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)=∑i≥0li(t,p,q)si and l′L(t,p,q)=∑i≥0l′i(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×G→K∖{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,g3∈G. It is obvious that
ε(g,0)=ε(0,g)=1,ε(g,g)=±1,∀g∈G. |
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×L→L, with even linear maps βi:L→L for i=1,2, which is called twisted maps, satisfying for all elements t,p,q,u,v∈L,
[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=⊕g∈GLg over F, a trilinear map [⋅,⋅,⋅]:L×L×L→L, with even linear maps βi:L→L for i=1,2, which is called twisted maps, satisfying for all elements t,p,q,u,v∈L, 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,j∈Z2.
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,q∈L.
Example 2.4. Suppose (L,[⋅,⋅,⋅]) is a Hom-δ-Jordan Lie color triple system, and l is an indeterminate. Assume L′={Σi≥0y⊗li|y∈L}. Then (L′,[⋅,⋅,⋅]′) is a Hom-δ-Jordan Lie color triple system satisfying a product [⋅,⋅,⋅]′ defined using
[p⊗li,q⊗lj,m⊗lk]′=[p,q,m]⊗li+j+k, |
where p⊗li,q⊗lj,m⊗lk∈L′, and |p⊗li|=|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′=(β′1,β′2)) 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 h∘βi=β′i∘h, 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 γ∈End(W). W is called a (L,[⋅,⋅,⋅])-module relating to γ if there is a bilinear map ϑ:L×L→End(W), (t,t1)↦ϑ(t,t1) satisfying for all elements t,t1,t2,t3∈L,
ϑ(β(t),β(t1))∘γ=γ∘ϑ(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:(L⊕W)×(L⊕W)×(L⊕W)→L⊕W 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:L⊕W→L⊕W by
(β+γ)(t,w)=(β(t),γ(w)), |
thus L⊕W 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, (L⊕W,[⋅,⋅,⋅]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:(L⊕W)×(L⊕W)×(L⊕W)→L⊕W, 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×⋯×L⏟ntimes→W satisfies
h(t1,⋯,t,p,tn)=−δε(|t|,|p|)h(t1,⋯,p,t,tn), |
ε(|t|,|q|)h(t1,⋯,tn−3,t,p,q)+ε(|p|,|t|)h(t1,⋯,tn−3,p,q,t)+ε(|q|,|p|)h(t1,⋯,tn−3,q,t,p)=0, |
thus h is called an n-cochain on L. Denote by Cnγ(L,W) the set of all n-cochains, ∀n≥1.
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 h∈C1(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 h∈C2(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 h∈C3(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 h∈C4(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 h∈Cnγ(L,W) is called an n-cocycle. The subspace Znγ(L,W) spanned by n-cocycle, and set Bnγ(L,W)=dn−2Cn−2γ(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×L→L is extended to be an F[[s]]-trilinear map h:L[[s]]×L[[s]]×L[[s]]→L[[s]] by
h(∑i≥0tisi,∑j≥0pjsj,∑k≥0qksk)=∑i,j,k≥0h(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)=∑i≥0li(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×L→L, 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,j≥0li(β(z),β(u),lj(t,p,q))=∑i,j≥0li(lj(z,u,t),β(p),β(q))+∑i,j≥0ε(|t|,|z|+|u|)li(β(t),lj(z,u,p),β(q))+δ∑i,j≥0ε(|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×L→L which are extended to be F[[s]]-trilinear, we show a map f∘g:L[[s]]×L[[s]]×L[[s]]×L[[s]]×L[[s]]→L[[s]] using
f∘g(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=nli∘lj=0. |
If n=1, then l0∘l1+l1∘l0=0.
If n≥2, then −(l0∘ln+ln∘l0)=l1∘ln−1+l2∘ln−2+⋯+ln−1∘l1.
