# Tθ∗-Extensions of n-Lie Algebras

Authors: Ruipu Bai; Ying Li
Journal: ISRN Algebra
 (2011)

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## Abstract

The paper is mainly concerned withTθ∗-extensions of n-Lie algebras. The Tθ∗-extension Lθ(L∗) of an n-Lie algebra L by a cocycle θ is defined, and a class of cocycles is constructed by means of linear mappings from an n-Lie algebra on to its dual space. Finally all Tθ∗-extensions of (n+1)-dimensional n-Lie algebras are classified, and the explicit multiplications are given.


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## Body

## 1. Introduction

n-Lie algebras (or Lie n-algebra, Filippov algebra, Nambu-Poisson algebra, and so on) are a kind of multiple algebraic systems appearing in many fields in mathematics and mathematical physics (cf. [1–5]). Although the theory of n-Lie algebras has been widely studied ([6–14]), it is quite necessary to get more examples of n-Lie algebras and the method of constructing n-Lie algebras. However it is not easy due to the n-ary operation.Bordemann in [15] introduced the notion of T*-extension of a Lie algebra and showed that each solvable quadratic Lie algebra over an algebraically closed field of characteristic zero is either a T*-extension or a nondegenerate ideal of codimension 1 in a T*-extension of some Lie algebra. In [16], Figueroa-O'Farrill defined the notion of a double extension of a metric Lie n-algebra by another Lie n-algebra and proved that all metric Lie n-algebras are obtained from the simple and one-dimensional ones by iterating the operations of orthogonal direct sum and double extension. The paper [17] studied the Tθ-extension and T*-extension of metric 3-Lie algebras and provided a sufficient and necessary condition of a T*-extension of 3-Lie algebra admitting a metric.This paper defines theTθ*-extension of an n-Lie algebra L by the coadjoint module L* and a cocycle θ from L∧n on to the dual space L* of L. The main result of the paper is the complete classification of the Tθ*-extensions of (n+1)-dimensional n-Lie algebras.Throughout this paper,n-Lie algebras are of finite dimensions and over an algebraically closed field F of characteristic zero. Any multiplication of basis vectors which is not listed in the multiplication table of an n-Lie algebra is assumed to be zero, and the symbol x̂ means that x  is omitted. If L is a vector space over a field F with a basis e1,⋯,em, then V can be denoted by V=Fe1+⋯+Fem.


## 2.Tθ*-Extensions of n-Lie Algebras

To study theTθ*-extensions of n-Lie algebras, we need some definitions and basic facts.Ann-Lie algebraL is a vector space with an n-ary skew-symmetric operation satisfying(2.1)[x1,…,xn]=sgn⁡(σ)[xσ(1),…,xσ(n)],(2.2)[[x1,…,xn],y2,…,yn]=∑i=1n[x1,…,[xi,y2,…,yn],…,xn]
for every x1,…,xn,y2,…,yn∈L and every permutation σ∈Sn. Identity (2.2) is called the generalized Jacobi identity. A subspace B of L is referred to as a subalgebra (ideal) of L if [B,…,B]⊆B ([B,L,…,L]⊆B). In particular, the subalgebra generated by [x1,…,xn] for all x1,…,xn∈L is called the derived algebra of L and is denoted by L1.Ann-Lie algebra L is called solvable if L(s)=0 for some s≥0, where L(0)=L and L(s) is defined as L(s+1)=[L(s),L(s),L,…,L] for s≥0. An ideal L is called nilpotent if Ls=0 for some s≥0, where L0=L and Ls is defined as Ls=[Ls-1,L,…,L], for s≥1. An n-Lie algebra L is called abelian if L1=0.LetL be an n-Lie algebra over the field F and V a vector space. If there exists a multilinear mapping ρ:L∧(n-1)→End(V) satisfying(2.3)ρ([x1,…,xn],y2,…,yn-1)=∑i=1n(-1)n-iρ(x1,…,x̂i,…,xn)ρ(xi,y2,…,yn-2)(2.4)[ρ(x1,…,xn-1),ρ(y1,…,yn-1)]=ρ(x1,…,xn-1)ρ(y1,…,yn-1)-ρ(y1,…,yn-1)ρ(x1,…,xn-1)=∑i=1nρ(y1,…,[x1,…,xn-1,yi],…,yn-1)
for all xi,yi∈L,i=1,…,n, then (V,ρ) is called a representation of L or V is an  L-module.Letρ(x1,…,xn-1)=ad(x1,…,xn-1) for x1,…,xn-1∈L. Then (L,ad) is an L-module and is called the adjoint module of L. If (V,ρ) is an L-module, then the dual space V* of V is an L-module in the following way. For f∈V*,v∈V,x1,…,xn-1∈L, defines ρ*:L∧n-1→End(V*),(2.5)ρ*(x1,…,xn-1)(f)(v)=-f(ρ(x1,…,xn-1)(v)).
