define the twisted map α+A:TC⟶TC by
(9)α+At1,a=αt1,Aa.
ByDt1,t2=θt2,t1−δθt1,t2, we get
(10)t1,a,t2,b,t3,cV=t1,t2,t3,θt2,t3a−δθt1,t3b+δDt1,t2c=−δt2,t1,t3,θt1,t3b−δθt2,t3a−Dt1,t2c=−δt2,t1,t3,θt1,t3b−δθt2,t3a+δDt2,t1c=−δt2,b,t1,a,t3,cV,t1,a,t2,b,t3,cV+t2,b,t3,c,t1,aV+t3,c,t1,a,t2,bV=t1,t2,t3,θt2,t3a−δθt1,t3b+δDt1,t2c+t2,t3,t1,θt3,t1b−δθt2,t1c+δDt2,t3a+t3,t1,t2,θt1,t2c−δθt3,t2a+δDt3,t1b=0,θt2,t3a−δθt3,t2a+δDt2,t3a+θt3,t1b−δθt1,t3b+δDt3,t1b+θt1,t2c−δθt2,t1c+δDt1,t2c=0,0.
By (6), (7), and (8), we have
(11)t1,a,t2,b,u,cV,α+Av,d,α+Aw,eV=t1,t2,u,θt2,ua−δθt1,ub+δDt1,t2c,αv,Ad,αw,AeV=t1,t2,u,αv,αw,θαv,αwθt2,ua−δθt1,ub+δDt1,t2c−δθt1,t2,u,αwAd+δDt1,t2,u,αvAe,α+Au,c,t1,a,t2,b,v,dV,α+Aw,eV=αu,Ac,t1,t2,v,θt2,va−δθt1,vb+δDt1,t2d,αw,AeV=αu,t1,t2,v,αw,θt1,t2,v,αwAc−δθαu,αwθt2,va−δθt1,vb+δDt1,t2d+δDαu,t1,t2,vAe,δα+Au,c,α+Av,d,t1,a,t2,b,w,eVV=δαu,Ac,αv,Ad,t1,t2,w,θt2,wa−δθt1,wb+δDt1,t2eV=δαu,αv,t1,t2,w,θαv,t1,t2,wAc−δθαu,t1,t2,wAd+δDαu,αvθt2,wa−δθt1,wb+δDt1,t2e,α+At1,a,α+At2,b,u,c,v,d,w,eVV=αt1,Aa,αt2,Ab,u,v,w,θv,wc−δθu,wd+δDu,veV=αt1,αt2,u,v,w,θαt2,u,v,wAa−δθαt1,u,v,wAb+δDαt1,αt2θv,wc−δθu,wd+δDu,ve.The calculation above shows that (2), (3), and (4) hold.By (5) and the linearity of α+A,
(12)α+At1,a,t2,b,t3,cV=α+At1,t2,t3,θt2,t3a−δθt1,t3b+δDt1,t2c=αt1,t2,t3,A∘θt2,t3a−δθt1,t3b+δDt1,t2c=αt1,αt2,αt3,θαt2,αt3Aa−δθαt1,αt3Ab+δDαt1,αt2Ac=αt1,Aa,αt2,Ab,αt3,AcV=α+At1,a,α+At2,b,α+At3,cV.Hence,U,·,·,·v,a+A is a multiplicative Hom-δ-Jordan Lie triple system.Suppose thatf:T×⋯×T⏟ntimes⟶V is an n-linear map, which satisfies
(13)Aft1,⋯,tn=fαt1,⋯,αtn,ft1,⋯,x,y,tn=−δft1,⋯,y,x,tn,ft1,⋯,tn−3,x,y,z+ft1,⋯,tn−3,y,z,x+ft1,⋯,tn−3,z,x,y=0,where f is said to be an n-Hom-cochain on T. The set of all n-Hom-cochains is denoted by Cα,AnT,V, for all n≥1.
