𝑇^∗𝜃-Extensions of 𝑛-Lie Algebras

Abstract

The paper is mainly concerned with 𝑇^∗𝜃-extensions of 𝑛-Lie algebras. The 𝑇^∗𝜃-extension 𝐿_𝜃(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 𝑇^∗𝜃-extensions of (𝑛+1)-dimensional 𝑛-Lie algebras are classified, and the explicit multiplications are given.

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 𝑇_𝜃-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 the 𝑇^∗𝜃-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 𝑇^∗𝜃-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,⋯,𝑒_𝑚, then 𝑉 can be denoted by 𝑉=𝐹𝑒₁+⋯+𝐹𝑒_𝑚.

2. 𝑇^∗𝜃-Extensions of 𝑛-Lie Algebras

To study the 𝑇^∗𝜃-extensions of 𝑛-Lie algebras, we need some definitions and basic facts.

An 𝑛-Lie algebra 𝐿 is a vector space with an 𝑛-ary skew-symmetric operation satisfying𝑥₁,…,𝑥_𝑛=sgn(𝜎)𝑥_𝜎(1),…,𝑥_𝜎(𝑛),(2.1)𝑥₁,…,𝑥_𝑛,𝑦₂,…,𝑦_𝑛=^𝑛_𝑖=1𝑥₁,…,𝑥_𝑖,𝑦₂,…,𝑦_𝑛,…,𝑥_𝑛(2.2) for every 𝑥₁,…,𝑥_𝑛,𝑦₂,…,𝑦_𝑛∈𝐿 and every permutation 𝜎∈𝑆_𝑛. 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,…,𝑥_𝑛] for all 𝑥₁,…,𝑥_𝑛∈𝐿 is called the derived algebra of 𝐿 and is denoted by 𝐿¹.

An 𝑛-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 𝐿^𝑠=0 for some 𝑠≥0, where 𝐿⁰=𝐿 and 𝐿^𝑠 is defined as 𝐿^𝑠=[𝐿^𝑠−1,𝐿,…,𝐿], for 𝑠≥1. An 𝑛-Lie algebra 𝐿 is called abelian if 𝐿¹=0.

Let 𝐿 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) satisfying𝜌𝑥₁,…,𝑥_𝑛,𝑦₂,…,𝑦_𝑛−1=^𝑛_𝑖=1(−1)^𝑛−𝑖𝜌𝑥₁,…,𝑥̂_𝑖,…,𝑥_𝑛𝜌𝑥_𝑖,𝑦₂,…,𝑦_𝑛−2(2.3)−𝜌𝑦𝜌₁𝑥₁,…,𝑥_𝑛−1,𝜌𝑦₁,…,𝑦_𝑛−1=𝜌𝑥₁,…,𝑥_𝑛−1𝜌𝑦₁,…,𝑦_𝑛−1,…,𝑦_𝑛−1𝜌𝑥₁,…,𝑥_𝑛−1=^𝑛_𝑖=1𝜌𝑦₁,…,𝑥₁,…,𝑥_𝑛−1,𝑦_𝑖,…,𝑦_𝑛−1(2.4) for all 𝑥_𝑖,𝑦_𝑖∈𝐿,𝑖=1,…,𝑛, then (𝑉,𝜌) is called a representation of 𝐿 or 𝑉 is an 𝐿-module.

Let 𝜌(𝑥₁,…,𝑥_𝑛−1)=ad(𝑥₁,…,𝑥_𝑛−1) for 𝑥₁,…,𝑥_𝑛−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,…,𝑥_𝑛−1∈𝐿, defines 𝜌^∗∶𝐿^∧𝑛−1→𝐸𝑛𝑑(𝑉^∗),𝜌^∗𝑥₁,…,𝑥_𝑛−1(𝑓)(𝑣)=−𝑓𝜌𝑥₁,…,𝑥_𝑛−1(𝑣).(2.5)   (𝑉^∗,𝜌^∗) is called the dual module of 𝑉. If 𝑉=𝐿 and 𝜌=ad, that is, ad^∗(𝑥₁,…,𝑥_𝑛−1)(𝑓)(𝑥)=−𝑓([𝑥₁,…,𝑥_𝑛−1,𝑥]), (𝐿^∗,ad^∗) is called the coadjoint module of 𝐿.

Definition 2.1. Let 𝐿 be an 𝑛-Lie algebra. If the 𝑛-linear mapping 𝜃∶𝐿^∧𝑛→𝐿^∗ satisfying for all 𝑥_𝑖,𝑦_𝑗∈𝐿,1≤𝑖≤𝑛,2≤𝑗≤𝑛, ^𝑛_𝑖=1+𝜃_𝑛𝑥₁,…,𝑥_𝑖,𝑦₂,…,𝑦_𝑛,…,𝑥_𝑛−𝜃𝑥₁,…,𝑥_𝑛,𝑦₂,…,𝑦_𝑛_𝑖=1(−1)^𝑛−𝑖𝑥₁,…,𝑥̂_𝑖,…,𝑥_𝑛,𝜃𝑥_𝑖,𝑦₂,…,𝑦_𝑛+(−1)^𝑛𝑦₂,…,𝑦_𝑛,𝜃𝑥₁,…,𝑥_𝑛=0,(2.6) then 𝜃 is called a cocycle of 𝐿.

Theorem 2.2. Let 𝐿 be an 𝑛-Lie algebra over 𝐹, and let 𝜃∶𝐿^∧𝑛→𝐿^∗ be a cocycle of 𝐿. Then 𝐿_𝜃(𝐿^∗)=𝐿⊕𝐿^∗ is an 𝑛-Lie algebra in the following multiplication: 𝑦₁+𝑓₁,…,𝑦_𝑛+𝑓_𝑛_𝜃=𝑦₁,…,𝑦_𝑛_𝐿+𝜃𝑦₁,…,𝑦_𝑛+^𝑛_𝑖=1(−1)^𝑛−𝑖ad^∗𝑦₁,…,𝑦_𝑖̂,…,𝑦_𝑛𝑓_𝑖,(2.7) where 𝑦_𝑖∈𝐿,𝑓_𝑖∈𝐿^∗,1≤𝑖≤𝑛.