(V*,ρ*) is called the dual module of V. If V=L and ρ=ad, that is, ad*(x1,…,xn-1)(f)(x)=-f([x1,…,xn-1,x]), (L*,ad*) is called the coadjoint module of L.Definition 2.1.
LetL be an n-Lie algebra. If the n-linear mapping θ:L∧n→L* satisfying for all xi,yj∈L,1≤i≤n,2≤j≤n,
(2.6)∑i=1nθ(x1,…,[xi,y2,…,yn],…,xn)-θ([x1,…,xn],y2,…,yn)+∑i=1n(-1)n-i[x1,…,x̂i,…,xn,θ(xi,y2,…,yn)]+(-1)n[y2,…,yn,θ(x1,…,xn)]=0,
then θ is called a cocycle of L.Theorem 2.2.
LetL be an n-Lie algebra over F, and let  θ:L∧n→L* be a cocycle of L. Then Lθ(L*)=L⊕L* is an n-Lie algebra in the following multiplication:
(2.7)[y1+f1,…,yn+fn]θ=[y1,…,yn]L+θ(y1,…,yn)+∑i=1n(-1)n-iad*(y1,…,ŷi,…,yn)fi,
where yi∈L,fi∈L*,1≤i≤n.Proof.
It suffices to verify the Jacobi identity (2.2) for Lθ(L*). For all yi∈L,fi∈L*,1≤i≤2n-1, set zi=yi+fi, and by identity (2.7) we have
(2.8)[[z1,…,zn]θ,zn+1,…,z2n-1]θ=[[y1+f1,…,yn+fn]θ,yn+1+fn+1,…,y2n-1+f2n-1]θ=[[y1,…,yn]L,yn+1,…,y2n-1]L+θ([y1,…,yn]L,yn+1,…,y2n-1)+(-1)n-1ad*(yn+1,…,y2n-1)θ(y1,…,yn)+ad*(yn+1,…,y2n-1)∑i=1n(-1)i+1ad*(y1,…,ŷi,…,yn)fi+∑j=1n-1(-1)n-j-1ad*([y1,…,yn]L,yn+1,…,ŷn+j,…,y2n-1)fn+j;
and for every 1≤i,k≤n,
(2.9)[z1,…,zk-1,[zk,zn+1,…,z2n-1]θ,zk+1,…,zn]θ=[y1,…,yk-1,[yk,yn+1,…,y2n-1]L,yk+1,…,yn]L+θ(y1,…,yk-1,[yk,yn+1,…,y2n-1]L,yk+1,…,yn)+∑i=1k-1(-1)n-iρ(y1,…,ŷi,…,yk-1,[yk,yn+1,…,y2n-1]L,yk+1,…,yn)fi+(-1)n-kad*(y1,…,yk-1,yk+1,…,yn)θ(yk,yn+1,…,y2n-1)+(-1)k+1ad*(y1,…,yk-1,yk+1,…,yn)ρ(yn+1,…,y2n-1)fk+ad*(y1,…,yk-1,yk+1,…,yn)∑i=1n-1(-1)k+i+1ad*(yk,yn+1,…,ŷn+i,…,y2n-1)fn+i+∑i=k+1n(-1)n-iad*(y1,…,yk-1,[yk,yn+1,…,y2n-1]L,yk+1,…,yn)fi.