(i)
Iff∈Cδ1T,V, then
(14)dhom1ft1,t2,t3=θt2,t3ft1−δθt1,t3ft2+δDt1,t2ft3−ft1,t2,t3.(ii)
Iff∈Cδ2T,V, then
(15)dhom2fy,t1,t2,t3=θαt2,αt3fy,t1−δθαt1,αt3fy,t2+δDαt1,αt2fy,t3−fαy,t1,t2,t3.(iii)
Iff∈Cδ3T,V, then
(16)dhom3ft1,t2,t3,t4,t5=θαt4,αt5ft1,t2,t3−δθαt3,αt5ft1,t2,t4−δDαt1,αt2ft3,t4,t5+Dαt3,αt4ft1,t2,t5+ft1,t2,t3,αt4,αt5+fαt3,t1,t2,t4,αt5+δfαt3,αt4,t1,t2,t5−fαt1,αt2,t3,t4,t5.(iv)
Iff∈Cδ4T,V, then
(17)dhom4fy,t1,t2,t3,t4,t5=θα2t4,α2t5fy,t1,t2,t3−δθα2t3,α2t5fy,t1,t2,t4−δDα2t1,α2t2fy,t3,t4,t5+Dα2t3,α2t4fy,t1,t2,t5+fαy,t1,t2,t3,αt4,αt5+fαy,αt3,t1,t2,t4,αt5+δfαy,αt3,αt4,t1,t2,t5−fαy,αt1,αt2,t3,t4,t5.Definition 6 [20].
Forn=1,2,3,4, the coboundary operator dhomn:Cα,AnT,V⟶Cα,An+2T,V is defined as follows.The mappingf∈Cα,AnT,V is said to be an n-Hom-cocycle if dhomnf=0,n=1,2,3,4. Denote by Zα,AnT,V the subspace spanned by n-Hom-cocycles. For n≥3, Bα,AnT,V=dhomn−2Cα,An−2T,V.Sincedhomn+2dhomn=0, Bα,AnT,V⊆Zα,AnT,V. Define a cohomology space:
(18)Hα,AnT,V=Zα,AnT,VBα,AnT,V.


## 3. Central Extensions of Hom-δ-Jordan Lie Triple Systems

LetT,·,·,·,δ,α be a multiplicative Hom-δ-Jordan Lie triple system and V be a trivial T,·,·,·,δ,α-module with respect to αV. Then, V,0,αV is an abelian multiplicative Hom-δ-Jordan Lie triple system with the trivial product. A multiplicative Hom-δ-Jordan Lie triple system TC,·,·,·C,δ,αC is said to be a central extension of T,·,·,·,δ,α by V,0,δ,αV if the following commutative diagram holds with the exact rows of Hom-δ-Jordan Lie triple systems.whereαC∘ι=ι∘αV,α∘π=π∘αC, s is a linear map satisfying πs=idT and αC∘s=s∘α, and ιV⊆ZTC=x∈TC∣x,TC,TCC=0. Two central extensions TC,·,·,·C,δ,αC and TC′,·,·,·C′,δ,αC′ are equivalent, if the following commutative diagram holds.whereφ:TC,·,·,·C,δ,αC⟶TC′,·,·,·C′,δ,αC′ is an isomorphism.Theorem 7.
There is bijective mapping betweenHα,αV3T,V and equivalent classes of central extensions of T,·,·,·,δ,α by V,0,δ,αV.Proof.
First, we show that there is a bijective mapping betweenZα,αV3T,V and central extensions of T,·,·,·,δ,α by V,0,δ,αV.Suppose thatTC,·,·,·C,δ,αC is a central extension of T,·,·,·,δ,α by V,0,δ,αV. Then, the following commutative diagram holds:withαC∘ι=ι∘αV, α∘π=π∘αC, and a linear map s satisfying αC∘s=s∘α and πs=idT.Fort1,t2,t3∈T, since πst1,st2,st3C−πst1,t2,t3=πst1,πst2,πst3−t1,t2,t3=0, it follows that st1,st2,st3C−st1,t2,t3∈Kerπ=ιV. Define a trilinear map g:T×T×T⟶V by
(19)ιgt1,t2,t3=st1,st2,st3C−st1,t2,t3.Sinceι is injective, g is well defined, and it follows from ιV⊆ZTC that
(20)st1,st2,st3C,u,vC=st1,t2,t3,u,vC,∀u,v∈TC.Note thatg satisfies gt1,t2,t3=−δgt2,t1,t3,gt1,t2,t3+gt2,t3,t1+gt3,t1,t2=0 and
(21)ιgαt1,αt2,αt3=sαt1,sαt2,sαt3C−sαt1,αt2,αt3=αCst1,αCst2,αCst3C−αCst1,t2,t3=αCst1,st2,st3C−st1,t2,t3=αCιgt1,t2,t3=ιαVgt1,t2,t3.Hence,g∈Cα,αV3T,V. Moreover, g∈Zα,αV3T,V since