Proof. It suffices to verify the Jacobi identity (2.2) for 𝐿_𝜃(𝐿^∗). For all 𝑦_𝑖∈𝐿,𝑓_𝑖∈𝐿^∗,1≤𝑖≤2𝑛−1, set 𝑧_𝑖=𝑦_𝑖+𝑓_𝑖, and by identity (2.7) we have 𝑧₁,…,𝑧_𝑛_𝜃,𝑧_𝑛+1,…,𝑧_2𝑛−1_𝜃=𝑦₁+𝑓₁,…,𝑦_𝑛+𝑓_𝑛_𝜃,𝑦_𝑛+1+𝑓_𝑛+1,…,𝑦_2𝑛−1+𝑓_2𝑛−1_𝜃=+(𝑦−₁,1…),_𝑛−1𝑦_𝑛_𝐿,𝑦_𝑛+1,…,𝑦_2𝑛−1_𝐿+𝜃𝑦₁,…,𝑦_𝑛_𝐿,𝑦_𝑛+1,…,𝑦_2𝑛−1+ad^∗𝑦_𝑛+1,…,𝑦_2𝑛−1𝜃𝑦₁,…,𝑦_𝑛ad^∗𝑦_𝑛+1,…,𝑦_2𝑛−1^𝑛_𝑖=1(−1)^𝑖+1ad^∗𝑦₁,…,𝑦_𝑖̂,…,𝑦_𝑛𝑓_𝑖+^𝑛−1_𝑗=1(−1)^{𝑛−𝑗−1}ad^∗𝑦₁,…,𝑦_𝑛_𝐿,𝑦_𝑛+1,…,𝑦̂_𝑛+𝑗,…,𝑦_2𝑛−1𝑓_𝑛+𝑗;(2.8) and for every 1≤𝑖,𝑘≤𝑛, 𝑧₁,…,𝑧_𝑘−1,𝑧_𝑘,𝑧_𝑛+1,…,𝑧_2𝑛−1_𝜃,𝑧_𝑘+1,…,𝑧_𝑛_𝜃=𝑦₁,…,𝑦_𝑘−1,𝑦_𝑘,𝑦_𝑛+1,…,𝑦_2𝑛−1_𝐿,𝑦_𝑘+1,…,𝑦_𝑛_𝐿++𝜃_𝑘−1𝑦₁,…,𝑦_𝑘−1,𝑦_𝑘,𝑦_𝑛+1,…,𝑦_2𝑛−1_𝐿,𝑦_𝑘+1,…,𝑦_𝑛_𝑖=1(−1)^𝑛−𝑖𝜌𝑦₁,…,𝑦_𝑖̂,…,𝑦_𝑘−1,𝑦_𝑘,𝑦_𝑛+1,…,𝑦_2𝑛−1_𝐿,𝑦_𝑘+1,…,𝑦_𝑛𝑓_𝑖++((−−11))^𝑛−𝑘_𝑘+1ad^∗𝑦₁,…,𝑦_𝑘−1,𝑦_𝑘+1,…,𝑦_𝑛𝜃𝑦_𝑘,𝑦_𝑛+1,…,𝑦_2𝑛−1ad^∗𝑦₁,…,𝑦_𝑘−1,𝑦_𝑘+1,…,𝑦_𝑛𝜌𝑦_𝑛+1,…,𝑦_2𝑛−1𝑓_𝑘+ad^∗𝑦₁,…,𝑦_𝑘−1,𝑦_𝑘+1,…,𝑦_𝑛^𝑛−1_𝑖=1(−1)^𝑘+𝑖+1ad^∗𝑦_𝑘,𝑦_𝑛+1,…,𝑦̂_𝑛+𝑖,…,𝑦_2𝑛−1𝑓_𝑛+𝑖+^𝑛_𝑖=_𝑘+1(−1)^𝑛−𝑖ad^∗𝑦₁,…,𝑦_𝑘−1,𝑦_𝑘,𝑦_𝑛+1,…,𝑦_2𝑛−1_𝐿,𝑦_𝑘+1,…,𝑦_𝑛𝑓_𝑖.(2.9) Thanks for identity (2.5), for 1≤𝑚≤𝑛, ad^∗𝑦_𝑛+1,…,𝑦_2𝑛−1ad^∗𝑦₁,…,𝑦_𝑚̂,…,𝑦_𝑛𝑓_𝑚=(−1)^𝑛𝑛−1_𝑗≠_𝑚,𝑗=1ad^∗𝑦₁,…,𝑦_𝑗,𝑦_𝑛+1,…,𝑦_2𝑛−1_𝐿,…,𝑦_𝑚̂,…,𝑦_𝑛𝑓_𝑚.(2.10) For 1≤𝑚≤𝑛−1, by identity (2.3), (−1)^{𝑛−𝑚−1}ad^∗𝑦₁,…,𝑦_𝑛_𝐿,𝑦_𝑛+1,…,𝑦̂_𝑛+𝑚,…,𝑦_2𝑛−1𝑓_𝑛+𝑚=^𝑛_𝑖=1(−1)^{−𝑚−𝑖−1}ad^∗𝑦₁,…,𝑦_𝑖̂,…,𝑦_𝑛ad^∗𝑦_𝑖,𝑦_𝑛+1,…,𝑦̂_𝑛+𝑚,…,𝑦_2𝑛−1𝑓_𝑛+𝑚.(2.11) Therefore, the multiplication of 𝐿_𝜃(𝐿^∗) defined by identity (2.7) satisfies 𝑧₁,…,𝑧_𝑛_𝜃,𝑧_𝑛+1,…,𝑧_2𝑛−1_𝜃=^𝑛_𝑘=1𝑧₁,…,𝑧_𝑘−1,𝑧_𝑘,𝑧_𝑛+1,…,𝑧_2𝑛−1_𝜃,𝑧_𝑘+1,…,𝑧_𝑛_𝜃(2.12) for every 𝑧_𝑖∈𝐿_𝜃(𝐿^∗),1≤𝑖≤2𝑛−1.