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 li∈C3(L,L). Considering the definition 2.5, we show d3ln=l0∘ln+ln∘βl0, for n=0,1,2⋯. Therefore, the deformation Eq (3.4) can be shown as
d3l1=0,−d3ln=l1∘ln−1+l2∘ln−2+⋯+ln−1∘l1. |
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)=∑i≥0li(t,p,q)si and l′L(t,p,q)=∑i≥0l′i(t,p,q)si are two 1-parameter formal deformations on (L,[⋅,⋅,⋅]). lL and l′L are known as equivalent, denoted by lL∼l′L, if there is a formal isomorphism of F[[s]]-modules
φt(t)=∑i≥0φi(t)si:(L[[s]],mL)⟶(L[[s]],m′L), |
any φi:L→L 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
φt∘β=β∘φt,φt∘lL(t,p,q)=l′L(φt(t),φt(p),φt(q)). |
In the case l1=l2=⋯=0, lL=l0 is known as the null deformation. If lL∼l0, 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)=∑i≥0li(t,p,q)si and l′L(t,p,q)=∑i≥0l′i(t,p,q)si are equivalent 1-parameter formal deformations on (L,[⋅,⋅,⋅]). Then l1 and l′1 belong to the same cohomology class in H3(L,L).
Proof. Suppose that φt(t)=∑i≥0φi(t)si is the formal F[[s]]-module isomorphism, satisfying φt∘β=β∘φt, and
∑i≥0φi(∑j≥0lj(t,p,q)sj)si=∑i≥0l′i(∑k≥0φk(t)sk,∑l≥0φl(p)sl,∑l≥0φm(q)sm)si. |
Then
∑i+j=nφi(lj(t,p,q))si+j=∑i+k+l+m=nl′i(φk(t),φl(p),φm(q))si+k+l+m. |
In especial,
∑i+j=1φi(lj(t,p,q))=∑i+k+l+m=1l′i(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)]+l′1(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)+l′1(t,p,q). |
We give that l1−l′1=d1φ1∈B3(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+∑i≥nlisi. Hence,
d3ln=l1∘ln−1+l2∘ln−2+⋯+ln−1∘l1=0, |
moreover, ln∈Z3(L,L)=B3(L,L). It is proved that there is hn∈C1(L,L) satisfying ln=d1hn.
Assume that φt is even, and φt=idL−hnsn:(L[[s]],lL)⟶(L[[s]],l′L). Notice that
φt∘∑i≥0hinsin=∑i≥0hinsin∘φt=idL[[s]]. |
It is easy to see that φt is a linear isomorphism, and φt∘β=β∘φt.
We shall consider l′L(t,p,q)=φ−1slL(φt(t),φt(p),φt(q)). The following fact is clear to prove that l′L is a 1-parameter formal deformation on (L,[⋅,⋅,⋅]). Indeed,
l′L(t,p,q)=φ−1slL(φt(t),φt(p),φt(q))=−δε(|t|,|p|)φ−1slL(φt(p),φt(t),φt(q))=−δε(|t|,|p|)l′L(p,t,q). |
ε(|t|,|q|)l′L(t,p,q)+ε(|p|,|t|)l′L(p,q,t)+ε(|q|,|p|)l′L(q,t,p)=ε(|t|,|q|)φ−1slL(φt(t),φt(p),φt(q))+ε(|p|,|t|)φ−1slL(φt(p),φt(q),φt(t))+ε(|q|,|p|)φ−1slL(φt(q),φt(t),φt(p))=φ−1s(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. |
l′L(β(t1),β(t2),l′L(t,p,q))=l′L(β(t1),β(t2),φ−1slL(φt(t),φt(p),φt(q)))=φ−1slL(φt(β(t1)),φt(β(t2)),lL(φt(t),φt(p),φt(q)))=φ−1slL(β(φt(t1)),β(φt(t2)),lL(φt(t),φt(p),φt(q))). |
l′L(l′L(t1,t2,t),β(p),β(q))=l′L(φ−1slL(φt(t1),φt(t2),φt(t)),β(p),β(q))=φ−1slL(lL(φt(t1),φt(t2),φt(t)),β(φt(p)),β(φt(q))). |
ε(|t|,|t1|+|t2|)l′L(β(t),l′L(t1,t2,p),β(q))=ε(|t|,|t1|+|t2|)l′L(β(t),φ−1slL(φt(t1),φt(t2),φt(p)),β(q))=ε(|t|,|t1|+|t2|)φ−1slL(φt(β(t)),lL(φt(t1),φt(t2),φt(p)),φt(β(q)))=ε(|t|,|t1|+|t2|)φ−1slL(β(φt(t)),lL(φt(t1),φt(t2),φt(p)),β(φt(q))). |
