Research article

Cohomology and its applications on multiplicative Hom-δ-Jordan Lie color triple systems

  • Received: 01 June 2024 Revised: 16 August 2024 Accepted: 27 August 2024 Published: 06 September 2024
  • MSC : 17B56, 17B61, 17B75

  • The theory of cohomologies on multiplicative Hom-δ-Jordan Lie color triple systems is given. As an application, deformations and extensions on the multiplicative Hom-δ-Jordan Lie color triple system are characterized in view of relevant cohomology.

    Citation: Lili Ma, Qiang Li. Cohomology and its applications on multiplicative Hom-δ-Jordan Lie color triple systems[J]. AIMS Mathematics, 2024, 9(9): 25936-25955. doi: 10.3934/math.20241267

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  • The theory of cohomologies on multiplicative Hom-δ-Jordan Lie color triple systems is given. As an application, deformations and extensions on the multiplicative Hom-δ-Jordan Lie color triple system are characterized in view of relevant cohomology.



    Lie triple systems were introduced in connection with the symmetric space. In the study of constant curvature spaces and their tangent spaces, the Lie triple system is one of the important tools. We know also that Lie triple systems are closely related to the totally geodesic submani-fold and elementary theoretical physics[1,2]. The study on Lie triple systems has experienced a vigorous development. For example, the controlling cohomology, cohomology spaces, and several deformations and extensions on Lie triple systems were discussed[3,4,5]. As a promotion structure on Lie triple systems, that is, Lie supertriple systems, was studied[6]. In 1997, Okubo and Kamiya got a new kind of Lie supertriple system, that is, δ-Jordan Lie supertriple systems[7]. Later, they gave several meaningful results on δ-Jordan Lie supertriple systems in[8,9]. Recently, cohomologies on δ-Jordan Lie triple systems were developed[10]. Then, Wang, Zhang, and Guo obtained structures of the derivation and the deformation on δ-Jordan Lie supertriple systems[11].

    Some Hom type algebraic structures have been determined[12,13,14,15,16,17,18,19,20,21,22]. For instance, Yau introduced the concept of Hom Lie triple systems[14]. In recent years, the authors studied the cohomology and several deformations on Hom Lie triple systems, respectively[16,17,18]. In 2023, Li and Ma obtained two kinds of extensions on Hom-δ-Jordan Lie supertriple systems[19]. Li and Ma also studied Lie color triple systems, and got relevant operators and extensions[23,24]. This paper is a continuation based on[23,24].

    The organization of this paper is as follows: In Section 2, it is necessary to give definitions and examples concerning Hom-δ-Jordan Lie color triple systems. We define the representation ϑ and construct a kind of Hom-δ-Jordan Lie color triple system by the representation ϑ, then we obtain the suitable cohomology structure on Hom-δ-Jordan Lie color triple systems. Applying the results given in Section 2, we obtain that equivalent 1-parameter formal deformations lL(t,p,q)=i0li(t,p,q)si and lL(t,p,q)=i0li(t,p,q)si are in the same cohomology class in H3(T,T) in Section 3. In Section 4, we prove that there is the same representation between two equivalent abelian extensions; moreover, we also construct a kind of multiplicative Hom-δ-Jordan Lie color triple system using a 3-Hom-cocycle and obtain the necessary and sufficient condition for two equivalent abelian extensions.

    Throughout this article, we study an algebraically closed field F of characteristic 0.

    Definition 2.1. [25] Suppose that G is an abelian group. A bi-character on G is a map ε:G×GK{0} and

    ε(g1,g2)ε(g2,g1)=1,
    ε(g1,g2+g3)=ε(g1,g2)ε(g1,g3),
    ε(g1+g3,g2)=ε(g1,g2)ε(g3,g2),

    where g1,g2,g3G. It is obvious that

    ε(g,0)=ε(0,g)=1,ε(g,g)=±1,gG.

    Definition 2.2. [14] The Hom Lie triple system (L,[,,],β=(β1,β2)) consists of an F-vector space L, a trilinear map [,,]:L×L×LL, with even linear maps βi:LL for i=1,2, which is called twisted maps, satisfying for all elements t,p,q,u,vL,

    [t,t,p]=0,
    [t,p,q]+[p,q,t]+[q,t,p]=0,
    [β1(u),β2(v),[t,p,q]]=[[u,v,t],β1(p),β2(q)]+[β1(t),[u,v,p],β2(q)]+[β1(t),β2(p),[u,v,q]].

    If |t| appears in some aspect, we usually think of t as a homogeneous element, meanwhile |t| as the homogeneous degree of t.

    Definition 2.3. The Hom-δ-Jordan Lie color triple system (L,[,,],β=(β1,β2)) consists of a G-graded vector space L=gGLg over F, a trilinear map [,,]:L×L×LL, with even linear maps βi:LL for i=1,2, which is called twisted maps, satisfying for all elements t,p,q,u,vL, δ=±1,

    ε(|t|,|p|)=ε(|p|,|t|), (2.1)
    [t,p,q]=δε(|t|,|p|)[p,t,q], (2.2)
    ε(|t|,|q|)[t,p,q]+ε(|p|,|t|)[p,q,t]+ε(|q|,|p|)[q,t,p]=0, (2.3)
    [β1(u),β2(v),[t,p,q]]=[[u,v,t],β1(p),β2(q)]+ε(|t|,|u|+|v|)[β1(t),[u,v,p],β2(q)]+δε(|u|+|v|,|t|+|p|)[β1(t),β2(p),[u,v,q]]. (2.4)

    Example 2.1. Hom-δ-Jordan Lie triple systems are examples of Hom-δ-Jordan Lie color triple systems with G={0}, and ε(|0|,|0|)=1.

    Example 2.2. Hom-δ-Jordan Lie supertriple systems are examples of Hom-δ-Jordan Lie color triple systems with G={Z2}={ˉ0,ˉ1}, and ε(|p|,|q|)=(1)ij for any i,jZ2.

    Example 2.3. Suppose (L,[,]) is a Hom-δ-Jordan Lie color algebra, thus (L,[,,]) defines a Hom-δ-Jordan Lie color triple system by [t,p,q]=[[t,p],q], for any t,p,qL.

    Example 2.4. Suppose (L,[,,]) is a Hom-δ-Jordan Lie color triple system, and l is an indeterminate. Assume L={Σi0yli|yL}. Then (L,[,,]) is a Hom-δ-Jordan Lie color triple system satisfying a product [,,] defined using

    [pli,qlj,mlk]=[p,q,m]li+j+k,

    where pli,qlj,mlkL, and |pli|=|p|.

    The Hom-δ-Jordan Lie color triple system is called be multiplicative if β1=β2=β and β([t,p,q])=[β(t),β(p),β(q)].

    The morphism h:(L,[,,],β=(β1,β2))(L,[,,],β=(β1,β2)) on multiplicative Hom-δ-Jordan Lie color triple system is a linear map such that h([t,p,q])=[h(t),h(p),h(q)] and hβi=βih, where i=1,2. The bijective morphism is called the isomorphism.

    Definition 2.4. Suppose that (L,[,,]) is a multiplicative Hom-δ-Jordan Lie color triple system. W is a G-graded vector space over F and γEnd(W). W is called a (L,[,,])-module relating to γ if there is a bilinear map ϑ:L×LEnd(W), (t,t1)ϑ(t,t1) satisfying for all elements t,t1,t2,t3L,

    ϑ(β(t),β(t1))γ=γϑ(t,t1), (2.5)
    ε(|t|+|t1|,|t2|+|t3|)ϑ(β(t2),β(t3))ϑ(t,t1)δε(|t|,|t1|)ε(|t3|,|t|+|t2|)ϑ(β(t1),β(t3))ϑ(t,t2)ϑ(β(t),[t1,t2,t3])γ+ε(|t|,|t1|+|t2|)D(β(t1),β(t2))ϑ(t,t3)=0, (2.6)
    δε(|t|+|t1|,|t2|+|t3|)ϑ(β(t2),β(t3))D(t,t1)δD(β(t),β(t1))ϑ(t2,t3)+ϑ([t,t1,t2],β(t3))γ+δε(|t2|,|t|+|t1|)ϑ(β(t2),[t,t1,t3])γ=0, (2.7)
    δε(|t|+|t1|,|t2|+|t3|)D(β(t2),β(t3))D(t,t1)D(β(t),β(t1))D(t2,t3)+δD([t,t1,t2],β(t3))γ+δε(|t2|,|t|+|t1|)D(β(t2),[t,t1,t3])γ=0, (2.8)

    where D(t,t1)=ε(|t|,|t1|)ϑ(t1,t)δϑ(t,t1).