Definition 2.3. The 𝑛-Lie algebra 𝐿_𝜃(𝐿^∗)=𝐿⊕𝐿^∗ with multiplication (2.7) is called the 𝑇^∗𝜃-extension of 𝐿. In particular, the 𝑇^∗0-extension corresponding to 𝜃=0 is called the trivial extension of 𝐿 and is denoted by 𝐿₀(𝐿^∗).

Then the multiplication of 𝐿₀(𝐿^∗) is as follows:𝑦₁+𝑓₁,…,𝑦_𝑛+𝑓_𝑛₀=𝑦₁,…,𝑦_𝑛_𝐿+^𝑛_𝑖=1(−1)^𝑛−𝑖ad^∗𝑦₁,…,𝑦_𝑖̂,…,𝑦_𝑛𝑓_𝑖,(2.13) where 𝑦_𝑖∈𝐿,𝑓_𝑖∈𝑉,1≤𝑖≤𝑛.

Theorem 2.4. Let 𝐿 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 𝑇^∗𝜃-extension.(2)If 𝐿 is solvable, then the 𝑇^∗𝜃-extension 𝐿_𝜃(𝐿^∗) is solvable.(3)If 𝐿 is a nilpotent 𝑛-Lie algebra, then every 𝑇^∗𝜃-extension is nilpotent.(4)If 𝜃≠0, then 𝐿_𝜃(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 𝐿_𝜃(𝐿^∗) since [𝐿^∗,𝐿^∗,𝐿_𝜃(𝐿^∗),…,𝐿_𝜃(𝐿^∗)]_𝜃=0, and [𝐿^∗,𝐿_𝜃(𝐿^∗),…,𝐿_𝜃(𝐿^∗)]_𝜃⊆𝐿^∗.
Now let 𝐿 be solvable and 𝐿^(𝑠)=0. By induction on 𝑟, we have 𝐿^𝜃(𝑟+1)𝐿^∗=[𝐿^𝜃(𝑟)𝐿^∗,𝐿^𝜃(𝑟)𝐿^∗,𝐿_𝜃𝐿^∗,…,𝐿_𝜃𝐿^∗_𝜃⊆𝐿^(𝑟+1)+𝜃𝐿^(𝑟),𝐿^(𝑟),𝐿,…,𝐿+𝐿^∗.(2.14) Then we have 𝐿^𝜃(𝑠+1)(𝐿^∗)⊆𝐿^∗. Thanks to result (1), 𝐿^𝜃(𝑠+2)(𝐿^∗)=0. Result (2) follows.
(3) Since 𝐿 is nilpotent, 𝐿^𝑠=[𝐿^𝑠−1,𝐿,…,𝐿]_𝐿=0 for some nonnegative integer 𝑠. For every cocycle 𝜃∶𝐿^𝑛→𝐿^∗, by identity (2.6), 𝐿^1𝜃𝐿^∗⊆𝐿¹+𝜃(𝐿,…,𝐿)+ad^∗(𝐿,…,𝐿)𝐿^∗⊆𝐿¹+𝐿^∗.(2.15)
Inductively, we have 𝐿^𝑠𝜃(𝐿^∗)⊆𝐿^(𝑠)+𝐿^∗=𝐿^∗ since 𝐿^𝑠=0. Then we have 𝐿^𝜃2𝑠(𝐿^∗)⊆ad^∗𝑠(𝐿,…,𝐿)(𝐿^∗). Note that for 𝑓∈ad^∗𝑠(𝐿,…,𝐿)(𝐿^∗), we have 𝑓(𝐿)⊆𝑓(𝐿^𝑠)=0. Thus, 𝐿^𝜃2𝑠(𝐿^∗)=0, that is, 𝐿_𝜃(𝐿^∗) is a nilpotent 𝑛-Lie algebra.
It follows from result (4) that 𝐿 is a subalgebra of 𝐿_𝜃(𝐿^∗) if 𝜃=0.

For constructing 𝑇^∗𝜃-extensions of an 𝑛-Lie algebra 𝐿, we give the following method to get cocycles.

Theorem 2.5. Let 𝐿 be an 𝑛-Lie algebra. Then for every linear mapping 𝜎∶𝐿→𝐿^∗, the skew-symmetric mapping 𝜃_𝜎∶𝐿^∧𝑛→𝐿^∗ given by, for all 𝑥₁,…,𝑥_𝑛∈𝐿, 𝜃_𝜎𝑥₁,…,𝑥_𝑛=𝜎𝑥₁,…,𝑥_𝑛_𝐿−^𝑛_𝑖=1(−1)^𝑛−𝑖ad^∗𝑥₁,…,𝑥̂_𝑖,…,𝑥_𝑛𝜎𝑥_𝑖(2.16) is a cocycle.