Thanks for identity (2.5), for 1≤m≤n,
(2.10)ad*(yn+1,…,y2n-1)ad*(y1,…,ŷm,…,yn)fm=(-1)n-1∑j≠m,j=1nad*(y1,…,[yj,yn+1,…,y2n-1]L,…,ŷm,…,yn)fm.
For 1≤m≤n-1, by identity (2.3),
(2.11)(-1)n-m-1ad*([y1,…,yn]L,yn+1,…,ŷn+m,…,y2n-1)fn+m=∑i=1n(-1)-m-i-1ad*(y1,…,ŷi,…,yn)ad*(yi,yn+1,…,ŷn+m,…,y2n-1)fn+m.
Therefore, the multiplication of Lθ(L*) defined by identity (2.7) satisfies
(2.12)[[z1,…,zn]θ,zn+1,…,z2n-1]θ=∑k=1n[z1,…,zk-1,[zk,zn+1,…,z2n-1]θ,zk+1,…,zn]θ
for every zi∈Lθ(L*),1≤i≤2n-1.Definition 2.3.
Then-Lie algebra Lθ(L*)=L⊕L* with multiplication (2.7) is called the Tθ*-extension of L. In particular, the T0*-extension corresponding to θ=0 is called the  trivial extension of L and is denoted by L0(L*).Then the multiplication ofL0(L*) is as follows:(2.13)[y1+f1,…,yn+fn]0=[y1,…,yn]L+∑i=1n(-1)n-iad*(y1,…,ŷi,…,yn)fi,
where yi∈L,fi∈V,1≤i≤n.Theorem 2.4.
LetL be an n-Lie algebra, and let θ:L∧n→L* be a cocycle of L. Then one has the following results.(1)
L* is an abelian ideal of the Tθ*-extension.(2)
IfL is solvable, then the Tθ*-extension Lθ(L*) is solvable.(3)
IfL is a nilpotent n-Lie algebra, then every Tθ*-extension is nilpotent.(4)
Ifθ≠0, then Lθ(L*) is an essential extension of L by the module L*. If θ=0, L0(L*) is a nonessential extension of L.Proof.
From identity (2.7), L* is an abelian ideal of Lθ(L*) since [L*,L*,Lθ(L*),…,Lθ(L*)]θ=0, and [L*,Lθ(L*),…,Lθ(L*)]θ⊆L*.
Now letL be solvable and L(s)=0. By induction on r, we have
(2.14)[Lθ(r+1)(L*)=[Lθ(r)(L*),Lθ(r)(L*),Lθ(L*),…,Lθ(L*)]θ⊆L(r+1)+θ(L(r),L(r),L,…,L)+L*.
Then we have Lθ(s+1)(L*)⊆L*. Thanks to result (1), Lθ(s+2)(L*)=0. Result (2) follows.
(3) SinceL is nilpotent, Ls=[Ls-1,L,…,L]L=0 for some nonnegative integer s. For every cocycle θ:Ln→L*, by identity (2.6),
(2.15)Lθ1(L*)⊆L1+θ(L,…,L)+ad*(L,…,L)(L*)⊆L1+L*.
Inductively, we haveLθs(L*)⊆L(s)+L*=L* since Ls=0. Then we have Lθ2s(L*)⊆ad*s(L,…,L)(L*). Note that for f∈ad*s(L,…,L)(L*), we have f(L)⊆f(Ls)=0. Thus, Lθ2s(L*)=0, that is, Lθ(L*) is a nilpotent n-Lie algebra.
It follows from result (4) thatL is a subalgebra of Lθ(L*) if θ=0.For constructingTθ*-extensions of an n-Lie algebra L, we give the following method to get cocycles.Theorem 2.5.
LetL be an n-Lie algebra. Then for every linear mapping σ:L→L*, the skew-symmetric mapping θσ:L∧n→L* given by, for all x1,…,xn∈L,
(2.16)θσ(x1,…,xn)=σ([x1,…,xn]L)-∑i=1n(-1)n-iad*(x1,…,x̂i,…,xn)σ(xi)
is a cocycle.Proof.