    Then ϑ is said to be the representation of (L,[,,]) on W relating to γ. When ϑ=0, W is said to be the trivial (L,[,,])-module relating to γ.

    In particular, set W=L, and ϑ(t,p)(q)=ε(|q|,|t|+|p|)[q,t,p]. Hence D(t,p)(q)=δ[t,p,q] and (2.5)–(2.8) hold. Under this situation, L is shown to be the adjoint (L,[,,])-module, and ϑ is said to be the adjoint representation of (L,[,,]).

    Proposition 2.1. Suppose that ϑ is a representation of a multiplicative Hom-δ-Jordan Lie color triple system (L,[,,]) on W relating to γ. Define the calculation [,,]W:(LW)×(LW)×(LW)LW by

    [(t,w),(t1,w1),(t2,w2)]W=([t,t1,t2],ε(|t|,|t1|+|t2|)ϑ(t1,t2)(w)δε(|t1|,|t2|)ϑ(t,t2)(w1)+δD(t,t1)(w2)),

    and assume the twisted map β+γ:LWLW by

    (β+γ)(t,w)=(β(t),γ(w)),

    thus LW is a multiplicative Hom-δ-Jordan Lie color triple system.

    Proof. By D(t,t1)=ε(|t|,|t1|)ϑ(t1,t)δϑ(t,t1), it follows that

    [(t,w),(t1,w1),(t2,w2)]W=([t,t1,t2],ε(|t|,|t1|+|t2|)ϑ(t1,t2)(w)δε(|t1|,|t2|)ϑ(t,t2)(w1)+δD(t,t1)(w2))=δε(|t|,|t1|)([t1,t,t2],ε(|t1|,|t|+|t2|)ϑ(t,t2)(w1)δε(|t|,|t2|)ϑ(t1,t2)(w)ε(|t|,|t1|)D(t,t1)(w2))=δε(|t|,|t1|)([t1,t,t2],ε(|t1|,|t|+|t2|)ϑ(t,t2)(w1)δε(|t|,|t2|)ϑ(t1,t2)(w)+δD(t1,t)(w2))=δε(|t|,|t1|)[(t1,w1),(t,w),(t2,w2)]W,

    and

    ε(|t|,|t2|)[(t,w),(t1,w1),(t2,w2)]W+ε(|t1|,|t|)[(t1,w1),(t2,w2),(t,w)]W+ε(|t2|,|t1|)[(t2,w2),(t,w),(t1,w1)]W=(ε(|t|,|t2|)[t,t1,t2],ε(|t|,|t1|)ϑ(t1,t2)(w)δε(|t2|,|t|+|t1|)ϑ(t,t2)(w1)+δε(|t|,|t2|)D(t,t1)(w2))+(ε(|t1|,|t|)[t1,t2,t],ε(|t1|,|t2|)ϑ(t2,t)(w1)δε(|t|,|t1|+|t2|)ϑ(t1,t)(w2)+δε(|t1|,|t|)D(t1,t2)(w))+(ε(|t2|,|t1|)[t2,t,t1],ε(|t2|,|t|)ϑ(t,t1)(w2)δε(|t1|,|t|+|t2|)ϑ(t2,t1)(w)+δε(|t2|,|t1|)D(t2,t)(w1))=(0,ε(|t|,|t1|)ϑ(t1,t2)(w)δε(|t1|,|t|+|t2|)ϑ(t2,t1)(w)+δε(|t1|,|t|)D(t1,t2)(w)+ε(|t1|,|t2|)ϑ(t2,t)(w1)δε(|t2|,|t|+|t1|)ϑ(t,t2)(w1)+δε(|t2|,|t1|)D(t2,t)(w1)+ε(|t2|,|t|)ϑ(t,t1)(w2)δε(|t|,|t1|+|t2|)ϑ(t1,t)(w2)+δε(|t|,|t2|)D(t,t1)(w2))=(0,0).

    By (2.6)–(2.8), it is clear that

    [[(t,w),(t1,w1),(t2,w2)]W,(β+γ)(t3,w3),(β+γ)(t4,w4)]W=[([t,t1,t2],ε(|t|,|t1|+|t2|)ϑ(t1,t2)(w)δε(|t1|,|t2|)ϑ(t,t2)(w1)+δD(t,t1)(w2)),(β(t3),γ(w3)),(β(t4),γ(w4))]W=([[t,t1,t2],β(t3),β(t4)],ε(|t|+|t1|+|t2|,|t3|+|t4|)ϑ(β(t3),β(t4))(ε(|t|,|t1|+|t2|)ϑ(t1,t2)(w)δε(|t1|,|t2|)ϑ(t,t2)(w1)+δD(t,t1)(w2))δε(|t3|,|t4|)ϑ([t,t1,t2],β(t4))(γ(w3))+δD([t,t1,t2],β(t3))(γ(w4))),
    ε(|t2|,|t|+|t1|)[(β+γ)(t2,w2),[(t,w),(t1,w1),(t3,w3)]W,(β+γ)(t4,w4)]W=ε(|t2|,|t|+|t1|)[(β(t2),γ(w2)),([t,t1,t3],ε(|t|,|t1|+|t3|)ϑ(t1,t3)(w)δε(|t1|,|t3|)ϑ(t,t3)(w1)+δD(t,t1)(w3)),(β(t4),γ(w4))]W=ε(|t2|,|t|+|t1|)([β(t2),[t,t1,t3],β(t4)],ε(|t2|,|t|+|t1|+|t3|+|t4|)ϑ([t,t1,t3],β(t4))(γ(w2))δε(|t|+|t1|+|t3|,|t4|)ϑ(β(t2),β(t4))(ε(|t|,|t1|+|t3|)ϑ(t1,t3)(w)δε(|t1|,|t3|)ϑ(t,t3)(w1)+δD(t,t1)(w3))+δD(β(t2),[t,t1,t3])(γ(w4))),
    δε(|t|+|t1|,|t2|+|t3|)[(β+γ)(t2,w2),(β+γ)(t3,w3),[(t,w),(t1,w1),(t4,w4)]W]W=δε(|t|+|t1|,|t2|+|t3|)[(β(t2),γ(w2)),(β(t3),γ(w3)),([t,t1,t4],ε(|t|,|t1|+|t4|)ϑ(t1,t4)(w)δε(|t1|,|t4|)ϑ(t,t4)(w1)+δD(t,t1)(w4))]W=δε(|t|+|t1|,|t2|+|t3|)([β(t2),β(t3),[t,t1,t4]],ε(|t2|,|t3|+|t|+|t1|+|t4|)ϑ(β(t3),[t,t1,t4])(γ(w2))δε(|t3|,|t|+|t1|+|t4|)ϑ(β(t2),[t,t1,t4])(γ(w3))+δD(β(t2),β(t3))(ε(|t|,|t1|+|t4|)ϑ(t1,t4)(w)δε(|t1|,|t4|)ϑ(t,t4)(w1)+D(t,t1)(w4))),
    [(β+γ)(t,w),(β+γ)(t1,w1),[(t2,w2),(t3,w3),(t4,w4)]W]W=[(β(t),γ(w)),(β(t1),γ(w1)),([t2,t3,t4],ε(|t2|,|t3|+|t4|)ϑ(t3,t4)(w2)δε(|t3|,|t4|)ϑ(t2,t4)(w3)+δD(t2,t3)(w4))]W=([β(t),β(t1),[t2,t3,t4]],ε(|t|,|t1|+|t2|+|t3|+|t4|)ϑ(β(t1),[t2,t3,t4])(γ(w))δε(|t1|,|t2|+|t3|+|t4|)ϑ(β(t),[t2,t3,t4])(γ(w1))+δD(β(t),β(t1))(ε(|t2|,|t3|+|t4|)ϑ(t3,t4)(w2)δε(|t3|,|t4|)ϑ(t2,t4)(w3)+D(t2,t3)(w4))).

    The results above show that (2.2)–(2.4) hold.