Proof. A tedious calculation shows that, for every 𝑥_𝑖,𝑦_𝑖∈𝐿,1≤𝑖,𝑘≤𝑛, 𝜃_𝜎=𝑥𝜎₁,…𝑥,₁𝑥_𝑘−1,𝑦₂,…,𝑦_𝑛,𝑥_𝑘_𝐿,𝑥_𝑘+1,…,𝑥_𝑛+_𝑘−1,…,𝑥_𝑘−1,𝑦₂,…,𝑦_𝑛,𝑥_𝑘_𝐿,𝑥_𝑘+1,…,𝑥_𝑛_𝐿_𝑖=1+(−1()−1)^{𝑛−𝑖−1}_{𝑛−𝑘−1}ad^∗𝑥₁,…,𝑥̂_𝑖,…,𝑥_𝑘−1,𝑦₂,…,𝑦_𝑛,𝑥_𝑘_𝐿,𝑥_𝑘+1,…,𝑥_𝑛𝜎𝑥_𝑖+_𝑛ad^∗𝑥₁,…,𝑥_𝑘−1,𝑥_𝑘+1,…,𝑥_𝑛𝜎𝑥_𝑘_𝑗=_𝑘+1𝜃_𝜎(−1)^{𝑛−𝑗−1}ad^∗𝑥₁,…,𝑦₂,…,𝑦_𝑛,𝑥_𝑘_𝐿,𝑥_𝑘+1,…,𝑥̂_𝑗,…,𝑥_𝑛𝜎𝑥_𝑗;+𝑥₁,_𝑛…,𝑥_𝑛_𝐿,𝑦₂,…,𝑦_𝑛=𝜎𝑥₁,…,𝑥_𝑛_𝐿,𝑦₂,…,𝑦_𝑛_𝐿_𝑖=2+(−1()−_𝑛1)^{𝑛−𝑖−1}ad^∗𝑥₁,…,𝑥_𝑛_𝐿,𝑦₂,…,𝑦_𝑖̂,…,𝑦_𝑛𝜎𝑦_𝑖ad^∗𝑦₂,…,𝑦_𝑛𝜎𝑥₁,…,𝑥_𝑛_𝐿;ad^∗=𝑥₁,…,𝑥̂_𝑘,…,𝑥_𝑛𝜃_𝜎𝑦₂,…,𝑦_𝑛,𝑥_𝑘+ad^∗𝑥₁,…,𝑥̂_𝑘,…,𝑥_𝑛𝜎𝑦₂,…,𝑦_𝑛,𝑥_𝑘_𝐿ad^∗𝑥₁,…,𝑥̂_𝑘,…,𝑥_𝑛^𝑛_𝑖=2−(−1)^𝑛−𝑖ad^∗𝑦₂,…,𝑦_𝑖̂,…,𝑦_𝑛,𝑥_𝑘𝜎𝑦_𝑖ad^∗𝑥₁,…,𝑥̂_𝑘,…,𝑥_𝑛ad^∗𝑦₂,…,𝑦_𝑛𝜎𝑥_𝑘;ad^∗=𝑦₂,…,𝑦_𝑛𝜃_𝑓𝑥₁,…,𝑥_𝑛ad^∗𝑦₂,…,𝑦_𝑛𝜎𝑥₁,…,𝑥_𝑛_𝐿−ad^∗𝑦₂,…,𝑦_𝑛^𝑛_𝑖=1(−1)^𝑛−𝑖ad^∗𝑥₁,…,𝑥̂_𝑖,𝑥_𝑛𝜎𝑥_𝑖.(2.17) Therefore, 𝜃_𝑓 satisfies identity (2.6). The proof is completed.

Theorem 2.6. Let 𝐿 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^∗Γ∶𝐿_𝜃𝐿^∗⟶𝐿_{𝜃+𝜃𝜎}𝐿^∗,Γ(𝑦+𝑓)=𝑦+𝜎(𝑦)+𝑓,(2.18) is an 𝑛-Lie algebra isomorphism.

Proof. It is clear that Γ is a linear isomorphism of the vector space 𝐿⊕𝐿^∗ to itself. Next, for every 𝑓_𝑖∈𝐿^∗,𝑦_𝑖∈𝐿,1≤𝑖≤𝑛, Γ𝑦=₁+Γ𝑓₁,𝑦…₁,𝑦_𝑛+𝑓_𝑛_𝜃,…,𝑦_𝑛_𝐿+𝜃𝑦₁,…,𝑦_𝑛+^𝑛_𝑖=1(−1)^𝑛−𝑖ad^∗𝑦₁,…,𝑦_𝑖,…,𝑦_𝑛𝑓_𝑖=𝑦₁,…,𝑦_𝑛_𝐿+𝜃𝑦₁,…,𝑦_𝑛+𝜎𝑦₁,…,𝑦_𝑛_𝐿+^𝑛_𝑖=1Γ𝑦₁(−1)^𝑛−𝑖ad^∗𝑦₁,…,𝑦_𝑖,…,𝑦_𝑛𝑓_𝑖.+𝑣₁,…,Γ𝑦_𝑛+𝑣_𝑛_{𝜃+𝜃𝜎}=𝑦₁+𝜎𝑦₁+𝑓₁,…,𝑦_𝑛+𝜎𝑦_𝑛+𝑓_𝑛_{𝜃+𝜃𝜎}=𝑦₁,…,𝑦_𝑛_𝐿+𝜃+𝜃_𝜎𝑦₁,…,𝑦_𝑛+^𝑛_𝑖=1=𝑦₁(−1)^𝑛−𝑖ad^∗𝑦₁,…,𝑦_𝑖,…,𝑦_𝑛𝜎𝑦_𝑖+𝑓_𝑖,…,𝑦_𝑛_𝐿+𝜃𝑦₁,…,𝑦_𝑛+𝜎𝑦₁,…,𝑦_𝑛_𝐿+^𝑛_𝑖=1(−1)^𝑛−𝑖ad^∗𝑦₁,…,𝑦_𝑖,…,𝑦_𝑛𝑓_𝑖=Γ𝑦₁+𝑓₁,…,𝑦_𝑛+𝑓_𝑛_𝜃.(2.19) the result follows.

Corollary 2.7. Let 𝐿 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=𝜃_𝜎, then the 𝑇^∗𝜃₁-extension 𝐿_𝜃1(𝐿^∗) is isomorphic to the 𝑇^∗𝜃₂-extension 𝐿_𝜃2(𝐿^∗) of 𝐿.