A tedious calculation shows that, for everyxi,yi∈L,1≤i,k≤n,
(2.17)θσ(x1,…,xk-1,[y2,…,yn,xk]L,xk+1,…,xn)=σ([x1,…,xk-1,[y2,…,yn,xk]L,xk+1,…,xn]L)+∑i=1k-1(-1)n-i-1ad*(x1,…,x̂i,…,xk-1,[y2,…,yn,xk]L,xk+1,…,xn)σ(xi)+(-1)n-k-1ad*(x1,…,xk-1,xk+1,…,xn)σ(xk)+∑j=k+1n(-1)n-j-1ad*(x1,…,[y2,…,yn,xk]L,xk+1,…,x̂j,…,xn)σ(xj);θσ([x1,…,xn]L,y2,…,yn)=σ([[x1,…,xn]L,y2,…,yn]L)+∑i=2n(-1)n-i-1ad*([x1,…,xn]L,y2,…,ŷi,…,yn)σ(yi)+(-1)nad*(y2,…,yn)σ([x1,…,xn]L);ad*(x1,…,x̂k,…,xn)θσ(y2,…,yn,xk)=ad*(x1,…,x̂k,…,xn)σ([y2,…,yn,xk]L)+ad*(x1,…,x̂k,…,xn)∑i=2n(-1)n-iad*(y2,…,ŷi,…,yn,xk)σ(yi)-ad*(x1,…,x̂k,…,xn)ad*(y2,…,yn)σ(xk);ad*(y2,…,yn)θf(x1,…,xn)=ad*(y2,…,yn)σ([x1,…,xn]L)-ad*(y2,…,yn)∑i=1n(-1)n-iad*(x1,…,x̂i,xn)σ(xi).
Therefore, θf satisfies identity (2.6). The proof is completed.Theorem 2.6.
LetL be an n-Lie algebra, and let θ:L∧n→L* be a cocycle. Then for every linear mapping σ:L→L*, for all y∈L,f∈L*(2.18)Γ:Lθ(L*)⟶Lθ+θσ(L*),Γ(y+f)=y+σ(y)+f,
is an n-Lie algebra isomorphism.Proof.
It is clear thatΓ is a linear isomorphism of the vector space L⊕L* to itself. Next, for every fi∈L*,yi∈L,1≤i≤n,
(2.19)Γ([y1+f1,…,yn+fn]θ)=Γ([y1,…,yn]L+θ(y1,…,yn)+∑i=1n(-1)n-iad*(y1,…,yî,…,yn)fi)=[y1,…,yn]L+θ(y1,…,yn)+σ([y1,…,yn]L)+∑i=1n(-1)n-iad*(y1,…,yî,…,yn)fi.[Γ(y1+v1),…,Γ(yn+vn)]θ+θσ=[y1+σ(y1)+f1,…,yn+σ(yn)+fn]θ+θσ=[y1,…,yn]L+(θ+θσ)(y1,…,yn)+∑i=1n(-1)n-iad*(y1,…,yî,…,yn)(σ(yi)+fi)=[y1,…,yn]L+θ(y1,…,yn)+σ([y1,…,yn]L)+∑i=1n(-1)n-iad*(y1,…,yî,…,yn)fi=Γ([y1+f1,…,yn+fn]θ).
the result follows.Corollary 2.7.
LetL be an n-Lie algebra, and let θ1,θ2:L∧n→L* be cocycles. If there exists a linear mapping σ:L→L* such that θ1-θ2=θσ, then the Tθ1*-extension Lθ1(L*) is isomorphic to the Tθ2*-extension Lθ2(L*) of L.Proof.
If there is a linear mappingσ:L→L* such that θ1=θ2+θσ, by Theorem 2.6, the Tθ1*-extension Lθ1(L*)=Lθ2+θσ(L*) is isomorphic to the Tθ2*-extension Lθ2(L*).


## 3. TheTθ*-Extension of (n+1)-Dimensional n-Lie Algebras

In this section, we study theTθ*-extension of (n+1)-dimensional n-Lie algebras over F. First, we recall the classification theorem of (n+1)-dimensional n-Lie algebras.Lemma 3.1 (see [6]).