    Since β+γ is an even linear map and in view of (2.5), it follows that

    (β+γ)[(t,w),(t1,w1),(t2,w2)]W=(β+γ)([t,t1,t2],ε(|t|,|t1|+|t2|)ϑ(t1,t2)(w)δε(|t1|,|t2|)ϑ(t,t2)(w1)+δD(t,t1)(w2))=(β([t,t1,t2]),γ(ε(|t|,|t1|+|t2|)ϑ(t1,t2)(w)δε(|t1|,|t2|)ϑ(t,t2)(w1)+δD(t,t1)(w2)))=([β(t),β(t1),β(t2)],ε(|t|,|t1|+|t2|)ϑ(β(t1),β(t2))γ(w)δε(|t1|,|t2|)ϑ(β(t),β(t2))γ(w1)+δD(β(t),β(t1))γ(w2))=[(β(t),γ(w)),(β(t1),γ(w1)),(β(t2),γ(w2))]W=[(β+γ)(t,w),(β+γ)(t1,w1),(β+γ)(t2,w2)]W.

    Moreover, (LW,[,,]W) is a multiplicative Hom-δ-Jordan Lie color triple system.

    Remark 2.1. Given a representation ϑ of the multiplicative Hom-δ-Jordan Lie color triple system (L,[,,]) on W, define the calculation [,,]W:(LW)×(LW)×(LW)LW, we construct a kind of multiplicative Hom-δ-Jordan Lie color triple system and regard Proposition 2.1 as an example of multiplicative Hom-δ-Jordan Lie color triple systems.

    Suppose that ϑ is a representation of (L,[,,]) on W relating to γ. Assume that an n-linear map h:L××LntimesW satisfies

    h(t1,,t,p,tn)=δε(|t|,|p|)h(t1,,p,t,tn),
    ε(|t|,|q|)h(t1,,tn3,t,p,q)+ε(|p|,|t|)h(t1,,tn3,p,q,t)+ε(|q|,|p|)h(t1,,tn3,q,t,p)=0,

    thus h is called an n-cochain on L. Denote by Cnγ(L,W) the set of all n-cochains, n1.

    Definition 2.5. For n=1,2,3,4, the definition of the coboundary operator dn:Cnγ(L,W)Cn+2γ(L,W) is given as follows:

    If hC1(L,W), then

    d1h(w1,w2,w3)=ε(|h|+|w1|,|w2|+|w3|)ϑ(w2,w3)h(w1)δε(|w2|,|w3|)ε(|h|,|w1|+|w3|)ϑ(w1,w3)h(w2)+δε(|h|,|w1|+|w2|)D(w1,w2)h(w3)h([w1,w2,w3]).

    If hC2(L,W), then

    d2h(y,w1,w2,w3)=ε(|h|+|y|+|w1|,|w2|+|w3|)ϑ(β(w2),β(w3))h(y,w1)δε(|w2|,|w3|)ε(|h|+|y1|,|w1|+|w3|)ϑ(β(w1),β(w3))h(y,w2)+δε(|h|+|y|,|w1|+|w2|)D(β(w1),β(w2))h(y,w3)h(β(y),[w1,w2,w3]).

    If hC3(L,W), then

    d3h(w1,w2,w3,w4,w5)=ε(|h|+|w1|+|w2|+|w3|,|w4|+|w5|)ϑ(β(w4),β(w5))h(w1,w2,w3)δε(|h|+|w1|+|w2|,|w3|+|w5|)ε(|w4|,|w5|)ϑ(β(w3),β(w5))h(w1,w2,w4)δε(|h|,|w1|+|w2|)D(β(w1),β(w2))h(w3,w4,w5)+ε(|h|+|w1|+|w2|,|w3|+|w4|)D(β(w3),β(w4))h(w1,w2,w5)+h([w1,w2,w3],β(w4),β(w5))+ε(|w3|,|w1|+|w2|)h(β(w3),[w1,w2,w4],β(w5))+δε(|w1|+|w2|,|w3|+|w4|)h(β(w3),β(w4),[w1,w2,w5])h(β(w1),β(w2),[w3,w4,w5]).

    If hC4(L,W), then

    d4h(y,w1,w2,w3,w4,w5)=ε(|h|+|y|+|w1|+|w2|+|w3|,|w4|+|w5|)ϑ(β2(w4),β2(w5))h(y,w1,w2,w3)δε(|h|+|y|+|w1|+|w2|,|w3|+|w5|)ε(|w4|,|w5|)ϑ(β2(w3),β2(w5))h(y,w1,w2,w4)δε(|h|+|y|,|w1|+|w2|)D(β2(w1),β2(w2))h(y,w3,w4,w5)+ε(|h|+|y|+|w1|+|w2|,|w3|+|w4|)D(β2(w3),β2(w4))h(y,w1,w2,w5)+h(β(y),[w1,w2,w3],β(w4),β(w5))+ε(|w3|,|w1|+|w2|)h(β(y),β(w3),[w1,w2,w4],β(w5))+δε(|w1|+|w2|,|w3|+|w4|)h(β(y),β(w3),β(w4),[w1,w2,w5])h(β(y),β(w1),β(w2),[w3,w4,w5]).

    Theorem 2.1. About the coboundary operator dn defined above, we have dn+2dn=0, n=1,2.

    Proof. Using Definition 2.5, it is obvious immediately that d3d1=0 implies d4d2=0. Then we only need to prove d3d1=0. In fact, by (2.5)–(2.8), we obtain