Proof. If there is a linear mapping 𝜎∶𝐿→𝐿^∗ such that 𝜃₁=𝜃₂+𝜃_𝜎, by Theorem 2.6, the 𝑇^∗𝜃₁-extension 𝐿_𝜃1(𝐿^∗)=𝐿_{𝜃2+𝜃𝜎}(𝐿^∗) is isomorphic to the 𝑇^∗𝜃₂-extension 𝐿_𝜃2(𝐿^∗).

3. The 𝑇^∗𝜃-Extension of (𝑛+1)-Dimensional 𝑛-Lie Algebras

In this section, we study the 𝑇^∗𝜃-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]). Let 𝐿 be an (𝑛+1)-dimensional 𝑛-Lie algebra over 𝐹 and 𝑒₁,𝑒₂,…,𝑒_𝑛+1 a basis of 𝐿 (𝑛≥3). Then one and only one of the following possibilities hold up to isomorphisms. (a) If dim𝐿¹=0, then 𝐿 is an abelian 𝑛-Lie algebra.(b)If dim𝐿¹=1 and letting 𝐿¹=𝐹𝑒₁,in the case that 𝐿¹⊆𝑍(𝐿), (𝑏₁)[𝑒₂,…,𝑒_𝑛+1]=𝑒₁; if 𝐿¹ is not contained in 𝑍(𝐿), (𝑏₂)[𝑒₁,…,𝑒_𝑛]=𝑒₁. (c) If dim𝐿¹=2 and letting 𝐿¹=𝐹𝑒₁+𝐹𝑒₂,(𝑐₁)[𝑒₂,…,𝑒_𝑛+1]=𝑒_1,[𝑒₁,𝑒₃,…,𝑒_𝑛+1]=𝑒₂; (𝑐₂)[𝑒₂,…,𝑒_𝑛+1]=𝛼𝑒₁+𝑒₂,[𝑒₁,𝑒₃,…,𝑒_𝑛+1]=𝑒₂; (𝑐₃)[𝑒₁,𝑒₃,…,𝑒_𝑛+1]=𝑒₁,[𝑒₂,…,𝑒_𝑛+1]=𝑒₂,𝛼∈𝐹,𝛼≠0. (d)If dim𝐿¹=𝑟, 3≤𝑟≤𝑛+1, let 𝐿¹=𝐹𝑒₁+𝐹𝑒₂+…+𝐹𝑒_𝑟. Then(𝑑_𝑟)[𝑒₁,…,𝑒_𝑖̂,…,𝑒_𝑛+1]=𝑒_𝑖,1≤𝑖≤𝑟, where symbol 𝑒_𝑖̂ means that 𝑒_𝑖 is omitted.

We first introduce some notations. Let 𝐿 be an (𝑛+1)-dimensional 𝑛-Lie algebra in the Lemma 3.1, and let 𝑓₁, …,𝑓_𝑛+1 be the basis of 𝐿^∗ satisfying 𝑓_𝑖(𝑒_𝑗)=𝛿_𝑖𝑗,1≤𝑖,𝑗≤𝑛+1. For a cocycle 𝜃∶𝐿^∧𝑛→𝐿^∗𝜃𝑒₁,…,𝑒_𝑗̂,…,𝑒_𝑛+1=^𝑛+1_𝑠=1𝑎_𝑗𝑠𝑓_𝑠,𝑎_𝑗𝑠∈𝐹,1≤𝑗≤𝑛+1.(3.1)

The 𝑇^∗𝜃-extensions of the classes (𝑏_𝑖), (𝑐_𝑗), and (𝑑_𝑟) in Lemma 3.1 are denoted by (𝑏^∗𝑖), (𝑐^∗𝑗) and (𝑑^∗𝑟), respectively.