LetL be an (n+1)-dimensional n-Lie algebra over F and e1,e2,…,en+1 a basis of L (n≥3). Then one and only one of the following possibilities hold up to isomorphisms.(a)
IfdimL1=0, then L is an abelian n-Lie algebra.(b)
IfdimL1=1 and letting L1=Fe1,
in the case thatL1⊆Z(L),
(b1)
[e2,…,en+1]=e1;ifL1 is not contained in Z(L),
(b2)
[e1,…,en]=e1.(c)
IfdimL1=2 and letting L1=Fe1+Fe2,
(c1)
[e2,…,en+1]=e1,[e1,e3,…,en+1]=e2;(c2)
[e2,…,en+1]=αe1+e2,[e1,e3,…,en+1]=e2;(c3)
[e1,e3,…,en+1]=e1,[e2,…,en+1]=e2,α∈F,α≠0.(d)
IfdimL1=r, 3≤r≤n+1, let L1=Fe1+Fe2+…+Fer. Then(dr)
[e1,…,êi,…,en+1]=ei,1≤i≤r, where symbol êi means that ei is omitted.We first introduce some notations. LetL be an (n+1)-dimensional n-Lie algebra in the Lemma 3.1, and let f1, …,fn+1 be the basis of L* satisfying fi(ej)=δij,1≤i,j≤n+1. For a cocycle θ:L∧n→L*(3.1)θ(e1,…,êj,…,en+1)=∑s=1n+1ajsfs,ajs∈F,1≤j≤n+1.TheTθ*-extensions of the classes (bi), (cj), and (dr) in Lemma 3.1 are denoted by (bi*), (cj*) and (dr*), respectively.Theorem 3.2.
LetL be an (n+1)-dimensional n-Lie algebra in the Lemma 3.1. Then up to isomorphisms the Tθ*-extensions of L are only of the following possibilities:(a*)
Lθ(L*)is abelian(b1*)
[e2,…,en+1]θ=e1,[e1,e2,…,êj,…,en+1]θ=∑s=2n+1ajsfs,[e2,…,êj,…,en+1,f1]θ=(-1)n+1+jfj,ajs∈F,2≤j≤n+1.(b2*)
[e1,…,en]θ=e1,[e1,…,êj,…,en+1]θ=∑s=2n+1ajsfs,[e1,…,êj,…,en,f1]θ=(-1)n-j+1fj,ajs∈F,1≤j≤n.(c1*)
[e1,e3,…,en+1]θ=e2,[e2,e3,…,en+1]θ=e1,[e3,…,en+1,f2]θ=(-1)nf1,[e1,e2,…,êj…,en+1]θ=∑s=3n+1ajsfs,[e1,e3,…,êj,…,en+1,f2]θ=(-1)n-jfj,[e2,…,êj,…,en+1,f1]θ=(-1)n-jfj,ajs∈F,3≤j≤n+1.(c2*)
[e1,e3,…,en+1]θ=e2,[e2,e3,…,en+1]θ=αe1+e2,[e2,e3,…,êi,…,en+1,f1]θ  =(-1)n-iαfi,[e2,e3,…,êj,…,en+1,f2]θ  =(-1)n-ifj,[e1,e3,…,êj,…,en+1,f2]θ  =(-1)n-jfj,[e3,…,…,en+1,f2]θ  =(-1)n(f2+f1),where α∈F,α≠0,2≤i≤n+1,3≤j≤n+1.(c3*)
[e1,e3,…,en+1]θ=e1,[e2,e3,…,en+1]θ=e2,[e3,…,en+1,f1]θ=(-1)nf1,[e1,e3,…,êj,…,en+1,f1]θ  =(-1)n-jfj,3≤j≤n+1,[e2,…,êi,…,en+1,f2]θ=(-1)n-ifi,2≤i≤n+1.(dr*)
[e1,…,êj,…,en+1]θ=ej,1≤j≤r,[e1,e2,…,er,…,êj,…,en+1]θ  =∑s=r+1n+1ajsfs,ajs∈F,ajs∈F,r<j,[e1,…,êj,…,êi,…,en+1,fi]θ  =(-1)n-j+1fj,1≤j<i≤r,[e1,…,êi,…,êj,…,en+1,fi]θ  =(-1)n-jfj,1≤i<j≤r,where 3≤r≤n+1.Proof.
