Abstract

The paper is mainly concerned with π‘‡βˆ—πœƒ-extensions of 𝑛-Lie algebras. The π‘‡βˆ—πœƒ-extension πΏπœƒ(πΏβˆ—) of an 𝑛-Lie algebra 𝐿 by a cocycle ΞΈ 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 π‘‡βˆ—πœƒ-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 π‘‡βˆ—-extension of a Lie algebra and showed that each solvable quadratic Lie algebra over an algebraically closed field of characteristic zero is either a π‘‡βˆ—-extension or a nondegenerate ideal of codimension 1 in a π‘‡βˆ—-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 π‘‡πœƒ-extension and π‘‡βˆ—-extension of metric 3-Lie algebras and provided a sufficient and necessary condition of a π‘‡βˆ—-extension of 3-Lie algebra admitting a metric.

This paper defines the π‘‡βˆ—πœƒ-extension of an 𝑛-Lie algebra 𝐿 by the coadjoint module πΏβˆ— and a cocycle πœƒ from πΏβˆ§π‘› on to the dual space πΏβˆ— of 𝐿. The main result of the paper is the complete classification of the π‘‡βˆ—πœƒ-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 Μ‚π‘₯ means that π‘₯ is omitted. If 𝐿 is a vector space over a field 𝐹 with a basis 𝑒1,β‹―,π‘’π‘š, then 𝑉 can be denoted by 𝑉=𝐹𝑒1+β‹―+πΉπ‘’π‘š.

2. π‘‡βˆ—πœƒ-Extensions of 𝑛-Lie Algebras

To study the π‘‡βˆ—πœƒ-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 satisfyingξ€Ίπ‘₯1,…,π‘₯𝑛π‘₯=sgn(𝜎)𝜎(1),…,π‘₯𝜎(𝑛)ξ€»,(2.1)π‘₯ξ€Ίξ€Ί1,…,π‘₯𝑛,𝑦2,…,𝑦𝑛=𝑛𝑖=1ξ€Ίπ‘₯1ξ€Ίπ‘₯,…,𝑖,𝑦2,…,𝑦𝑛,…,π‘₯𝑛(2.2) for every π‘₯1,…,π‘₯𝑛,𝑦2,…,π‘¦π‘›βˆˆπΏ and every permutation πœŽβˆˆπ‘†π‘›. Identity (2.2) is called the generalized Jacobi identity. A subspace 𝐡 of 𝐿 is referred to as a subalgebra (ideal) of 𝐿 if [𝐡,…,𝐡]βŠ†π΅ ([𝐡,𝐿,…,𝐿]βŠ†π΅). In particular, the subalgebra generated by [π‘₯1,…,π‘₯𝑛] for all π‘₯1,…,π‘₯π‘›βˆˆπΏ 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 𝑠β‰₯0, where 𝐿(0)=𝐿 and 𝐿(𝑠) is defined as 𝐿(𝑠+1)=[𝐿(𝑠),𝐿(𝑠),𝐿,…,𝐿] for 𝑠β‰₯0. An ideal 𝐿 is called nilpotent if 𝐿𝑠=0 for some 𝑠β‰₯0, where 𝐿0=𝐿 and 𝐿𝑠 is defined as 𝐿𝑠=[πΏπ‘ βˆ’1,𝐿,…,𝐿], for 𝑠β‰₯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∧(π‘›βˆ’1)→𝐸𝑛𝑑(𝑉) satisfying𝜌π‘₯ξ€·ξ€Ί1,…,π‘₯𝑛,𝑦2,…,π‘¦π‘›βˆ’1ξ€Έ=𝑛𝑖=1(βˆ’1)π‘›βˆ’π‘–πœŒξ€·π‘₯1,…,Μ‚π‘₯𝑖,…,π‘₯π‘›ξ€ΈπœŒξ€·π‘₯𝑖,𝑦2,…,π‘¦π‘›βˆ’2ξ€Έ(2.3)ξ€ΊπœŒξ€·π‘₯1,…,π‘₯π‘›βˆ’1𝑦,𝜌1,…,π‘¦π‘›βˆ’1ξ€·π‘₯ξ€Έξ€»=𝜌1,…,π‘₯π‘›βˆ’1ξ€ΈπœŒξ€·π‘¦1,…,π‘¦π‘›βˆ’1ξ€Έξ€·π‘¦βˆ’πœŒ1,…,π‘¦π‘›βˆ’1ξ€ΈπœŒξ€·π‘₯1,…,π‘₯π‘›βˆ’1ξ€Έ=𝑛𝑖=1πœŒξ€·π‘¦1ξ€Ίπ‘₯,…,1,…,π‘₯π‘›βˆ’1,𝑦𝑖,…,π‘¦π‘›βˆ’1ξ€Έ(2.4) for all π‘₯𝑖,π‘¦π‘–βˆˆπΏ,𝑖=1,…,𝑛, then (𝑉,𝜌) is called a representation of 𝐿 or 𝑉 is an 𝐿-module.