Theorem 3.2. Let 𝐿 be an (𝑛+1)-dimensional 𝑛-Lie algebra in the Lemma 3.1. Then up to isomorphisms the 𝑇^∗𝜃-extensions of 𝐿 are only of the following possibilities: (𝑎^∗)𝐿_𝜃(𝐿^∗)is abelian (𝑏^∗1)[𝑒₂,…,𝑒_𝑛+1]_𝜃=𝑒₁,[𝑒₁,𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1]_𝜃=∑^𝑛+1𝑠=2𝑎_𝑗𝑠𝑓_𝑠,[𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₁]_𝜃=(−1)^𝑛+1+𝑗𝑓_𝑗,𝑎_𝑗𝑠∈𝐹,2≤𝑗≤𝑛+1. (𝑏^∗2)[𝑒₁,…,𝑒_𝑛]_𝜃=𝑒₁,[𝑒₁,…,𝑒_𝑗̂,…,𝑒_𝑛+1]_𝜃=∑^𝑛+1𝑠=2𝑎_𝑗𝑠𝑓_𝑠,[𝑒₁,…,𝑒_𝑗̂,…,𝑒_𝑛,𝑓₁]_𝜃=(−1)^{𝑛−𝑗+1}𝑓_𝑗,𝑎_𝑗𝑠∈𝐹,1≤𝑗≤𝑛. (𝑐^∗1)[𝑒₁,𝑒₃,…,𝑒_𝑛+1]_𝜃=𝑒₂,[𝑒₂,𝑒₃,…,𝑒_𝑛+1]_𝜃=𝑒₁,[𝑒₃,…,𝑒_𝑛+1,𝑓₂]_𝜃=(−1)^𝑛𝑓₁,[𝑒₁,𝑒₂,…,𝑒_𝑗̂…,𝑒_𝑛+1]_𝜃=∑^𝑛+1𝑠=3𝑎_𝑗𝑠𝑓_𝑠,[𝑒₁,𝑒₃,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₂]_𝜃=(−1)^𝑛−𝑗𝑓_𝑗,[𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₁]_𝜃=(−1)^𝑛−𝑗𝑓_𝑗,𝑎_𝑗𝑠∈𝐹,3≤𝑗≤𝑛+1.(𝑐^∗2)[𝑒₁,𝑒₃,…,𝑒_𝑛+1]_𝜃=𝑒₂,[𝑒₂,𝑒₃,…,𝑒_𝑛+1]_𝜃=𝛼𝑒₁+𝑒₂,[𝑒₂,𝑒₃,…,𝑒_𝑖̂,…,𝑒_𝑛+1,𝑓₁]_𝜃  =(−1)^𝑛−𝑖𝛼𝑓_𝑖,[𝑒₂,𝑒₃,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₂]_𝜃  =(−1)^𝑛−𝑖𝑓_𝑗,[𝑒₁,𝑒₃,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₂]_𝜃  =(−1)^𝑛−𝑗𝑓_𝑗,[𝑒₃,…,…,𝑒_𝑛+1,𝑓₂]_𝜃  =(−1)^𝑛(𝑓₂+𝑓₁),where 𝛼∈𝐹,𝛼≠0,2≤𝑖≤𝑛+1,3≤𝑗≤𝑛+1.(𝑐^∗3)[𝑒₁,𝑒₃,…,𝑒_𝑛+1]_𝜃=𝑒₁,[𝑒₂,𝑒₃,…,𝑒_𝑛+1]_𝜃=𝑒₂,[𝑒₃,…,𝑒_𝑛+1,𝑓₁]_𝜃=(−1)^𝑛𝑓₁,[𝑒₁,𝑒₃,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₁]_𝜃  =(−1)^𝑛−𝑗𝑓_𝑗,3≤𝑗≤𝑛+1,[𝑒₂,…,𝑒_𝑖̂,…,𝑒_𝑛+1,𝑓₂]_𝜃=(−1)^𝑛−𝑖𝑓_𝑖,2≤𝑖≤𝑛+1.(𝑑^∗𝑟)[𝑒₁,…,𝑒_𝑗̂,…,𝑒_𝑛+1]_𝜃=𝑒_𝑗,1≤𝑗≤𝑟,[𝑒₁,𝑒₂,…,𝑒_𝑟,…,𝑒_𝑗̂,…,𝑒_𝑛+1]_𝜃  =∑^{𝑛+1𝑠=𝑟+1}𝑎_𝑗𝑠𝑓_𝑠,𝑎_𝑗𝑠∈𝐹,𝑎_𝑗𝑠∈𝐹,𝑟<𝑗,[𝑒₁,…,𝑒_𝑗̂,…,𝑒_𝑖̂,…,𝑒_𝑛+1,𝑓_𝑖]_𝜃  =(−1)^{𝑛−𝑗+1}𝑓_𝑗,1≤𝑗<𝑖≤𝑟,[𝑒₁,…,𝑒_𝑖̂,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓_𝑖]_𝜃  =(−1)^𝑛−𝑗𝑓_𝑗,1≤𝑖<𝑗≤𝑟,where 3≤𝑟≤𝑛+1.