(22)ιdhom3gt1,t2,t3,t4,t5=ιgt1,t2,t3,αt4,αt5+gαt3,t1,t2,t4,αt5+δgαt3,αt4,t1,t2,t5−gαt1,αt2,t3,t4,t5=st1,t2,t3,sαt4,sαt5C−st1,t2,t3,αt4,αt5+sαt3,st1,t2,t4,sαt5C−sαt3,t1,t2,t4,αt5+δsαt3,sαt4,st1,t2,t5C−δsαt3,αt4,t1,t2,t5−sαt1,sαt2,st3,t4,t5C+sαt1,αt2,t3,t4,t5=st1,st2,st3c,αCst4,αCst5C+αCst3,st1,st2,st4C,αCst5C+δαCst3,αCst4,st1,st2,st5CC−αCst1,αCst2,st3,st4,st5CC=0.On the other hand, letg∈Zα,αV3T,V and TC=T⊕V with
(23)t1,a,t2,b,t3,cC=t1,t2,t3,gt1,t2,t3;αCt1,a=αt1,αVa.Thus,αC is linear, and
(24)αCt1,a,t2,b,t3,cC=αCt1,t2,t3,gt1,t2,t3=αt1,t2,t3,αVgt1,t2,t3=αt1,αt2,αt3,gαt1,αt2,αt3=αt1,αVa,αt2,αVb,αt3,αVcC=αCt1,a,αCt2,b,αCt3,cC.Since that(25)αCt1,a,αCt2,b,u,c,v,d,w,eCC=αt1,αVa,αt2,αVb,u,v,w,gu,v,wC=αt1αt2u,v,w,gαt1,αt2,u,v,w=t1,t2,uαvαw,gt1,t2,u,αv,αw+αut1,t2,vαw,gαu,t1,t2,v,αw+δαuαvt1,t2,w,δgαu,αv,t1,t2,w=t1,a,t2,b,u,cC,αCv,d,αCw,eC+αCu,c,t1,a,t2,b,v,dC,αCw,eC+δαCu,c,αCv,d,t1,a,t2,b,w,eCC.We haveTC,·,·,·c,δ,αC which is a multiplicative Hom-δ-Jordan Lie triple system.Define three mappingsι:V⟶TC, π:TC⟶T, and s:T⟶TC by ιa=0,a, πt,a=t, and st=t,0, respectively. Then,
(26)αC∘ιa=αC0,a=0,αVa=ι∘αVa,π∘αCt,a=παt,αVa=αt=α∘πt,a,πs=idT,αCst=αCt,0=αt,0=sαt.It is clear thatιV is a subspace of ZTC. Hence, TC,·,·,·C,δ,αC is a central extension of T,·,·,·,δ,α by V,0,δ,αV.Assume thatTC,·,·,·C,δ,αC and TC′,·,·,·C′,δ,αC′ are equivalent central extensions of T,·,·,·,δ,α by V,0,δ,αV. Then, the following commutative diagram holds:such thatπ=π′∘φ and φ∘ι=ι′ with an isomorphism φ and πs=π′s′=idT. For their corresponding 3-Hom-cocycles g and g′ as above, we have
(27)ιgt1,t2,t3=st1,st2,st3C−st1,t2,t3,ι′g′t1,t2,t3=s′t1,s′t2,s′t3C′−s′t1,t2,t3,ι′gt1,t2,t3=φιgt1,t2,t3=φst1,st2,st3C−φst1,t2,t3.We haveg−g′∈Bα,αV3T,V. In fact, since
(28)π′s′t1−π′φst1=t1−πst1=0,there exists a linear mapping f:T⟶V by ι′ft1=s′t1−φst1, for all t1∈T. Then,
(29)ι′fαt1=s′αt1−φsαt1=αC′s′t1−φαCst1=αC′s′t1−αC′φst1=αC′ι′ft1=ι′αVft1,that is, f∈Cα,αV1T,V. By s′t1−φst1=ι′ft1∈ZTC′,
(30)s′t1,s′t2,s′t3C′=φst1,φst2,φst3C′=φst1,st2,st3C.Then,(31)ι′g′−gt1,t2,t3=−ι′ft1,t2,t3=ι′dhom1ft1,t2,t3,so g′−g=dhom1f∈Bα,αV3T,V.Supposeg,g′∈Zα,αV3T,V and g′−g∈Bα,αV3T,V; i.e., there is f∈Cα,αV1T,V satisfying g′−g=dhom1f. Then, g′−gt1,t2,t3=−ft1,t2,t3. Let TC,·,·,·C,δ,αC and TC′,·,·,·C′,δ,αC, which are defined as above with respect to g and g′, be two central extensions of T,·,·,·,δ,α by V,0,δ,αV. Then, ιa=0,a=ι′a and πt,a=t=π′t,a. There is a linear map:
(32)φ:TC,·,·,·C,δ,αC⟶TC′,·,·,·C′,δ,αC,t,a↦t,a−ft,such that φιa=ι′a and π′φt,a=π′t,a−ft=t=πt,a. The following commutative diagram holds:The sufficiency ofφ is an isomorphism that is proven.Ifφt,a=φt~,a~, then t,a−ft=t~,a~−ft~; that is, t=t~ and a−ft=a~−ft~; then, a=a~; hence, φ is injective; φ is obviously surjective. Note that
(33)φαCt,a=φαt,αVa=αt,αVa−fαt=αt,αVa−αVft=αC′t,a−ft=αC′φt,a,φt1,a,t2,b,t3,cC=φt1,t2,t3,gt1,t2,t3=t1,t2,t3,gt1,t2,t3−ft1,t2,t3=t1,t2,t3,g′t1,t2,t3=t1,a−ft1,t2,b−ft2,t3,c−ft3C′=φt1,a,φt2,b,φt3,cC′.The equivalence ofTC,·,·,·C,δ,αC and TC′,·,·,·C′,δ,αC′ is proven.