    d3(d1h)(w1,w2,w3,w4,w5)=ε(|h|+|w1|+|w2|+|w3|,|w4|+|w5|)ϑ(β(w4),β(w5))(d1h)(w1,w2,w3)δε(|h|+|w1|+|w2|,|w3|+|w5|)ε(|w4|,|w5|)ϑ(β(w3),β(w5))(d1h)(w1,w2,w4)δε(|h|,|w1|+|w2|)D(β(w1),β(w2))(d1h)(w3,w4,w5)+ε(|h|+|w1|+|w2|,|w3|+|w4|)D(β(w3),β(w4))(d1h)(w1,w2,w5)+(d1h)([w1,w2,w3],β(w4),β(w5))+ε(|w3|,|w1|+|w2|)(d1h)(β(w3),[w1,w2,w4],β(w5))+δε(|w1|+|w2|,|w3|+|w4|)(d1h)(β(w3),β(w4),[w1,w2,w5])(d1h)(β(w1),β(w2),[w3,w4,w5])=ε(|h|+|w1|+|w2|+|w3|,|w4|+|w5|)ϑ(β(w4),β(w5))(ε(|h|+|w1|,|w2|+|w3|)ϑ(w2,w3)h(w1)δε(|w2|,|w3|)ε(|h|,|w1|+|w3|)ϑ(w1,w3)h(w2)+δε(|h|,|w1|+|w2|)D(w1,w2)h(w3)h([w1,w2,w3]))δε(|h|+|w1|+|w2|,|w3|+|w5|)ε(|w4|,|w5|)ϑ(β(w3),β(w5))(ε(|h|+|w1|,|w2|+|w4|)ϑ(w2,w4)h(w1)δε(|w2|,|w4|)ε(|h|,|w1|+|w4|)ϑ(w1,w4)h(w2)+δε(|h|,|w1|+|w2|)D(w1,w2)h(w4)h([w1,w2,w4]))δε(|h|,|w1|+|w2|)D(β(w1),β(w2))(ε(|h|+|w3|,|w4|+|w5|)ϑ(w4,w5)h(w3)δε(|w4|,|w5|)ε(|h|,|w3|+|w5|)ϑ(w3,w5)h(w4)+δε(|h|,|w3|+|w4|)D(w3,w4)h(w5)h([w3,w4,w5]))+ε(|h|+|w1|+|w2|,|w3|+|w4|)D(β(w3),β(w4))(ε(|h|+|w1|,|w2|+|w5|)ϑ(w2,w5)h(w1)δε(|w2|,|w5|)ε(|h|,|w1|+|w5|)ϑ(w1,w5)h(w2)+δε(|h|,|w1|+|w2|)D(w1,w2)h(w5)h([w1,w2,w5]))+ε(|h|+|w1|+|w2|+|w3|,|w4|+|w5|)(ϑ(β(w4),β(w5))h([w1,w2,w3])δε(|w4|,|w5|)ε(|h|,|w1|+|w2|+|w3|+|w5|)ϑ([w1,w2,w3],β(w5))h(β(w4))+δε(|h|,|w1|+|w2|+|w3|+|w4|)D([w1,w2,w3],β(w4))h(β(w5))h([[w1,w2,w3],β(w4),β(w5)]))+ε(|w3|,|w1|+|w2|)ε(|h|+|w3|,|w1|+|w2|+|w4|+|w5|)(ϑ([w1,w2,w4],β(w5))h(β(w3))δε(|h|+|w1|+|w2|,|w3|+|w5|)ε(w4|,|w5|)ϑ(β(w3),β(w5))h([w1,w2,w4])+δε(|w3|,|w1|+|w2|)ε(|h|,|w1|+|w2|+|w3|+|w4|)D(β(w3),[w1,w2,w4])h(β(w5))ε(|w3|,|w1|+|w2|)h([β(w3),[w1,w2,w4],β(w5)]))+δε(|w1|+|w2|,|w3|+|w4|)(ε(|h|+|w3|,|w1|+|w2|+|w4|+|w5|)ϑ(β(w4),[w1,w2,w5])h(β(w3))δε(|w4|,|w1|+|w2|+|w5|)ε(|h|,|w1|+|w2|+|w3|+|w5|)ϑ(β(w3),[w1,w2,w5])h(β(w4))+δε(|h|,|w3|+|w4|)D(β(w3),β(w4))h([w1,w2,w5])h([β(w3),β(w4),[w1,w2,w5]]))(ε(|h|+|w1|,|w2|+|w3|+|w4|+|w5|)ϑ(β(w2),[w3,w4,w5])h(β(w1))δε(|h|,|w1|+|w3|+|w4|+|w5|)ε(|w2|,|w3|+|w4|+|w5|)ϑ(β(w1),[w3,w4,w5])h(β(w2))+δε(|h|,|w1|+|w2|)D(β(w1),β(w2))h([w3,w4,w5])h([β(w1),β(w2),[w3,w4,w5]]))=h([[w1,w2,w3],β(w4),β(w5)])ε(|w3|,|w1|+|w2|)h([β(w3),[w1,w2,w4],β(w5)])δε(|w1|+|w2|,|w3|+|w4|)h([β(w3),β(w4),[w1,w2,w5]])+h([β(w1),β(w2),[w3,w4,w5]])+ε(|h|+|w1|+|w2|+|w3|,|w4|+|w5|)ε(|h|+|w1|,|w2|+|w3|)ϑ(β(w4),β(w5))ϑ(w2,w3)h(w1)δε(|h|+|w1|+|w2|,|w3|+|w5|)ε(|w4|,|w5|)ε(|h|+|w1|,|w2|+|w4|)ϑ(β(w3),β(w5))ϑ(w2,w4)h(w1)+ε(|h|+|w1|+|w2|,|w3|+|w4|)ε(|h|+|w1|,|w2|+|w5|)D(β(w3),β(w4))ϑ(w2,w5)h(w1)ε(|h|+|w1|,|w2|+|w3|+|w4|+|w5|)ϑ(β(w2),[w3,w4,w5])h(β(w1))δε(|h|+|w1|+|w2|+|w3|,|w4|+|w5|)ε(|w2|,|w3|)ε(|h|,|w1|+|w3|)ϑ(β(w4),β(w5))ϑ(w1,w3)h(w2)+ε(|h|+|w1|+|w2|,|w3|+|w5|)ε(|w4|,|w2|+|w5|)ε(|h|,|w1|+|w4|)ϑ(β(w3),β(w5))ϑ(w1,w4)h(w2)δε(|h|+|w1|+|w2|,|w3|+|w4|)ε(|w2|,|w5|)ε(|h|,|w1|+|w5|)D(β(w3),β(w4))ϑ(w1,w5)h(w2)+δε(|h|,|w1|+|w3|+|w4|+|w5|)ε(|w2|,|w3|+|w4|+|w5|)ϑ(β(w1),[w3,w4,w5])h(β(w2))+δε(|h|+|w1|+|w2|+|w3|,|w4|+|w5|)ε(|h|,|w1|+|w2|)ϑ(β(w4),β(w5))D(w1,w2)h(w3)δε(|h|,|w1|+|w2|)ε(|h|+|w3|,|w4|+|w5|)D(β(w1),β(w2))ϑ(w4,w5)h(w3)+ε(|w3|,|w1|+|w2|)ε(|h|+|w3|,|w1|+|w2|+|w4|+|w5|)ϑ([w1,w2,w4],β(w5))h(β(w3))+δε(|w1|+|w2|,|w3|+|w4|)ε(|h|+|w3|,|w1|+|w2|+|w4|+|w5|)ϑ(β(w4),[w1,w2,w5])h(β(w3))ε(|h|+|w1|+|w2|,|w3|+|w5|)ε(|w4|,|w5|)ε(|h|,|w1|+|w2|)ϑ(β(w3),β(w5))D(w1,w2)h(w4)+ε(|h|,|w1|+|w2|)ε(|w4|,|w5|)ε(|h|,|w3|+|w5|)D(β(w1),β(w2))ϑ(w3,w5)h(w4)δε(|w4|,|w5|)ε(|h|,|w1|+|w2|+|w3|+|w5|)ϑ([w1,w2,w3],β(w5))h(β(w4))ε(|w1|+|w2|,|w3|+|w4|)ε(|w4|,|w1|+|w2|+|w5|)ε(|h|,|w1|+|w2|+|w3|+|w5|)ϑ(β(w3),[w1,w2,w5])h(β(w4))ε(|h|,|w1|+|w2|+|w3|+|w4|)D(β(w1),β(w2))D(w3,w4)h(w5)+δε(|h|+|w1|+|w2|,|w3|+|w4|)ε(|h|,|w1|+|w2|)D(β(w3),β(w4))D(w1,w2)h(w5)+δε(|h|,|w1|+|w2|+|w3|+|w4|)D([w1,w2,w3],β(w4))h(β(w5))+δε(|w3|,|w1|+|w2|)ε(|h|,|w1|+|w2|+|w3|+|w4|)D(β(w3),[w1,w2,w4])h(β(w5))ε(|h|+|w1|+|w2|+|w3|,|w4|+|w5|)ϑ(β(w4),β(w5))h([w1,w2,w3])+δε(|h|+|w1|+|w2|,|w3|+|w5|)ε(|w4|,|w5|)ϑ(β(w3),β(w5))h([w1,w2,w4])+δε(|h|,|w1|+|w2|)D(β(w1),β(w2))h([w3,w4,w5])ε(|h|,|w3|+|w4|)ε(|w1|+|w2|,|w3|+|w4|)D(β(w3),β(w4))h([w1,w2,w5])+ε(|h|+|w1|+|w2|+|w3|,|w4|+|w5|)ϑ(β(w4),β(w5))h([w1,w2,w3])δε(|h|+|w1|+|w2|,|w3|+|w5|)ε(|w4|,|w5|)ϑ(β(w3),β(w5))h([w1,w2,w4])+ε(|h|,|w3|+|w4|)ε(|w1|+|w2|,|w3|+|w4|)D(β(w3),β(w4))h([w1,w2,w5])δε(|h|,|w1|+|w2|)D(β(w1),β(w2))h([w3,w4,w5])=0.

    Therefore, the proof is complete.

    For n=1,2,3,4. If dnh=0, then hCnγ(L,W) is called an n-cocycle. The subspace Znγ(L,W) spanned by n-cocycle, and set Bnγ(L,W)=dn2Cn2γ(L,W).

    Since dn+2dn=0, Bnγ(L,W) is a subspace of Znγ(L,W). Therefore, we can determine a cohomology space Hnγ(L,W) of (L,[,,]) as the factor space Znγ(L,W)/Bnγ(L,W).

    Suppose that (L,[,,]) is a multiplicative Hom-δ-Jordan Lie color triple system, and F[[s]] is the ring of formal power series over F. Suppose that L[[s]] is the set of formal power series over L. An F-trilinear map h:L×L×LL is extended to be an F[[s]]-trilinear map h:L[[s]]×L[[s]]×L[[s]]L[[s]] by

    h(i0tisi,j0pjsj,k0qksk)=i,j,k0h(ti,pj,qk)si+j+k.