Let 𝜌(π‘₯1,…,π‘₯π‘›βˆ’1)=ad(π‘₯1,…,π‘₯π‘›βˆ’1) for π‘₯1,…,π‘₯π‘›βˆ’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 π‘‰βˆ— of 𝑉 is an 𝐿-module in the following way. For π‘“βˆˆπ‘‰βˆ—,π‘£βˆˆπ‘‰,π‘₯1,…,π‘₯π‘›βˆ’1∈𝐿, defines πœŒβˆ—βˆΆπΏβˆ§π‘›βˆ’1→𝐸𝑛𝑑(π‘‰βˆ—),πœŒβˆ—ξ€·π‘₯1,…,π‘₯π‘›βˆ’1ξ€Έξ€·πœŒξ€·π‘₯(𝑓)(𝑣)=βˆ’π‘“1,…,π‘₯π‘›βˆ’1ξ€Έξ€Έ.(𝑣)(2.5)   (π‘‰βˆ—,πœŒβˆ—) is called the dual module of 𝑉. If 𝑉=𝐿 and 𝜌=ad, that is, adβˆ—(π‘₯1,…,π‘₯π‘›βˆ’1)(𝑓)(π‘₯)=βˆ’π‘“([π‘₯1,…,π‘₯π‘›βˆ’1,π‘₯]), (πΏβˆ—,adβˆ—) is called the coadjoint module of 𝐿.

Definition 2.1. Let 𝐿 be an 𝑛-Lie algebra. If the 𝑛-linear mapping πœƒβˆΆπΏβˆ§π‘›β†’πΏβˆ— satisfying for all π‘₯𝑖,π‘¦π‘—βˆˆπΏ,1≀𝑖≀𝑛,2≀𝑗≀𝑛, 𝑛𝑖=1πœƒξ€·π‘₯1ξ€Ίπ‘₯,…,𝑖,𝑦2,…,𝑦𝑛,…,π‘₯𝑛π‘₯βˆ’πœƒξ€·ξ€Ί1,…,π‘₯𝑛,𝑦2,…,𝑦𝑛+𝑛𝑖=1(βˆ’1)π‘›βˆ’π‘–ξ€Ίπ‘₯1,…,Μ‚π‘₯𝑖,…,π‘₯𝑛π‘₯,πœƒπ‘–,𝑦2,…,𝑦𝑛+(βˆ’1)𝑛𝑦2,…,𝑦𝑛π‘₯,πœƒ1,…,π‘₯𝑛=0,(2.6) then πœƒ is called a cocycle of 𝐿.

Theorem 2.2. Let 𝐿 be an 𝑛-Lie algebra over 𝐹, and let πœƒβˆΆπΏβˆ§π‘›β†’πΏβˆ— be a cocycle of 𝐿. Then πΏπœƒ(πΏβˆ—)=πΏβŠ•πΏβˆ— is an 𝑛-Lie algebra in the following multiplication: 𝑦1+𝑓1,…,𝑦𝑛+π‘“π‘›ξ€»πœƒ=𝑦1,…,𝑦𝑛𝐿𝑦+πœƒ1,…,𝑦𝑛+𝑛𝑖=1(βˆ’1)π‘›βˆ’π‘–adβˆ—ξ€·π‘¦1,…,̂𝑦𝑖,…,𝑦𝑛𝑓𝑖,(2.7) where π‘¦π‘–βˆˆπΏ,π‘“π‘–βˆˆπΏβˆ—,1≀𝑖≀𝑛.