## 4. Nijenhuis Operators of Hom-δ-Jordan Lie Triple Systems

In this section, the deformation of Hom-δ-Jordan Lie triple systems is studied. The notion of Nijenhuis operators of Hom-δ-Jordan Lie triple systems is introduced, and the trivial deformations of this kind of operators are shown.LetT,·,·,·,δ,α be a Hom-δ-Jordan Lie triple system and ψ:T3⟶T be a trilinear mapping. Consider a λ-parametrized family of linear operations:
(34)t1,t2,t3λ=t1,t2,t3+λψt1,t2,t3,where λ is a formal variable.We call thatψ generates a λ-parameter infinitesimal deformation of the Hom-δ-Jordan Lie triple system, if ·,·,·λ endow T with the Hom-δ-Jordan Lie triple system structure which is denoted by Tλ.
(i)
ψ itself defines a Hom-δ-Jordan Lie triple system structure on T(ii)
ψ is a 3-cocycle of TTheorem 8.
ψ generates a λ-parameter infinitesimal deformation of the Hom-δ-Jordan Lie triple system T; then, the following two conclusions hold:Proof.
(35)t1,t2,t3λ=t1,t2,t3+λψt1,t2,t3,−δt2,t1,t3λ=−δt2,t1,t3−δλψt2,t1,t3.We have
(36)ψt1,t2,t3=−δψt2,t1,t3.
From the equality(37)0=t1,t2,t3λ+t2,t3,t1λ+t3,t1,t2λ=t1,t2,t3+t2,t3,t1+t3,t1,t2+λψt1,t2,t3+ψt2,t3,t1+ψt3,t1,t2,it follows that
(38)ψt1,t2,t3+ψt2,t3,t1+ψt3,t1,t2=0.
For the equality(39)αt1,αt2,r1,r2,r3λλ=t1,t2,r1λ,αr2,αr3λ+αr1,t1,t2,r2λαr3λ+δαr1,αr2,t1,t2,r3λλ,the left hand side is equal to
(40)αt1,αt2,r1,r2,r3+λψr1,r2,r3λ=αt1,αt2,r1,r2,r3+λψαt1,αt2,r1,r2,r3+αt1,αt2,λψr1,r2,r3+λψαt1,αt2,λψr1,r2,r3=αt1,αt2,r1,r2,r3+λψαt1,αt2,r1,r2,r3+αt1,αt2,ψr1,r2,r3+λ2ψαt1,αt2,ψr1,r2,r3,and the right hand side is equal to
(41)t1,t2,r1+λψt1,t2,r1,αr2,αr3λ+αr1,t1,t2,r2+λψt1,t2,r2,αr3λ+δαr1,αr2,t1,t2,r3+λψt1,t2,r3λ=t1,t2,r1,αr2,αr3+αr1,t1,t2,r2,αr3+δαr1,αr2,t1,t2,r3+λψt1,t2,r1,αr2,αr3+ψt1,t2,r1,αr2,αr3+ψαr1,t1,t2,r2,αr3+αr1,ψt1,t2,r2,αr3+δψαr1,αr2,t1,t2,r3+δαr1,αr2,ψt1,t2,r3+λ2ψψt1,t2,r1,αr2,αr3+ψαr1,ψt1,t2,r2,αr3+δψαr1,αr2,ψt1,t2,r3.