    Definition 3.1. Suppose that (L,[,,]) is a multiplicative Hom-δ-Jordan Lie color triple system over F. A 1-parameter formal deformation on (L,[,,]) is a formal power series lL:L[[s]]×L[[s]]×L[[s]]L[[s]] of the form

    lL(t,p,q)=i0li(t,p,q)si=l0(t,p,q)+l1(t,p,q)t+l2(t,p,q)s2+,

    any li is an F-trilinear map li:L×L×LL, which is extended to be an F[[s]]-trilinear map, and l0(t,p,q)=[t,p,q], satisfying the following relations hold

    lL(β(t),β(p),β(q))=βlL(t,p,q), (3.1)
    lL(t,p,q)=δε(|t|,|p|)lL(p,t,q), (3.2)
    ε(|t|,|q|)lL(t,p,q)+ε(|p|,|t|)lL(p,q,t)+ε(|q|,|p|)lL(q,t,p)=0, (3.3)
    lL(β(z),β(u),lL(t,p,q))=lL(lL(z,u,t),β(p),β(q))+ε(|t|,|z|+|u|)lL(β(t),lL(z,u,p),β(q))+δε(|z|+|u|,|t|+|p|)lL(β(t),β(p),lL(z,u,q)). (3.4)

    Equations (3.2)–(3.4) are said to be the deformation equations of a multiplicative Hom-δ-Jordan Lie color triple system.

    Since L[[s]] is a module over F[[s]] and lL shows the trilinear on L[[s]], we get Lt=(L[[s]],lL) is a multiplicative Hom-δ-Jordan Lie color triple system. In the following, we study the deformation Eqs (3.2)–(3.4).

    Equations (3.2) and (3.3) are equivalent to the following relations:

    li(β(t),β(p),β(q))=βli(t,p,q), (3.5)
    li(t,p,q)=δε(|t|,|p|)li(p,t,q), (3.6)
    ε(|t|,|q|)li(t,p,q)+ε(|p|,|t|)li(p,q,t)+ε(|q|,|p|)li(q,t,p)=0, (3.7)

    respectively, for i=0,1,2,. The Eq (3.4) can be written as

    i,j0li(β(z),β(u),lj(t,p,q))=i,j0li(lj(z,u,t),β(p),β(q))+i,j0ε(|t|,|z|+|u|)li(β(t),lj(z,u,p),β(q))+δi,j0ε(|z|+|u|,|t|+|p|)li(β(t),β(p),lj(z,u,q)).

    Then

    i+j=n(li(lj(z,u,t),β(p),β(q))+ε(|t|,|z|+|u|)li(β(t),lj(z,u,p),β(q))+δε(|z|+|u|,|t|+|p|)li(β(t),β(p),lj(z,u,q))li(β(z),β(u),lj(t,p,q)))=0,n=0,1,2.

    Two F-trilinear maps f,g:L×L×LL which are extended to be F[[s]]-trilinear, we show a map fg:L[[s]]×L[[s]]×L[[s]]×L[[s]]×L[[s]]L[[s]] using

    fg(z,u,t,p,q)=f(g(z,u,t),β(p),β(q))+ε(|t|,|z|+|u|)f(β(t),g(z,u,p),β(q))+δε(|z|+|u|,|t|+|p|)f(β(t),β(p),g(z,u,q))f(β(z),β(u),g(t,p,q)).

    Hence, the deformation Eq (3.4) can be obtained as

    i+j=nlilj=0.

    If n=1, then l0l1+l1l0=0.

    If n2, then (l0ln+lnl0)=l1ln1+l2ln2++ln1l1.

    Section 2 gives that L is the adjoint (L,[,,],β)-module by assuming ϑ(t,p)(q)=ε(|q|,|t|+|p|)[q,t,p]. Under this situation, using (3.6) and (3.7), we get liC3(L,L). Considering the definition 2.5, we show d3ln=l0ln+lnβl0, for n=0,1,2. Therefore, the deformation Eq (3.4) can be shown as

    d3l1=0,d3ln=l1ln1+l2ln2++ln1l1.

    Thus, l1 is a 3-Hom-cocycle and is said to be the infinitesimal of lL.

    Definition 3.2. Suppose that (L,[,,]) is a multiplicative Hom-δ-Jordan Lie color triple system. Assume that lL(t,p,q)=i0li(t,p,q)si and lL(t,p,q)=i0li(t,p,q)si are two 1-parameter formal deformations on (L,[,,]). lL and lL are known as equivalent, denoted by lLlL, if there is a formal isomorphism of F[[s]]-modules

    φt(t)=i0φi(t)si:(L[[s]],mL)(L[[s]],mL),

    any φi:LL is an F-linear map that is extended to be an F[[s]]-linear map, and φ0 is the identity mapping of L. We obtain

    φtβ=βφt,φtlL(t,p,q)=lL(φt(t),φt(p),φt(q)).

    In the case l1=l2==0, lL=l0 is known as the null deformation. If lLl0, then the 1-parameter formal deformation lL is said to be trivial. If any 1-parameter formal deformation lL is trivial, then the multiplicative Hom-δ-Jordan Lie color triple system (L,[,,]) is known as analytically rigid.

    Theorem 3.1. Suppose that lL(t,p,q)=i0li(t,p,q)si and lL(t,p,q)=i0li(t,p,q)si are equivalent 1-parameter formal deformations on (L,[,,]). Then l1 and l1 belong to the same cohomology class in H3(L,L).

    Proof. Suppose that φt(t)=i0φi(t)si is the formal F[[s]]-module isomorphism, satisfying φtβ=βφt, and

    i0φi(j0lj(t,p,q)sj)si=i0li(k0φk(t)sk,l0φl(p)sl,l0φm(q)sm)si.

    Then

    i+j=nφi(lj(t,p,q))si+j=i+k+l+m=nli(φk(t),φl(p),φm(q))si+k+l+m.

    In especial,

    i+j=1φi(lj(t,p,q))=i+k+l+m=1li(φk(t),φl(p),φm(q)),

    thus,

    l1(t,p,q)+φ1([t,p,q])=[φ1(t),p,q]+[t,φ1(p),q]+[t,p,φ1(q)]+l1(t,p,q)=ε(|t|,|p|+|q|)ϑ(p,q)φ1(t)δε(|p|,|q|)ϑ(t,q)φ1(p)+δD(t,p)φ1(q)+l1(t,p,q).

    We give that l1l1=d1φ1B3(L,L).

    Theorem 3.2. Suppose that (L,[,,]) is a multiplicative Hom-δ-Jordan Lie color triple system satisfying H3(L,L)=0. We obtain that (L,[,,]) is analytically rigid.

    Proof. Suppose that lL is a 1-parameter formal deformation on (L,[,,]). Assume that lL=l0+inlisi. Hence,

    d3ln=l1ln1+l2ln2++ln1l1=0,

    moreover, lnZ3(L,L)=B3(L,L). It is proved that there is hnC1(L,L) satisfying ln=d1hn.

    Assume that φt is even, and φt=idLhnsn:(L[[s]],lL)(L[[s]],lL). Notice that

    φti0hinsin=i0hinsinφt=idL[[s]].

    It is easy to see that φt is a linear isomorphism, and φtβ=βφt.

    We shall consider lL(t,p,q)=φ1slL(φt(t),φt(p),φt(q)). The following fact is clear to prove that lL is a 1-parameter formal deformation on (L,[,,]). Indeed,

    lL(t,p,q)=φ1slL(φt(t),φt(p),φt(q))=δε(|t|,|p|)φ1slL(φt(p),φt(t),φt(q))=δε(|t|,|p|)lL(p,t,q).
    ε(|t|,|q|)lL(t,p,q)+ε(|p|,|t|)lL(p,q,t)+ε(|q|,|p|)lL(q,t,p)=ε(|t|,|q|)φ1slL(φt(t),φt(p),φt(q))+ε(|p|,|t|)φ1slL(φt(p),φt(q),φt(t))+ε(|q|,|p|)φ1slL(φt(q),φt(t),φt(p))=φ1s(ε(|t|,|q|)lL(φt(t),φt(p),φt(q))+ε(|p|,|t|)lL(φt(p),φt(q),φt(t))+ε(|q|,|p|)lL(φt(q),φt(t),φt(p)))=0.
    lL(β(t1),β(t2),lL(t,p,q))=lL(β(t1),β(t2),φ1slL(φt(t),φt(p),φt(q)))=φ1slL(φt(β(t1)),φt(β(t2)),lL(φt(t),φt(p),φt(q)))=φ1slL(β(φt(t1)),β(φt(t2)),lL(φt(t),φt(p),φt(q))).
    lL(lL(t1,t2,t),β(p),β(q))=lL(φ1slL(φt(t1),φt(t2),φt(t)),β(p),β(q))=φ1slL(lL(φt(t1),φt(t2),φt(t)),β(φt(p)),β(φt(q))).
    ε(|t|,|t1|+|t2|)lL(β(t),lL(t1,t2,p),β(q))=ε(|t|,|t1|+|t2|)lL(β(t),φ1slL(φt(t1),φt(t2),φt(p)),β(q))=ε(|t|,|t1|+|t2|)φ1slL(φt(β(t)),lL(φt(t1),φt(t2),φt(p)),φt(β(q)))=ε(|t|,|t1|+|t2|)φ1slL(β(φt(t)),lL(φt(t1),φt(t2),φt(p)),β(φt(q))).
    δε(|t1|+|t2|,|t|+|p|)lL(β(t),β(p),lL(t1,t2,q))=δε(|t1|+|t2|,|t|+|p|)lL(β(t),β(p),φ1slL(φt(t1),φt(t2),φt(q)))=δε(|t1|+|t2|,|t|+|p|)φ1slL(φt(β(t)),φt(β(p)),lL(φt(t1),φt(t2),φt(q)))=δε(|t1|+|t2|,|t|+|p|)φ1slL(β(φt(t)),β(φt(p)),lL(φt(t1),φt(t2),φt(q))).