Case(a*) is trivial. If L is case (b1), let f1,…,fn+1 be a basis of L* satisfying fi(ej)=δij,1≤i,j≤n+1. By the direct computation, identity (2.6), and Lemma 3.1, for every cocycle θ0:L∧n→L*, we have θ0(e2,e3,…,en+1)=∑s=1n+1a1sfs,θ0(e1,e2,…,êj,…,en+1)=∑j=2sajsfs,ajs∈F,2≤j≤n+1. The multiplication of Lθ0(L*) in the basis e1,…,en+1,f1,…,fn+1 is
(3.2)[e2,…,en+1]θ0=e1+∑s=1n+1a1sfs,a1s∈F,[e1,e2,…,êj,…,en+1]θ0=∑j=2sajsfs,ajs∈F,2≤j≤n+1,[e2,,…,êj,…,en+1,f1]θ0=(-1)n+j+1fj,2≤j≤n+1.
By Theorem2.5, omitting the computation process, for every linear mapping σ:L→L*, the cocycle θσ:L∧n→L* satisfies θσ(e2,e3,…,en+1)=(n+1)σ(e1) and θη(e1,e2,…,êj,…,en+1)=0,2≤j≤n+1. Then, define
(3.3)σ(e1)=-1n+1θ0(e2,e3,…,en+1)=-∑s=1n+1a1sfs,
and σ(ei)=0,2≤i≤n+1. Follows Theorem 2.6 that Lθ0(L*) is isomorphic to Lθ0+θσ(L*) which with the multiplication (b1*).
In the case(b2), let θ0:L∧n→L* be a cocycle. Omitting the computation process, we have θ0(e2,e3,…,en+1)=a11f1+…+a1n+1fn+1,θ0(e1,e2,…,êj,…,en+1)=aj2f2+…+ajn+1fn+1,j∈2…,n+1. The multiplication table of Lθ0(L*) is as follows:
(3.4)[e2,…,en+1]θ0=e1+∑s=1n+1a1sfs,[e1,e2,…,êj,…,en+1]θ0=∑s=2n+1ajsfs,[e2,e3,…,êi,…,en+1,f1]θ0=(-1)n+i+1fi,2≤i≤n+1.
For every linear mappingσ:L→L*, the cocycle θσ: L∧n→L*, by Theorem 2.5, omitting the computation process, θη(e1,e2,…,êj,…,en+1)=0,2≤j≤n+1,θη(e2,…,en+1)=(n+1)η(e1). Then defining
(3.5)σ(e1)=-1n+1θ(e2,…,en+1)=a11f1+⋯+a1n+1fn+1,η(ej)=0,2≤j≤n+1,
we have Lθ0+θσ(L*) with the multiplication (b2*) which is isomorphic to Lθ0(L*).
In case(c1), for every cocycle θ0:L∧n→L*, omitting the computation process, we have θ0(e1,e3,…,en+1)=∑s=1n+1a2sfs,θ0(e2,…,en+1)=∑s=1n+1a1sfs,θ0(e1,e2,…,êj,…,en+1)=∑s=3n+1ajsfs,j=3,…,n+1. The multiplication table of Lθ0(L*) is as follows:
(3.6)[e1,e3,…,en+1]θ0=e2+∑s=1n+1a2sfs,[e2,…,en+1]θ0=e1+∑s=1n+1a1sfs,[e1,e2,e3,…,êj,…,en+1]θ0=∑s=3n+1ajsfs,3≤j≤n+1,[e1,e3,…,êj,…,en+1,f2]θ0=(-1)n-jfj,3≤j≤n+1,[e2,e3,…,êj,…,en+1,f1]θ0=(-1)n-jfj,3≤j≤n+1,[e3,…,en+1,f2]θ0=(-1)nf1.
Define the linear mappingσ:L→L*:σ(e2)=-(1/(n+1))θ0(e1,e3,…,en+1), σ(e1)=-(1/(n+1))θ0(e2,e3,…,en+1) and others are zero. By the direct computation
(3.7)θσ(e1,e3,…,en+1)=(n+1)η0(e2),θσ(e2,e3,…,en+1)=(n+1)η0(e1).