Proof. It suffices to verify the Jacobi identity (2.2) for πΏπœƒ(πΏβˆ—). For all π‘¦π‘–βˆˆπΏ,π‘“π‘–βˆˆπΏβˆ—,1≀𝑖≀2π‘›βˆ’1, set 𝑧𝑖=𝑦𝑖+𝑓𝑖, and by identity (2.7) we have 𝑧1,…,π‘§π‘›ξ€»πœƒ,𝑧𝑛+1,…,𝑧2π‘›βˆ’1ξ€»πœƒ=𝑦1+𝑓1,…,𝑦𝑛+π‘“π‘›ξ€»πœƒ,𝑦𝑛+1+𝑓𝑛+1,…,𝑦2π‘›βˆ’1+𝑓2π‘›βˆ’1ξ€»πœƒ=𝑦1,…,𝑦𝑛𝐿,𝑦𝑛+1,…,𝑦2π‘›βˆ’1𝐿𝑦+πœƒξ€·ξ€Ί1,…,𝑦𝑛𝐿,𝑦𝑛+1,…,𝑦2π‘›βˆ’1ξ€Έ+(βˆ’1)π‘›βˆ’1adβˆ—ξ€·π‘¦π‘›+1,…,𝑦2π‘›βˆ’1ξ€Έπœƒξ€·π‘¦1,…,𝑦𝑛+adβˆ—ξ€·π‘¦π‘›+1,…,𝑦2π‘›βˆ’1𝑛𝑖=1(βˆ’1)𝑖+1adβˆ—ξ€·π‘¦1,…,̂𝑦𝑖,…,𝑦𝑛𝑓𝑖+π‘›βˆ’1𝑗=1(βˆ’1)π‘›βˆ’π‘—βˆ’1adβˆ—π‘¦ξ€·ξ€Ί1,…,𝑦𝑛𝐿,𝑦𝑛+1,…,̂𝑦𝑛+𝑗,…,𝑦2π‘›βˆ’1𝑓𝑛+𝑗;(2.8) and for every 1≀𝑖,π‘˜β‰€π‘›, 𝑧1,…,π‘§π‘˜βˆ’1,ξ€Ίπ‘§π‘˜,𝑧𝑛+1,…,𝑧2π‘›βˆ’1ξ€»πœƒ,π‘§π‘˜+1,…,π‘§π‘›ξ€»πœƒ=𝑦1,…,π‘¦π‘˜βˆ’1,ξ€Ίπ‘¦π‘˜,𝑦𝑛+1,…,𝑦2π‘›βˆ’1𝐿,π‘¦π‘˜+1,…,𝑦𝑛𝐿𝑦+πœƒ1,…,π‘¦π‘˜βˆ’1,ξ€Ίπ‘¦π‘˜,𝑦𝑛+1,…,𝑦2π‘›βˆ’1𝐿,π‘¦π‘˜+1,…,𝑦𝑛+π‘˜βˆ’1𝑖=1(βˆ’1)π‘›βˆ’π‘–πœŒξ€·π‘¦1,…,̂𝑦𝑖,…,π‘¦π‘˜βˆ’1,ξ€Ίπ‘¦π‘˜,𝑦𝑛+1,…,𝑦2π‘›βˆ’1𝐿,π‘¦π‘˜+1,…,𝑦𝑛𝑓𝑖+(βˆ’1)π‘›βˆ’π‘˜adβˆ—ξ€·π‘¦1,…,π‘¦π‘˜βˆ’1,π‘¦π‘˜+1,…,π‘¦π‘›ξ€Έπœƒξ€·π‘¦π‘˜,𝑦𝑛+1,…,𝑦2π‘›βˆ’1ξ€Έ+(βˆ’1)π‘˜+1adβˆ—ξ€·π‘¦