Proof. Case (𝑎^∗) is trivial. If 𝐿 is case (𝑏₁), let 𝑓₁,…,𝑓_𝑛+1 be a basis of 𝐿^∗ satisfying 𝑓_𝑖(𝑒_𝑗)=𝛿_𝑖𝑗,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,…,𝑒_𝑛+1)=∑^𝑛+1𝑠=1𝑎_1𝑠𝑓_𝑠,𝜃₀(𝑒₁,𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1)=∑^𝑠𝑗=2𝑎_𝑗𝑠𝑓_𝑠,𝑎_𝑗𝑠∈𝐹,2≤𝑗≤𝑛+1. The multiplication of 𝐿_𝜃0(𝐿^∗) in the basis 𝑒₁,…,𝑒_𝑛+1,𝑓₁,…,𝑓_𝑛+1 is 𝑒𝑒₂,…,𝑒_𝑛+1_𝜃0=𝑒₁+∑^𝑛+1_𝑠=1𝑎_1𝑠𝑓_𝑠,𝑎_1𝑠∈𝐹,₁𝑒,𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1_𝜃0=∑^𝑠_𝑗=2𝑎_𝑗𝑠𝑓_𝑠,𝑎_𝑗𝑠∈𝐹,2≤𝑗≤𝑛+1,₂,,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₁_𝜃0=(−1)^𝑛+𝑗+1𝑓_𝑗,2≤𝑗≤𝑛+1.(3.2)
By Theorem 2.5, omitting the computation process, for every linear mapping 𝜎∶𝐿→𝐿^∗, the cocycle 𝜃_𝜎∶𝐿^∧𝑛→𝐿^∗ satisfies 𝜃_𝜎(𝑒₂,𝑒₃,…,𝑒_𝑛+1)=(𝑛+1)𝜎(𝑒₁) and 𝜃_𝜂(𝑒₁,𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1)=0,2≤𝑗≤𝑛+1. Then, define 𝜎𝑒₁=−1𝑛+1𝜃⁰𝑒₂,𝑒₃,…,𝑒_𝑛+1=−^𝑛+1_𝑠=1𝑎_1𝑠𝑓_𝑠,(3.3) and 𝜎(𝑒_𝑖)=0,2≤𝑖≤𝑛+1. Follows Theorem 2.6 that 𝐿_𝜃0(𝐿^∗) is isomorphic to 𝐿_{𝜃0+𝜃𝜎}(𝐿^∗) which with the multiplication (𝑏^∗1).
In the case (𝑏₂), let 𝜃₀∶𝐿^∧𝑛→𝐿^∗ be a cocycle. Omitting the computation process, we have 𝜃₀(𝑒₂,𝑒₃,…,𝑒_𝑛+1)=𝑎₁₁𝑓₁+…+𝑎_1𝑛+1𝑓_𝑛+1,𝜃₀(𝑒₁,𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1)=𝑎_𝑗2𝑓₂+…+𝑎_𝑗𝑛+1𝑓_𝑛+1,𝑗∈2…,𝑛+1. The multiplication table of 𝐿_𝜃0(𝐿^∗) is as follows: 𝑒₁𝑒₂,…,𝑒_𝑛+1_𝜃0=𝑒₁+^𝑛+1_𝑠=1𝑎_1𝑠𝑓_𝑠,,𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1_𝜃0=^𝑛+1_𝑠=2𝑒₂𝑎_𝑗𝑠𝑓_𝑠,,𝑒₃,…,𝑒_𝑖̂,…,𝑒_𝑛+1,𝑓₁_𝜃0=(−1)^𝑛+𝑖+1𝑓_𝑖,2≤𝑖≤𝑛+1.(3.4)
For every linear mapping 𝜎∶𝐿→𝐿^∗, the cocycle 𝜃_𝜎: 𝐿^∧𝑛→𝐿^∗, by Theorem 2.5, omitting the computation process, 𝜃_𝜂(𝑒₁,𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1)=0,2≤𝑗≤𝑛+1,𝜃_𝜂(𝑒₂,…,𝑒_𝑛+1)=(𝑛+1)𝜂(𝑒₁). Then defining 𝜎𝑒₁=−1𝑛+1𝜃𝑒²,…,𝑒_𝑛+1=𝑎₁₁𝑓₁+⋯+𝑎_1𝑛+1𝑓_𝑛+1,𝜂𝑒_𝑗=0,2≤𝑗≤𝑛+1,(3.5) we have 𝐿_{𝜃0+𝜃𝜎}(𝐿^∗) with the multiplication (𝑏^∗2) which is isomorphic to 𝐿_𝜃0(𝐿^∗).
In case (𝑐₁), for every cocycle 𝜃₀∶𝐿^∧𝑛→𝐿^∗, omitting the computation process, we have 𝜃₀(𝑒₁,𝑒₃,…,𝑒_𝑛+1)=∑^𝑛+1𝑠=1𝑎_2𝑠𝑓_𝑠,𝜃₀(𝑒₂,…,𝑒_𝑛+1)=∑^𝑛+1𝑠=1𝑎_1𝑠𝑓_𝑠,𝜃₀(𝑒₁,𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1)=∑^𝑛+1𝑠=3𝑎_𝑗𝑠𝑓_𝑠,𝑗=3,…,𝑛+1. The multiplication table of 𝐿_𝜃0(𝐿^∗) is as follows: 𝑒₁𝑒,₂𝑒₃,…,𝑒_𝑛+1_𝜃0=𝑒₂+^𝑛+1_𝑠=1𝑎_2𝑠𝑓_𝑠,,…,𝑒_𝑛+1_𝜃0=𝑒₁+^𝑛+1_𝑠=1𝑒₁𝑎_1𝑠𝑓_𝑠,,𝑒₂,𝑒₃,…,𝑒_𝑗̂,…,𝑒_𝑛+1_𝜃0=^𝑛+1_𝑠=3𝑒𝑎_𝑗𝑠𝑓_𝑠,3≤𝑗≤𝑛+1,₁𝑒,𝑒₃,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₂_𝜃0=(−1)^𝑛−𝑗𝑓_𝑗,3≤𝑗≤𝑛+1,₂,𝑒₃,…,𝑒_𝑗̂,…𝑒,𝑒_𝑛+1,𝑓₁_𝜃0=(−1)^𝑛−𝑗𝑓_𝑗,3≤𝑗≤𝑛+1,₃,…,𝑒_𝑛+1,𝑓₂_𝜃0=(−1)^𝑛𝑓₁.(3.6)