    The calculations above give that relations (3.2)–(3.4) hold. In view of Definition 3.2, we know lLlL. Assume that lL=i0lisi. Hence

    (idLhnsn)(i0li(t,p,q)si)=(l0+inlisi)(thn(t)sn,phn(p)sn,qhn(q)sn),

    i.e.,

    i0li(t,p,q)sii0hnli(t,p,q)si+n=[t,p,q]([hn(t),p,q]+[t,hn(p),q]+[t,p,hn(q)])sn+([hn(t),hn(p),q]+[t,hn(p),hn(q)]+[hn(t),p,hn(q)])s2n[hn(t),hn(p),hn(q)]s3n+inli(t,p,q)siin(li(hn(t),p,q)+li(t,hn(p),q)+li(t,p,hn(q)))si+n+in(li(hn(t),hn(p),q)+li(t,hn(p),hn(q))+li(hn(t),p,hn(q)))si+2ninli(hn(t),hn(p),hn(q))si+3n.

    Then, it follows l1==ln1=0 and

    ln(t,p,q)hn([t,p,q])=([hn(t),p,q]+[t,hn(p),q]+[t,p,hn(q)])+ln(t,p,q)=ε(|t|,|p|+|q|)ϑ(p,q)hn(t)+δε(|p|,|q|)ϑ(t,q)hn(p)δD(t,p)hn(q)+ln(t,p,q).

    Therefore, ln=lnd1hn=0 and lL=l0+in+1lisi. By induction, we know lLl0, i.e., (L,[,,]) is analytically rigid.

    A graded subspace I satisfying [I,L,L]I, which is said to be a graded ideal on the multiplicative Hom-δ-Jordan Lie color triple system L. If [L,I,I]=0, then the ideal I on a multiplicative Hom-δ-Jordan Lie color triple system is said to be an abelian ideal. Moreover, note that [L,I,I]=0 indicates [I,L,I]=0 and [I,I,L]=0.

    Definition 4.1. Suppose that (L,[,,]L), (W,[,,]W), and (ˆL,[,,]ˆL) are multiplicative Hom-δ-Jordan Lie color triple systems, and f:WˆL, g:ˆLL are homomorphisms. The following sequence on multiplicative Hom-δ-Jordan Lie color triple systems is a short exact sequence if Im(f)=Ker(g), Ker(f)=0, and Im(g)=L,

    0WfˆLgL0. (4.1)

    Under this circumstance, we show ˆL an extension of L by W, which is denoted by EˆL. It is termed an abelian extension if W is an abelian ideal of ˆL, that is, [w1,w2,]ˆL=[w1,,w2]ˆL=[,w1,w2]ˆL=0, for all w1,w2W.

    A section τ:LˆL of g:ˆLL is composed of the linear map τ:LˆL satisfying gτ=idL, and ˆβτ=τβ.

    Definition 4.2. Two extensions on multiplicative Hom-δ-Jordan Lie color triple systems EˆL:0WfˆLgL0 and E˜L:0Wf˜LgL0 are equivalent. If there is a multiplicative Hom-δ-Jordan Lie color triple system homomorphism F:ˆL˜L satisfying the following diagram commutes

    Suppose that ˆL is an abelian extension on L by W, and an even linear mapping τ:LˆL is a section. Assume maps LLEnd(W) by

    D(u1,u2)(v)=δ[τ(u1),τ(u2),v]ˆL, (4.2)
    ϑ(u1,u2)(v)=ε(|v|,|u1|+|u2|)[v,τ(u1),τ(u2)]ˆL. (4.3)

    It is obvious that the following result holds, that is,

    D(u1,u2)(v)=ε(|u1|,|u2|)ϑ(u2,u1)(v)δϑ(u1,u2)(v),

    for all (u1,u2)LL,vW.

    Theorem 4.1. Using the above symbols, it follows that (W,ϑ) is a representation of L, and ϑ is independent for the section τ. Furthermore, there is the same representation between equivalent abelian extensions.

    Proof. Suppose that another section τ:LˆL, we obtain

    p(τ(ui)τ(ui))=uiui=0τ(ui)τ(ui)Wτ(ui)=τ(ui)+vi,

    for some viW.

    Notice that [w1,w2,]ˆL=0=[w1,,w2]ˆL for all w1,w2W, this shows that

    [v,τ(u1),τ(u2)]ˆL=[v,τ(u1),τ(u2)]ˆL.

    Then, ϑ does not rely on the choice of the section τ.

    Now, we will obtain that (W,ϑ) is a representation of L.

    By the equation

    [ˆβ(v),ˆβ(τ(t1)),[τ(u1),τ(u2),τ(u3)]ˆL]ˆL=[[v,τ(t1),τ(u1)]ˆL,ˆβ(τ(u2)),ˆβ(τ(u3))]ˆL+ε(|u1|,|v|+|t1|)[ˆβ(τ(u1)),[v,τ(t1),τ(u2)]ˆL,ˆβ(τ(u3))]ˆL+δε(|v|+|t1|,|u1|+|u2|)[ˆβ(τ(u1)),ˆβ(τ(u2)),[v,τ(t1),τ(u3)]ˆL]ˆL,

    it follows that

    ε(|t1|+|u1|,|u2|+|u3|)ϑ(β(u2),β(u3))ϑ(t1,u1)(v)δε(|t1|,|u1|)ε(|u3|,|u2|+|t1|)ϑ(β(u1),β(u3))ϑ(t1,u2)(v)ϑ(β(t1),[u1,u2,u3])(ˆβ(v))+ε(|t1|,|u1|+|u2|)D(β(u1),β(u2))ϑ(t1,u3)(v)=0.

    Then we prove (2.6) holds.

    Similarly, by the equation

    [ˆβ(τ(t1)),ˆβ(τ(t2)),[v,τ(u1),τ(u2)]ˆL]ˆL=[[τ(t1),τ(t2),v]ˆL,ˆβ(τ(u1)),ˆβ(τ(u2))]ˆL+ε(|v|,|t1|+|t2|)[ˆβ(v),[τ(t1),τ(t2),τ(u1)]ˆL,ˆβ(τ(u2))]ˆL+δε(|t1|+|t2|,|v|+|u1|)[ˆβ(v),ˆβ(τ(u1)),[τ(t1),τ(t2),τ(u2)]ˆL]ˆL,

    we have

    δε(|t1|+|t2|,|u1|+|u2|)ϑ(β(u1),β(u2))D(t1,t2)(v)δD(β(t1),β(t2))ϑ(u1,u2)(v)+ϑ([t1,t2,u1],β(u2))(ˆβ(v))+δε(|u1|,|t1|+|t2|)ϑ(β(u1),[t1,t2,u2])(ˆβ(v))=0,

    by the equation

    [ˆβ(τ(t1)),ˆβ(τ(t2)),[τ(u1),τ(u2),v]ˆL]ˆL=[[τ(t1),τ(t2),τ(u1)]ˆL,ˆβ(τ(u2)),ˆβ(v)]ˆL+ε(|u1|,|t1|+|t2|)[ˆβ(τ(u1)),[τ(t1),τ(t2),τ(u2)]ˆL,ˆβ(v)]ˆL+δε(|t1|+|t2|,|u1|+|u2|)[ˆβ(τ(u1)),ˆβ(τ(u2)),[τ(t1),τ(t2),v]ˆL]ˆL,

    we obtain

    δε(|t1|+|t2|,|u1|+|u2|)D(β(u1),β(u2))D(t1,t2)(v)D(β(t1),β(t2))D(u1,u2)(v)+δD([t1,t2,u1],β(u2))(ˆβ(v))+δε(|u1|,|t1|+|t2|)D(β(u1),[t1,t2,u2])(ˆβ(v))=0.