1,…,π‘¦π‘˜βˆ’1,π‘¦π‘˜+1,…,π‘¦π‘›ξ€ΈπœŒξ€·π‘¦π‘›+1,…,𝑦2π‘›βˆ’1ξ€Έπ‘“π‘˜+adβˆ—ξ€·π‘¦1,…,π‘¦π‘˜βˆ’1,π‘¦π‘˜+1,…,π‘¦π‘›ξ€Έπ‘›βˆ’1𝑖=1(βˆ’1)π‘˜+𝑖+1adβˆ—ξ€·π‘¦π‘˜,𝑦𝑛+1,…,̂𝑦𝑛+𝑖,…,𝑦2π‘›βˆ’1𝑓𝑛+𝑖+𝑛𝑖=π‘˜+1(βˆ’1)π‘›βˆ’π‘–adβˆ—ξ€·π‘¦1,…,π‘¦π‘˜βˆ’1,ξ€Ίπ‘¦π‘˜,𝑦𝑛+1,…,𝑦2π‘›βˆ’1𝐿,π‘¦π‘˜+1,…,𝑦𝑛𝑓𝑖.(2.9) Thanks for identity (2.5), for 1β‰€π‘šβ‰€π‘›, adβˆ—ξ€·π‘¦π‘›+1,…,𝑦2π‘›βˆ’1ξ€Έadβˆ—ξ€·π‘¦1,…,Μ‚π‘¦π‘š,…,π‘¦π‘›ξ€Έπ‘“π‘š=(βˆ’1)π‘›π‘›βˆ’1ξ“π‘—β‰ π‘š,𝑗=1adβˆ—ξ€·π‘¦1𝑦,…,𝑗,𝑦𝑛+1,…,𝑦2π‘›βˆ’1𝐿,…,Μ‚π‘¦π‘š,…,π‘¦π‘›ξ€Έπ‘“π‘š.(2.10) For 1β‰€π‘šβ‰€π‘›βˆ’1, by identity (2.3), (βˆ’1)π‘›βˆ’π‘šβˆ’1adβˆ—π‘¦ξ€·ξ€Ί1,…,𝑦𝑛𝐿,𝑦𝑛+1,…,̂𝑦𝑛+π‘š,…,𝑦2π‘›βˆ’1𝑓𝑛+π‘š=𝑛𝑖=1(βˆ’1)βˆ’π‘šβˆ’π‘–βˆ’1adβˆ—ξ€·π‘¦1,…,̂𝑦𝑖,…,𝑦𝑛adβˆ—ξ€·π‘¦π‘–,𝑦𝑛+1,…,̂𝑦𝑛+π‘š,…,𝑦2π‘›βˆ’1𝑓𝑛+π‘š.(2.11) Therefore, the multiplication of πΏπœƒ(πΏβˆ—) defined by identity (2.7) satisfies 𝑧1,…,π‘§π‘›ξ€»πœƒ,𝑧𝑛+1,…,𝑧2π‘›βˆ’1ξ€»πœƒ=π‘›ξ“π‘˜=1𝑧1,…,π‘§π‘˜βˆ’1,ξ€Ίπ‘§π‘˜,𝑧𝑛+1,…,𝑧2π‘›βˆ’1ξ€»πœƒ,π‘§π‘˜+1,…,π‘§π‘›ξ€»πœƒ(2.12) for every π‘§π‘–βˆˆπΏπœƒ(πΏβˆ—),1≀𝑖≀2π‘›βˆ’1.