Define the linear mapping 𝜎∶𝐿→𝐿^∗∶𝜎(𝑒₂)=−(1/(𝑛+1))𝜃₀(𝑒₁,𝑒₃,…,𝑒_𝑛+1), 𝜎(𝑒₁)=−(1/(𝑛+1))𝜃₀(𝑒₂,𝑒₃,…,𝑒_𝑛+1) and others are zero. By the direct computation 𝜃_𝜎𝑒₁,𝑒₃,…,𝑒_𝑛+1=(𝑛+1)𝜂₀𝑒₂,𝜃_𝜎𝑒₂,𝑒₃,…,𝑒_𝑛+1=(𝑛+1)𝜂₀𝑒₁.(3.7) Then 𝐿_{𝜃0+𝜃𝜎}(𝐿^∗) has the multiplication (𝑐^∗1).
In the case (𝑐₂), for every cocycle 𝜃₀∶𝐿^∧𝑛→𝐿^∗, we have 𝜃₀(𝑒₁,𝑒₃,…,𝑒_𝑛+1)=∑^𝑛+1𝑠=1𝑎_2𝑠𝑓_𝑠,𝜃₀(𝑒₂,…,𝑒_𝑛+1)=∑^𝑛+1𝑠=1𝑎_1𝑠𝑓_𝑠,𝜃₀(𝑒₁,𝑒₂,…,𝑒_𝑗̂,…,𝑒_𝑛+1)=∑^𝑛+1𝑠=3𝑎_𝑗𝑠𝑓_𝑠,𝑗=3,…,𝑛+1. The multiplication table of 𝐿_𝜃0(𝐿^∗) is as follows: 𝑒₂𝑒₁,𝑒₃,…,𝑒_𝑛+1_𝜃0=𝑒₂+^𝑛+1_𝑠=1𝑎_2𝑠𝑓_𝑠,,…,𝑒_𝑛+1_𝜃0=𝛼𝑒₁+𝑒₂+^𝑛+1_𝑠=1𝑒₁𝑎_1𝑠𝑓_𝑠,,𝑒₂,𝑒₃,…,𝑒_𝑗̂,…,𝑒_𝑛+1_𝜃0=^𝑛+1_𝑠=3𝑒𝑎_𝑗𝑠𝑓_𝑠,3≤𝑗≤𝑛+1,₂𝑒,…,𝑒_𝑖̂,…,𝑒_𝑛+1,𝑓₁_𝜃0=(−1)^𝑛+𝑖𝛼𝑓_𝑖,2≤𝑖≤𝑛+1,₂𝑒,𝑒₃,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₂_𝜃0=(−1)^𝑛−𝑗𝑓_𝑗,3≤𝑗≤𝑛+1,₁,𝑒₃,…𝑒,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₂_𝜃0=(−1)^𝑛−𝑗𝑓_𝑗,3≤𝑗≤𝑛+1,₃,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓₂_𝜃0=(−1)^𝑛𝑓₂+𝑓₁.(3.8)
Define linear mapping 𝜎∶𝐿→𝐿^∗∶𝜎(𝑒₂)=−(1/(𝑛+1))𝜃₀(𝑒₁,𝑒₃,…,𝑒_𝑛+1), 𝜎(𝑒₁)=(1/𝛼(𝑛+1))(𝜃₀(𝑒₁,𝑒₃,…,𝑒_𝑛+1)−𝜃₀(𝑒₂,𝑒₃,…,𝑒_𝑛+1)). Then we obtain 𝜃_𝜎(𝑒₁,𝑒₃,…,𝑒_𝑛+1)=(𝑛+1)𝜎(𝑒₂)=−𝜃₀(𝑒₁,𝑒₃,…,𝑒_𝑛+1),𝜃_𝜎(𝑒₂,𝑒₃,…,𝑒_𝑛+1)=(𝑛+1)𝜎(𝛼𝑒₁+𝑒₂)=−𝜃₀(𝑒₂,𝑒₃,…,𝑒_𝑛+1) and others are zero. Therefore, 𝐿_{𝜃0+𝜃𝜎}(𝐿^∗) has the multiplication (𝑐^∗2) in the basis 𝑒₁,…,𝑒_𝑛+1,𝑓₁,…,𝑓_𝑛+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,…,𝑒_𝑛+1), 𝜂₀(𝑒₂)=−(1/(𝑛+1))𝜃₀(𝑒₂,𝑒₃,…,𝑒_𝑛+1), we have 𝜃_𝜎𝑒₁,𝑒₃,…,𝑒_𝑛+1=(𝑛+1)𝜎𝑒₁=−𝜃₀𝑒₁,𝑒₃,…,𝑒_𝑛+1,(3.9)  𝜃_𝜎(𝑒₂,𝑒₃,…,𝑒_𝑛+1)=(𝑛+1)𝜎(𝑒₂)=−𝜃₀(𝑒₂,𝑒₃,…,𝑒_𝑛+1) and others are zero. Then 𝐿_{𝜃0+𝜃𝜎}(𝐿^∗) has the multiplication (𝑐^∗3) in the basis 𝑒₁,…,𝑒_𝑛+1,𝑓₁,…,𝑓_𝑛+1.
Lastly, if 𝐿 is case (𝑑_𝑟), 3≤𝑟≤𝑛+1, for every cocycle 𝜃₀∶𝐿^∧𝑛→𝐿^∗, we have 𝜃₀(𝑒₁,,…,𝑒_𝑖̂,…,𝑒_𝑛+1)=∑^𝑛+1𝑠=1𝑎_𝑖𝑠𝑓_𝑠,1≤𝑖≤𝑟,𝜃₀(𝑒₁,…,𝑒_𝑟,…,𝑒_𝑗̂,…,𝑒_𝑛+1)=∑^{𝑛+1𝑗=𝑟+1}𝑎_𝑗𝑠𝑓_𝑠,𝑟<𝑗≤𝑛+1. By the direct computation, the multiplication of 𝐿_𝜃0(𝐿^∗) is as follows: 𝑒𝑒₁,…,𝑒_𝑖̂,…,𝑒_𝑛+1_𝜃0=𝑒_𝑖+^𝑛+1_𝑠=1𝑎_𝑖𝑠𝑓_𝑠,1≤𝑖≤𝑟,₁,…,𝑒_𝑟,…,𝑒_𝑗̂,…,𝑒_𝑛+1_𝜃0=^𝑛+1_𝑠_=𝑟+1𝑒𝑎_𝑗𝑠𝑓_𝑗,𝑟<𝑗≤𝑛+1,₁𝑒,…,𝑒_𝑗̂,…,𝑒_𝑖̂,…,𝑒_𝑛+1,𝑓_𝑖_𝜃0=(−1)^{𝑛−𝑗+1}𝑓_𝑗,1≤𝑗<𝑖≤𝑟,₁,…,𝑒_𝑖̂,…,𝑒_𝑗̂,…,𝑒_𝑛+1,𝑓_𝑖_𝜃0=(−1)^𝑛−𝑗𝑓_𝑗,1≤𝑖<𝑗≤𝑟.(3.10) Define linear mapping 𝜎∶𝐿→𝐿^∗∶𝜎(𝑒_𝑖)=−(1/(𝑛+1))𝜃₀(𝑒₁,…,𝑒_𝑖̂,…,𝑒_𝑛+1),1≤𝑖≤𝑟, and 𝜎(𝑒_𝑖)=0 if 𝑟<𝑖. Then we obtain 𝜃_𝜎(𝑒₁,…,𝑒_𝑖̂,…,𝑒_𝑛+1)=(𝑛+1)𝜎(𝑒_𝑖)=−𝜃₀(𝑒₁,…,𝑒_𝑖̂,…,𝑒_𝑛+1) for 1≤𝑖≤𝑟, and 𝜃_𝜎(𝑒₁,…,𝑒_𝑖̂,…,𝑒_𝑛+1)=0 if 𝑖>𝑟. Therefore, 𝐿_{𝜃0+𝜃𝜎}(𝐿^∗) with the multiplication (𝑑^∗𝑟) in the basis 𝑒₁,…,𝑒_𝑛+1,𝑓₁,…,𝑓_𝑛+1 and 𝐿_𝜃0(𝐿^∗) is isomorphic to 𝐿_{𝜃0+𝜃𝜎}(𝐿^∗).

Acknowledgments

This project partially supported by NSF (10871192) of China, NSF (A2010000194) of Hebei Province, China.