    Moreover, we obtain that (2.7) and (2.8) hold. Consequently, we have that (W,ϑ) is a representation of L.

    Then, the following result will give that equivalent abelian extensions show the same ϑ.

    Suppose that EˆL and E˜L are equivalent abelian extensions, and F:ˆL˜L is the multiplicative Hom-δ-Jordan Lie color triple system homomorphism such that Ff=f, gF=g. Considering linear sections τ and τ of g and g, we know gFτ(ui)=gτ(ui)=ui=gτ(ui), thus Fτ(ui)τ(ui)Ker(g)W. Hence,

    [v,τ(u1),τ(u2)]ˆL=[v,Fτ(u1),Fτ(u2)]ˆL=[v,τ(u1),τ(u2)]ˆL,

    completing the proof.

    Assume that τ:LˆL is a section of the abelian extension. Suppose that the map is as follows:

    ω(t1,t2,t3)=[τ(t1),τ(t2),τ(t3)]ˆLτ([t1,t2,t3]L), (4.4)

    for all t1,t2,t3L.

    Theorem 4.2. Suppose that 0WˆLL0 is an abelian extension on L by W, and the representation ϑ is determined using (4.3). Thus ω given using (4.4) is a 3-Hom-cocycle of L with coefficients in W.

    Proof. Considering the equation

    [ˆβ(τ(u1)),ˆβ(τ(u2)),[τ(t1),τ(t2),τ(t3)]ˆL]ˆL=[[τ(u1),τ(u2),τ(t1)]ˆL,ˆβ(τ(t2)),ˆβ(τ(t3))]ˆL+ε(|t1|,|u1|+|u2|)[ˆβ(τ(t1)),[τ(u1),τ(u2),τ(t2)]ˆL,ˆβ(τ(t3))]ˆL+δε(|u1|+|u2|,|t1|+|t2|)[ˆβ(τ(t1)),ˆβ(τ(t2)),[τ(u1),τ(u2),τ(t3)]ˆL]ˆL.

    The left side is equivalent to

    [ˆβ(τ(u1)),ˆβ(τ(u2)),[τ(t1),τ(t2),τ(t3)]ˆL]ˆL=[ˆβ(τ(u1)),ˆβ(τ(u2)),ω(t1,t2,t3)+τ([t1,t2,t3]L)]ˆL=δD(β(u1),β(u2))ω(t1,t2,t3)+[ˆβ(τ(u1)),ˆβ(τ(u2)),τ([t1,t2,t3]L)]ˆL=δD(β(u1),β(u2))ω(t1,t2,t3)+ω(β(u1),β(u2),[t1,t2,t3]L)+τ([β(u1),β(u2),[t1,t2,t3]L]L).

    On the other hand, the right side shows that

    [[τ(u1),τ(u2),τ(t1)]ˆL,ˆβ(τ(t2)),ˆβ(τ(t3))]ˆL+ε(|t1|,|u1|+|u2|)[ˆβ(τ(t1)),[τ(u1),τ(u2),τ(t2)]ˆL,ˆβ(τ(t3))]ˆL+δε(|u1|+|u2|,|t1|+|t2|)[ˆβ(τ(t1)),ˆβ(τ(t2)),[τ(u1),τ(u2),τ(t3)]ˆL]ˆL=[ω(u1,u2,t1)+τ([u1,u2,t1]L),ˆβ(τ(t2)),ˆβ(τ(t3))]ˆL+ε(|t1|,|u1|+|u2|)[ˆβ(τ(t1)),ω(u1,u2,t2)+τ([u1,u2,t2]L),ˆβ(τ(t3))]ˆL+δε(|u1|+|u2|,|t1|+|t2|)[ˆβ(τ(t1)),ˆβ(τ(t2)),ω(u1,u2,t3)+τ([u1,u2,t3]L)]ˆL=[ω(u1,u2,t1),ˆβ(τ(t2)),ˆβ(τ(t3))]ˆL+[τ([u1,u2,t1]L),ˆβ(τ(t2)),ˆβ(τ(t3))]ˆL+ε(|t1|,|u1|+|u2|)[ˆβ(τ(t1)),ω(u1,u2,t2),ˆβ(τ(t3))]ˆL+ε(|t1|,|u1|+|u2|)[ˆβ(τ(t1)),τ([u1,u2,t2]L),ˆβ(τ(t3))]ˆL+δε(|u1|+|u2|,|t1|+|t2|)[ˆβ(τ(t1)),ˆβ(τ(t2)),ω(u1,u2,t3)]ˆL+δε(|u1|+|u2|,|t1|+|t2|)[ˆβ(τ(t1)),ˆβ(τ(t2)),τ([u1,u2,t3]L)]ˆL=[ω(u1,u2,t1),τ(β(t2)),τ(β(t3))]ˆL+[τ([u1,u2,t1]L),τ(β(t2)),τ(β(t3))]ˆL+ε(|t1|,|u1|+|u2|)[τ(β(t1)),ω(u1,u2,t2),τ(β(t3))]ˆL+ε(|t1|,|u1|+|u2|)[τ(β(t1)),τ([u1,u2,t2]L),τ(β(t3))]ˆL+δε(|u1|+|u2|,|t1|+|t2|)[τ(β(t1)),τ(β(t2)),ω(u1,u2,t3)]ˆL+δε(|u1|+|u2|,|t1|+|t2|)[τ(β(t1)),τ(β(t2)),τ([u1,u2,t3]L)]ˆL=ε(|u1|+|u2|+|t1|,|t2|+|t3|)ϑ(β(t2),β(t3))ω(u1,u2,t1)+τ([[u1,u2,t1]L,β(t2),β(t3)]L)+ω([[u1,u2,t1]L,β(t2),β(t3)]L)δε(|t1|,|u1|+|u2|)ε(|t3|,|u1|+|u2|+|t2|)ϑ(β(t1),β(t3))ω(u1,u2,t2)+ε(|t1|,|u1|+|u2|)ω(β(t1),[u1,u2,t2]L,β(t3))+ε(|t1|,|u1|+|u2|)τ([β(t1),[u1,u2,t2]L,β(t3)]L)+ε(|u1|+|u2|,|t1|+|t2|)D(β(t1),β(t2))ω(u1,u2,t3)+δε(|u1|+|u2|,|t1|+|t2|)ω(β(t1),β(t2),[u1,u2,t3]L)+δε(|u1|+|u2|,|t1|+|t2|)τ([β(t1),β(t2),[u1,u2,t3]L]L).

    Thus, it follows that

    ω([u1,u2,t1]L,β(t2),β(t3))+ε(|t1|,|u1|+|u2|)ω(β(t1),[u1,u2,t2]L,β(t3))+δε(|u1|+|u2|,|t1|+|t2|)ω(β(t1),β(t2),[u1,u2,t3]L)+ε(|u1|+|u2|+|t1|,|t2|+|t3|)ϑ(β(t2),β(t3))ω(u1,u2,t1)δε(|u1|+|u2|,|t1|+|t3|)ε(t2,t3)ϑ(β(t1),β(t3))ω(u1,u2,t2)+ε(|u1|+|u2|,|t1|+|t2|)D(β(t1),β(t2))ω(u1,u2,t3)ω(β(u1),β(u2),[t1,t2,t3]L)δD(β(u1),β(u2))ω(t1,t2,t3)=0.

    Consequently, this proves the result.

    Theorem 4.3. Suppose that L is a multiplicative Hom-δ-Jordan Lie color triple system, (W,ϑ) is a L-module, and ω is a 3-Hom-cocycle. We have LW is a multiplicative Hom-δ-Jordan Lie color triple system together with the following operation:

    [u1+x1,u2+x2,u3+x3]ω=[u1,u2,u3]+ω(u1,u2,u3)+δD(u1,u2)(x3)δε(|u2|,|u3|)ϑ(u1,u3)(x2)+ε(|u1|,|u2|+|u3|)ϑ(u2,u3)(x1).

    And

    (β+γ)(u+x)=β(u)+γ(x).

    Proof. By a direct computation, the inclusion is straightforward.

    Theorem 4.4. Two abelian extensions on multiplicative Hom-δ-Jordan Lie color triple systems EˆL:0WfˆLgL0 and E˜L:0Wf˜LgL0 are equivalent if and only if ω and ω belong to the same cohomology class.