Definition 2.3. The 𝑛-Lie algebra πΏπœƒ(πΏβˆ—)=πΏβŠ•πΏβˆ— with multiplication (2.7) is called the π‘‡βˆ—πœƒ-extension of 𝐿. In particular, the π‘‡βˆ—0-extension corresponding to πœƒ=0 is called the trivial extension of 𝐿 and is denoted by 𝐿0(πΏβˆ—).

Then the multiplication of 𝐿0(πΏβˆ—) is as follows:𝑦1+𝑓1,…,𝑦𝑛+𝑓𝑛0=𝑦1,…,𝑦𝑛𝐿+𝑛𝑖=1(βˆ’1)π‘›βˆ’π‘–adβˆ—ξ€·π‘¦1,…,̂𝑦𝑖,…,𝑦𝑛𝑓𝑖,(2.13) where π‘¦π‘–βˆˆπΏ,π‘“π‘–βˆˆπ‘‰,1≀𝑖≀𝑛.

Theorem 2.4. Let 𝐿 be an 𝑛-Lie algebra, and let πœƒβˆΆπΏβˆ§π‘›β†’πΏβˆ— be a cocycle of 𝐿. Then one has the following results. (1)πΏβˆ— is an abelian ideal of the π‘‡βˆ—πœƒ-extension.(2)If 𝐿 is solvable, then the π‘‡βˆ—πœƒ-extension πΏπœƒ(πΏβˆ—) is solvable.(3)If 𝐿 is a nilpotent 𝑛-Lie algebra, then every π‘‡βˆ—πœƒ-extension is nilpotent.(4)If πœƒβ‰ 0, then πΏπœƒ(πΏβˆ—) is an essential extension of 𝐿 by the module πΏβˆ—. If πœƒ=0, 𝐿0(πΏβˆ—) is a nonessential extension of 𝐿.

Proof. From identity (2.7), πΏβˆ— is an abelian ideal of πΏπœƒ(πΏβˆ—) since [πΏβˆ—,πΏβˆ—,πΏπœƒ(πΏβˆ—),…,πΏπœƒ(πΏβˆ—)]πœƒ=0, and [πΏβˆ—,πΏπœƒ(πΏβˆ—),…,πΏπœƒ(πΏβˆ—)]πœƒβŠ†πΏβˆ—.
Now let 𝐿 be solvable and 𝐿(𝑠)=0. By induction on π‘Ÿ, we have ξ‚ƒπΏπœƒ(π‘Ÿ+1)ξ€·πΏβˆ—ξ€Έ=[πΏπœƒ(π‘Ÿ)ξ€·πΏβˆ—ξ€Έ,πΏπœƒ(π‘Ÿ)ξ€·πΏβˆ—ξ€Έ,πΏπœƒξ€·πΏβˆ—ξ€Έ,…,πΏπœƒξ€·πΏβˆ—ξ€Έξ‚„πœƒβŠ†πΏ(π‘Ÿ+1)𝐿+πœƒ(π‘Ÿ),𝐿(π‘Ÿ)ξ€Έ,𝐿,…,𝐿+πΏβˆ—.(2.14) Then we have πΏπœƒ(𝑠+1)(πΏβˆ—)βŠ†πΏβˆ—. Thanks to result (1), πΏπœƒ(𝑠+2)(πΏβˆ—)=0. Result (2) follows.
(3) Since 𝐿 is nilpotent, 𝐿𝑠=[πΏπ‘ βˆ’1,𝐿,…,𝐿]𝐿=0 for some nonnegative integer 𝑠. For every cocycle πœƒβˆΆπΏπ‘›β†’πΏβˆ—, by identity (2.6), 𝐿1πœƒξ€·πΏβˆ—ξ€ΈβŠ†πΏ1+πœƒ(𝐿,…,𝐿)+adβˆ—ξ€·πΏ(𝐿,…,𝐿)βˆ—ξ€ΈβŠ†πΏ1+πΏβˆ—.(2.15)
Inductively, we have πΏπ‘ πœƒ(πΏβˆ—)βŠ†πΏ(𝑠)+πΏβˆ—=πΏβˆ— since 𝐿𝑠=0. Then we have πΏπœƒ2𝑠(πΏβˆ—)βŠ†adβˆ—π‘ (𝐿,…,𝐿)(πΏβˆ—). Note that for π‘“βˆˆadβˆ—π‘ (𝐿,…,𝐿)(πΏβˆ—), we have 𝑓(𝐿)βŠ†π‘“(𝐿𝑠)=0. Thus, πΏπœƒ2𝑠(πΏβˆ—)=0, that is, πΏπœƒ(πΏβˆ—) is a nilpotent 𝑛-Lie algebra.
It follows from result (4) that 𝐿 is a subalgebra of πΏπœƒ(πΏβˆ—) if πœƒ=0.

For constructing π‘‡βˆ—πœƒ-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 πœŽβˆΆπΏβ†’πΏβˆ—, the skew-symmetric mapping πœƒπœŽβˆΆπΏβˆ§π‘›β†’πΏβˆ— given by, for all π‘₯1,…,π‘₯π‘›βˆˆπΏ, πœƒπœŽξ€·π‘₯1,…,π‘₯𝑛π‘₯=πœŽξ€·ξ€Ί1,…,π‘₯π‘›ξ€»πΏξ€Έβˆ’π‘›ξ“π‘–=1(βˆ’1)π‘›βˆ’π‘–adβˆ—ξ€·π‘₯1,…,Μ‚π‘₯𝑖,…,π‘₯π‘›ξ€ΈπœŽξ€·π‘₯𝑖(2.16) is a cocycle.