Then Lθ0+θσ(L*) has the multiplication (c1*).
In the case(c2), for every cocycle θ0:L∧n→L*, we have θ0(e1,e3,…,en+1)=∑s=1n+1a2sfs,θ0(e2,…,en+1)=∑s=1n+1a1sfs,θ0(e1,e2,…,êj,…,en+1)=∑s=3n+1ajsfs,j=3,…,n+1. The multiplication table of Lθ0(L*) is as follows:
(3.8)[e1,e3,…,en+1]θ0=e2+∑s=1n+1a2sfs,[e2,…,en+1]θ0=αe1+e2+∑s=1n+1a1sfs,[e1,e2,e3,…,êj,…,en+1]θ0=∑s=3n+1ajsfs,3≤j≤n+1,[e2,…,êi,…,en+1,f1]θ0=(-1)n+iαfi,2≤i≤n+1,[e2,e3,…,êj,…,en+1,f2]θ0=(-1)n-jfj,3≤j≤n+1,[e1,e3,…,êj,…,en+1,f2]θ0=(-1)n-jfj,3≤j≤n+1,[e3,…,êj,…,en+1,f2]θ0=(-1)n(f2+f1).
Define linear mappingσ:L→L*:σ(e2)=-(1/(n+1))θ0(e1,e3,…,en+1), σ(e1)=(1/α(n+1))(θ0(e1,e3,…,en+1)-θ0(e2,e3,…,en+1)). Then we obtain θσ(e1,e3,…,en+1)=(n+1)σ(e2)=-θ0(e1,e3,…,en+1),θσ(e2,e3,…,en+1)=(n+1)σ(αe1+e2)=-θ0(e2,e3,…,en+1) and others are zero. Therefore, Lθ0+θσ(L*) has the multiplication (c2*) in the basis e1,…,en+1,f1,…,fn+1.
In case(c3), in similar discussions to above, for every cocycle θ0:L∧n→L*, defining linear mapping σ:L→L*:σ(e1)=-(1/(n+1))θ0(e1,e3,…,en+1), η0(e2)=-(1/(n+1))θ0(e2,e3,…,en+1), we have
(3.9)θσ(e1,e3,…,en+1)=(n+1)σ(e1)=-θ0(e1,e3,…,en+1),
θσ(e2,e3,…,en+1)=(n+1)σ(e2)=-θ0(e2,e3,…,en+1) and others are zero. Then Lθ0+θσ(L*) has the multiplication (c3*) in the basis e1,…,en+1,f1,…,fn+1.
Lastly, ifL is case (dr), 3≤r≤n+1, for every cocycle θ0:L∧n→L*, we have θ0(e1,,…,êi,…,en+1)=∑s=1n+1aisfs,1≤i≤r,θ0(e1,…,er,…,êj,…,en+1)=∑j=r+1n+1ajsfs,r<j≤n+1. By the direct computation, the multiplication of Lθ0(L*) is as follows:
(3.10)[e1,…,êi,…,en+1]θ0=ei+∑s=1n+1aisfs,1≤i≤r,[e1,…,er,…,êj,…,en+1]θ0=∑s=r+1n+1ajsfj,r<j≤n+1,[e1,…,êj,…,êi,…,en+1,fi]θ0=(-1)n-j+1fj,1≤j<i≤r,[e1,…,êi,…,êj,…,en+1,fi]θ0=(-1)n-jfj,1≤i<j≤r.
Define linear mapping σ:L→L*:σ(ei)=-(1/(n+1))θ0(e1,…,êi,…,en+1),1≤i≤r, and σ(ei)=0 if r<i. Then we obtain θσ(e1,…,êi,…,en+1)=(n+1)σ(ei)=-θ0(e1,…,êi,…,en+1) for 1≤i≤r, and θσ(e1,…,êi,…,en+1)=0 if i>r. Therefore, Lθ0+θσ(L*) with the multiplication (dr*) in the basis e1,…,en+1,f1,…,fn+1 and Lθ0(L*) is isomorphic to Lθ0+θσ(L*).



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*Source: 381875-2011-09-11.xml*