    Proof. ) Suppose that H:LωWLωW is the corresponding homomorphism. Then

    H[u1,u2,u3]ω=[H(u1),H(u2),H(u3)]ω. (4.5)

    Since H is an equivalence of extensions, it follows ρ:LW satisfying

    H(ui+xi)=ui+ρ(ui)+xi,i=1,2,3. (4.6)

    The left side of (4.5) is equivalent to

    H([u1,u2,u3]+ω(u1,u2,u3))=[u1,u2,u3]+ω(u1,u2,u3)+ρ([u1,u2,u3]),

    and the right side of (4.5) shows that

    [u1+ρ(u1),u2+ρ(u2),u3+ρ(u3)]ω=[u1,u2,u3]+ω(u1,u2,u3)+δD(u1,u2)ρ(u3)δε(|u2|,|u3|)ϑ(u1,u3)ρ(u2)+ε(|u1|,|u2|+|u3|)ϑ(u2,u3)ρ(u1).

    Hence, we have

    (ωω)(u1,u2,u3)=δD(u1,u2)ρ(u3)δε(|u2|,|u3|)ϑ(u1,u3)ρ(u2)+ε(|u1|,|u2|+|u3|)ϑ(u2,u3)ρ(u1)ρ([u1,u2,u3]).

    Therefore, ωω=d1ρ, we obtain ω and ω are in the same cohomology class.

    ) On the other hand, we can assume that ωω=d1ρ, moreover, H defined by (4.6) is an equivalence.

    In this paper, we define the multiplicative Hom-δ-Jordan Lie color triple system and give its semidirect product. Then the representation and cohomology are characterized in order to discuss structures of deformations and extensions. Finally, we obtain that two equivalent 1-parameter formal deformations are in the same cohomology class and prove that there is the same representation between two equivalent abelian extensions. Moreover, it is shown that two abelian extensions on Hom-δ-Jordan Lie color triple systems are equivalent 3-Hom-cocycles that construct Hom-δ-Jordan Lie color triple systems are in the same cohomology class.

    L. Ma: Conceptualization, methodology, writing-original draft; Q. Li: Conceptualization, methodology, writing-review and editing. All authors have read and agreed to the published version of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This study is supported by National Natural Science Foundation of China (No.11801211), and the Fundamental Research Funds in Heilongjiang Provincial Universities (No.145209132).

    All authors declare no conflicts of interest in this paper.



    [1] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, New York: Academic Press, 1978. https://doi.org/10.1090/gsm/034
    [2] S. Okubo, Introdution to Octonion and other non-associative algebras in Physics, Cambridge: Cambridge University Press, 1995. https://doi.org/10.1017/CBO9780511524479
    [3] K. Yamaguti, On the cohomology space of Lie triple system, Kumamoto J. Sci. Ser. A, 5 (1960), 44–52.
    [4] F. Kubo, Y. Taniguchi, A controlling cohomology of the deformation theory of Lie triple systems, J. Algebra, 278 (2004), 242–250. https://doi.org/10.1016/j.jalgebra.2004.01.005 doi: 10.1016/j.jalgebra.2004.01.005
    [5] T. Zhang, Notes on cohomologies of Lie triple systems, J. Lie Theory, 24 (2014), 909–929.
    [6] S. Okubo, Parastatistics as Lie supertriple systems, J. Math. Phys., 35 (1994), 2785–2803. https://doi.org/10.1063/1.530486 doi: 10.1063/1.530486
    [7] S. Okubo, N. Kamiya, Jordan-Lie superalgebra and Jordan-Lie triple system, J. Algebra, 198 (1997), 388–411. https://doi.org/10.1006/jabr.1997.7144 doi: 10.1006/jabr.1997.7144
    [8] S. Okubo, N. Kamiya, Quasi-classical Lie superalgebras and Lie supertriple systems, Commun. Algebra, 30 (2002), 3825–3850. https://doi.org/10.1081/AGB-120005822 doi: 10.1081/AGB-120005822
    [9] N. Kamiya, S. Okubo, A construction of simple Jordan superalgebra of F type from a Jordan-Lie triple system, Ann. Mat. Pura. Appl. IV Ser., 181 (2002), 339–348. https://doi.org/10.1007/s102310100045 doi: 10.1007/s102310100045
    [10] L. Ma, L. Chen, On δ-Jordan Lie triple systems, Linear Multilinear A., 65 (2017), 731–751. https://doi.org/10.1080/03081087.2016.1202184 doi: 10.1080/03081087.2016.1202184
    [11] S. Wang, X. Zhang, S. Guo, Derivations and deformations of δ-Jordan Lie supertriple systems, Adv. Math. Phys., 2019 (2019), 3295462. https://doi.org/10.1155/2019/3295462 doi: 10.1155/2019/3295462
    [12] F. Ammar, S. Mabrouk, A. Makhlouf, Cohomology and deformations of Hom-algebras, J. Lie Theory, 21 (2011), 813–836.
    [13] L. Ma, L. Chen, J. Zhao, δ-Hom-Jordan Lie superalgebras, Commun. Algebra, 46 (2018), 1668–1697. https://doi.org/10.1080/00927872.2017.1354008 doi: 10.1080/00927872.2017.1354008
    [14] D. Yau, On n-ary Hom-Nambu and Hom-Nambu-Lie algebras, J. Geom. Phys., 62 (2012), 506–522. https://doi.org/10.1016/j.geomphys.2011.11.006 doi: 10.1016/j.geomphys.2011.11.006
    [15] B. Sun, L. Chen, Y. Liu, T-extensions and abelian extensions of Hom-Lie color algebras, Rev. Union. Mat. Argent., 59 (2018), 123–142. https://doi.org/10.33044/revuma.v59n1a06 doi: 10.33044/revuma.v59n1a06
    [16] L. Chen, Y. Yi, M. Chen, Y. Tang, Cohomology and 1-parameter formal deformations of Hom-δ-Lie triple systems, Adv. Appl. Clifford Algebras, 29 (2019), 63. https://doi.org/10.1007/s00006-019-0982-z doi: 10.1007/s00006-019-0982-z
    [17] Q. Li, L. Ma, Central extensions and Nijenhuis operators of Hom-δ-Jordan Lie triple systems, Adv. Math. Phys., 2022 (2022), 2706774. https://doi.org/10.1155/2022/2706774 doi: 10.1155/2022/2706774
    [18] Y. Ma, L. Chen, J. Lin, Central extensions and deformations of Hom-Lie triple systems, Commun. Algebra, 46 (2018), 1212–1230. https://doi.org/10.1080/00927872.2017.1339063 doi: 10.1080/00927872.2017.1339063
    [19] Q. Li, L. Ma, Nijenhuis operators and abelian extensions of Hom-δ-Jordan Lie supertriple systems, Mathematics, 11 (2023), 871. https://doi.org/10.3390/math11040871 doi: 10.3390/math11040871
    [20] D. Gaparayi, S. Attan, A. N. Issa, Hom-Lie-Yamaguti superalgebras, Korean J. Math., 27 (2019), 175–192. https://doi.org/10.11568/kjm.2019.27.1.175 doi: 10.11568/kjm.2019.27.1.175
    [21] J. Zhao, L. Chen, L. Ma, Representations and T-extensions of Hom-Jordan-Lie algebras, Commun. Algebra, 44 (2016), 2786–2812. https://doi.org/10.1080/00927872.2015.1065843 doi: 10.1080/00927872.2015.1065843
    [22] T. Zhang, J. Li, Representations and cohomologies of Hom-Lie-Yamaguti algebras with applications, Colloq. Math., 148 (2017), 131–155. https://doi.org/10.4064/cm6903-6-2016 doi: 10.4064/cm6903-6-2016
    [23] Q. Li, L. Ma, 1-parameter formal deformations and abelian extensions of Lie color triple systems, Electron. Res. Arch., 30 (2022), 2524–2539. https://doi.org/10.3934/era.2022129 doi: 10.3934/era.2022129
    [24] Q. Li, L. Ma, 1-parameter formal deformations, Nijenhuis operators and abelian extensions of δ-Lie color triple systems, Topol. Appl., 328 (2023), 108458. https://doi.org/10.1016/j.topol.2023.108458 doi: 10.1016/j.topol.2023.108458
    [25] J. Feldvoss, Representations of Lie color algebras, Adv. Math., 157 (2001), 95–137. https://doi.org/10.1006/aima.2000.1942 doi: 10.1006/aima.2000.1942
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