Thus, we have(42)ψαt1,αt2,r1,r2,r3+δDαt1,αt2ψr1,r2,r3=ψt1,t2,r1,αr2,αr3+θαr2,αr3ψt1,t2,r1+ψαr1,t1,t2,r2,αr3−δθαr1,αr3ψt1,t2,r2+δψαr1,αr2,t1,t2,r3+Dαr1,αr2ψt1,t2,r3,(43)ψαt1,αt2,ψr1,r2,r3=ψψt1,t2,r1,αr2,αr3+ψαr1,ψt1,t2,r2,αr3+δψαr1,αr2,ψt1,t2,r3.
Therefore,ψ defines a Hom-δ-Jordan Lie triple system structure on T by (36), (38), and (43). Furthermore, by (42), ψ is a 3-cocycle.A deformation is called trivial if there exists a linear mapN:T⟶T such that for φλ=id+λN:Tλ⟶T,
(44)φλt1,t2,t3λ=φλt1,φλt2,φλt3.It is clear that(45)φλt1,t2,t3λ=t1,t2,t3+λψt1,t2,t3+λNt1,t2,t3+λψt1,t2,t3=t1,t2,t3+λψt1,t2,t3+Nt1,t2,t3+λ2Nψt1,t2,t3,φλt1,φλt2,φλt3=t1+λNt1,t2+λNt2,t3+λNt3=t1,t2,t3+λNt1,t2,t3+t1,Nt2,t3+t1,t2,Nt3+λ2Nt1,Nt2,t3+Nt1,t2,Nt3+t1,Nt2,Nt3+λ3Nt1,Nt2,Nt3.Thus, we have(46)ψt1,t2,t3=Nt1,t2,t3+t1,Nt2,t3+t1,t2,Nt3−Nt1,t2,t3=θt2,t3Nt1−δθt1,t3Nt2+δDt1,t2Nt3−Nt1,t2,t3,(47)Nψt1,t2,t3=Nt1,Nt2,t3+Nt1,t2,Nt3+t1,Nt2,Nt3,(48)0=Nt1,Nt2,Nt3.By the cohomologies discussed in Section2, equation (46) can be represented in terms of 1-coboundary as ψ=dhom1N. Furthermore, the following condition holds for N by (46) and (47):
(49)N2t1,t2,t3=NNt1,t2,t3+Nt1,Nt2,t3+Nt1,t2,Nt3−Nt1,Nt2,t3+Nt1,t2,Nt3+t1,Nt2,Nt3.In the following, we denote by(50)ψt1,t2,t3=t1,t2,t3N,then, (47) is equivalent to 51
(51)Nt1,t2,t3N=Nt1,Nt2,t3+Nt1,t2,Nt3+t1,Nt2,Nt3.Definition 9.
A linear operatorN:T⟶T is said to be a Nijenhuis operator if and only if (48) and (49) hold.Theorem 10.
LetN be a Nijenhuis operator for T. Then, a deformation of T can be obtained if
(52)ψt1,t2,t3=θt2,t3Nt1−δθt1,t3Nt2+δDt1,t2Nt3−Nt1,t2,t3.
Moreover,ψ is a trivial deformation.Proof.
Clearly,ψ=dN and dψ=d2N=0. Then, ψ is a 3-cocycle of T. In the following, we show that (4) holds for ψ. By (46), (50), and (51), we have
(53)ψαt1,αt2,ψr1,r2,r3=αt1,αt2,Nr1,r2,r3+r1,Nr2,r3+r1,r2,Nr3−Nr1,r2,r3N=αt1,αt2,Nr1,r2,r3N+αt1,αt2,r1,Nr2,r3N+αt1,αt2,r1,r2,Nr3N−αt1,αt2,Nr1,r2,r3N=Nαt1,αt2,Nr1,r2,r3+αt1,Nαt2,Nr1,r2,r3+αt1,αt2,NNr1,r2,r3−Nαt1,αt2,Nr1,r2,r3+Nαt1,αt2,r1,Nr2,r3+αt1,Nαt2,r1,Nr2,r3+αt1,αt2,Nr1,Nr2,r3−Nαt1,αt2,r1,Nr2,r3+Nαt1,αt2,r1,r2,Nr3+αt1,Nαt2,r1,r2,Nr3+αt1,αt2,Nr1,r2,Nr3−Nαt1,αt2,r1,r2,Nr3−Nαt1,αt2,Nr1,r2,r3−αt1,Nαt2,Nr1,r2,r3−αt1,αt2,N2r1,r2,r3+Nαt1,αt2,Nr1,r2,r3=Nαt1,αt2,Nr1,r2,r3+αt1,Nαt2,Nr1,r2,r3−Nαt1,αt2,Nr1,r2,r3+Nαt1,αt2,r1,Nr2,r3+αt1,Nαt2,r1,Nr2,r3−Nαt1,αt2,r1,Nr2,r3+Nαt1,αt2,r1,r2,Nr3+αt1,Nαt2,r1,r2,Nr3−Nαt1,αt2,r1,r2,Nr3−Nαt1,αt2,Nr1,r2,r3−αt1,Nαt2,Nr1,r2,r3+Nαt1,αt2,Nr1,r2,r3+αt1,αt2,Nr1,Nr2,r3+αt1,αt2,Nr1,r2,Nr3+αt1,αt2,r1,Nr2,Nr3.