δε(|t1|+|t2|,|t|+|p|)l′L(β(t),β(p),l′L(t1,t2,q))=δε(|t1|+|t2|,|t|+|p|)l′L(β(t),β(p),φ−1slL(φt(t1),φt(t2),φt(q)))=δε(|t1|+|t2|,|t|+|p|)φ−1slL(φt(β(t)),φt(β(p)),lL(φt(t1),φt(t2),φt(q)))=δε(|t1|+|t2|,|t|+|p|)φ−1slL(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 lL∼l′L. Assume that l′L=∑i≥0l′isi. Hence
(idL−hnsn)(∑i≥0l′i(t,p,q)si)=(l0+∑i≥nlisi)(t−hn(t)sn,p−hn(p)sn,q−hn(q)sn), |
i.e.,
∑i≥0l′i(t,p,q)si−∑i≥0hn∘l′i(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+∑i≥nli(t,p,q)si−∑i≥n(li(hn(t),p,q)+li(t,hn(p),q)+li(t,p,hn(q)))si+n+∑i≥n(li(hn(t),hn(p),q)+li(t,hn(p),hn(q))+li(hn(t),p,hn(q)))si+2n−∑i≥nli(hn(t),hn(p),hn(q))si+3n. |
Then, it follows l′1=⋯=l′n−1=0 and
l′n(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, l′n=ln−d1hn=0 and l′L=l0+∑i≥n+1l′isi. By induction, we know lL∼l0, 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:W→ˆL, g:ˆL→L 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,
0⟶Wf⟶ˆLg⟶L⟶0. | (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,w2∈W.
A section τ:L→ˆL of g:ˆL→L is composed of the linear map τ:L→ˆL satisfying g∘τ=idL, and ˆβ∘τ=τ∘β.
Definition 4.2. Two extensions on multiplicative Hom-δ-Jordan Lie color triple systems EˆL:0⟶Wf⟶ˆLg⟶L⟶0 and E˜L:0⟶Wf′⟶˜Lg′⟶L⟶0 are equivalent. If there is a multiplicative Hom-δ-Jordan Lie color triple system homomorphism F:ˆL→˜L satisfying the following diagram commutes
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Suppose that ˆL is an abelian extension on L by W, and an even linear mapping τ:L→ˆL is a section. Assume maps L⊗L→End(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)∈L⊗L,v∈W.
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′:L→ˆL, we obtain
p(τ(ui)−τ′(ui))=ui−ui=0⇒τ(ui)−τ′(ui)∈W⇒τ′(ui)=τ(ui)+vi, |
for some vi∈W.
Notice that [w1,w2,⋅]ˆL=0=[w1,⋅,w2]ˆL for all w1,w2∈W, 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 F∘f=f′, g′∘F=g. Considering linear sections τ and τ′ of g and g′, we know g′Fτ(ui)=gτ(ui)=ui=g′τ′(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:L→ˆL is a section of the abelian extension. Suppose that the map is as follows:
ω(t1,t2,t3)=[τ(t1),τ(t2),τ(t3)]ˆL−τ([t1,t2,t3]L), | (4.4) |
for all t1,t2,t3∈L.
Theorem 4.2. Suppose that 0⟶W⟶ˆL⟶L⟶0 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 L⊕W 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:0⟶Wf⟶ˆLg⟶L⟶0 and E˜L:0⟶Wf′⟶˜Lg′⟶L⟶0 are equivalent if and only if ω and ω′ belong to the same cohomology class.
Proof. ⇒) Suppose that H:L⊕ωW→L⊕ω′W 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:L→W 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
(ω−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)−ρ([u1,u2,u3]). |
Therefore, ω−ω′=d1ρ, we obtain ω and ω′ are in the same cohomology class.
⇐) On the other hand, we can assume that ω−ω′=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.
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