Similarly, a direct computation shows that(54)ψψt1,t2,r1,αr2,αr3=Nt1,Nt2,r1,αr2,αr3+Nt1,t2,Nr1,αr2,αr3+t1,Nt2,Nr1,αr2,αr3+Nt1,t2,r1,Nαr2,αr3+Nt1,t2,r1,αr2,Nαr3−NNt1,t2,r1,αr2,αr3+t1,Nt2,r1,Nαr2,αr3+t1,Nt2,r1,αr2,Nαr3−Nt1,Nt2,r1,αr2,αr3+t1,t2,Nr1,Nαr2,αr3+t1,t2,Nr1,αr2,Nαr3−Nt1,t2,Nr1,αr2,αr3−Nt1,t2,r1,Nαr2,αr3−Nt1,t2,r1,αr2,Nαr3+NNt1,t2,r1,αr2,αr3,ψαr1,ψt1,t2,r2,αr3=Nαr1,Nt1,t2,r2,αr3+αr1,Nt1,t2,r2,Nαr3−Nαr1,Nt1,t2,r2,αr3+Nαr1,t1,Nt2,r2,αr3+αr1,t1,Nt2,r2,Nαr3−Nαr1,t1,Nt2,r2,αr3+Nαr1,t1,t2,Nr2,αr3+αr1,t1,t2,Nr2,Nαr3−Nαr1,t1,t2,Nr2,αr3−Nαr1,Nt1,t2,r2,αr3−αr1,Nt1,t2,r2,Nαr3+Nαr1,Nt1,t2,r2,αr3+αr1,Nt1,Nt2,r2,αr3+αr1,Nt1,t2,Nr2,αr3+αr1,t1,Nt2,Nr2,αr3,δψαr1,αr2,ψt1,t2,r3=δNαr1,αr2,Nt1,t2,r3+δαr1,Nαr2,Nt1,t2,r3−δNαr1,αr2,Nt1,t2,r3+δNαr1,αr2,t1,Nt2,r3+δαr1,Nαr2,t1,Nt2,r3−δNαr1,αr2,t1,Nt2,r3+δNαr1,αr2,t1,t2,Nr3+δαr1,Nαr2,t1,t2,Nr3−δNαr1,αr2,t1,t2,Nr3−δNαr1,αr2,Nt1,t2,r3−δαr1,Nαr2,Nt1,t2,r3+δNαr1,αr2,Nt1,t2,r3+δαr1,αr2,Nt1,Nt2,r3+δαr1,αr2,Nt1,t2,Nr3+δαr1,αr2,t1,Nt2,Nr3.Note thatN∈Cα1T; by (4), (51), and Theorem 8, it follows that
(55)ψαt1,αt2,ψr1,r2,r3−ψψt1,t2,r1,αr2,αr3−ψαr1,ψt1,t2,r2,αr3−δψαr1,αr2,ψt1,t2,r3=0.The conclusion of Theorem10 is proven.Remark 11.
LetN be a Nijenhuis operator. If k,m>0, then
(56)t1,t2,t3Nk+1=t1,t2,t3NkN,t1,t2,t3NkNm=t1,t2,t3NkNNm−1=t1,t2,t3Nk+1Nm−1=t1,t2,t3Nk+2Nm−2=⋯=t1,t2,t3Nk+m.Remark 12.
LetN be a Nijenhuis operator; by mathematical induction and (48), for any k>0, Nk is also a Nijenhuis operator.



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*Source: 2706774-2022-01-31.xml*