Proof. A tedious calculation shows that, for every π‘₯𝑖,π‘¦π‘–βˆˆπΏ,1≀𝑖,π‘˜β‰€π‘›, πœƒπœŽξ€·π‘₯1,…,π‘₯π‘˜βˆ’1,𝑦2,…,𝑦𝑛,π‘₯π‘˜ξ€»πΏ,π‘₯π‘˜+1,…,π‘₯𝑛π‘₯=𝜎1,…,π‘₯π‘˜βˆ’1,𝑦2,…,𝑦𝑛,π‘₯π‘˜ξ€»πΏ,π‘₯π‘˜+1,…,π‘₯𝑛𝐿+π‘˜βˆ’1𝑖=1(βˆ’1)π‘›βˆ’π‘–βˆ’1adβˆ—ξ€·π‘₯1,…,Μ‚π‘₯𝑖,…,π‘₯π‘˜βˆ’1,𝑦2,…,𝑦𝑛,π‘₯π‘˜ξ€»πΏ,π‘₯π‘˜+1,…,π‘₯π‘›ξ€ΈπœŽξ€·π‘₯𝑖+(βˆ’1)π‘›βˆ’π‘˜βˆ’1adβˆ—ξ€·π‘₯1,…,π‘₯π‘˜βˆ’1,π‘₯π‘˜+1,…,π‘₯π‘›ξ€ΈπœŽξ€·π‘₯π‘˜ξ€Έ+𝑛𝑗=π‘˜+1(βˆ’1)π‘›βˆ’π‘—βˆ’1adβˆ—ξ€·π‘₯1𝑦,…,2,…,𝑦𝑛,π‘₯π‘˜ξ€»πΏ,π‘₯π‘˜+1,…,Μ‚π‘₯𝑗,…,π‘₯π‘›ξ€ΈπœŽξ€·π‘₯𝑗;πœƒπœŽπ‘₯ξ€·ξ€Ί1,…,π‘₯𝑛𝐿,𝑦2,…,𝑦𝑛π‘₯=πœŽξ€Ίξ€Ί1,…,π‘₯𝑛𝐿,𝑦2,…,𝑦𝑛𝐿+𝑛𝑖=2(βˆ’1)π‘›βˆ’π‘–βˆ’1adβˆ—π‘₯ξ€·ξ€Ί1,…,π‘₯𝑛𝐿,𝑦2,…,̂𝑦𝑖,…,π‘¦π‘›ξ€ΈπœŽξ€·π‘¦π‘–ξ€Έ+(βˆ’1)𝑛adβˆ—ξ€·π‘¦2,…,π‘¦π‘›ξ€ΈπœŽπ‘₯ξ€·ξ€Ί1,…,π‘₯𝑛𝐿;adβˆ—ξ€·π‘₯1,…,Μ‚π‘₯π‘˜,…,π‘₯π‘›ξ€ΈπœƒπœŽξ€·π‘¦2,…,𝑦𝑛,π‘₯π‘˜ξ€Έ=adβˆ—ξ€·π‘₯1,…,Μ‚π‘₯π‘˜,…,π‘₯π‘›ξ€ΈπœŽπ‘¦ξ€·ξ€Ί2,…,𝑦𝑛,π‘₯π‘˜ξ€»πΏξ€Έ+adβˆ—ξ€·π‘₯1,…,Μ‚π‘₯π‘˜,…,π‘₯𝑛𝑛𝑖=2(βˆ’1)π‘›βˆ’π‘–adβˆ—ξ€·π‘¦2,…,̂𝑦𝑖,…,𝑦𝑛,π‘₯π‘˜ξ€ΈπœŽξ€·π‘¦π‘–ξ€Έβˆ’adβˆ—ξ€·π‘₯1,…,Μ‚π‘₯π‘˜,…,π‘₯𝑛adβˆ—ξ€·π‘¦2,…,π‘¦π‘›ξ€ΈπœŽξ€·π‘₯π‘˜ξ€Έ;adβˆ—ξ€·π‘¦2,…,π‘¦π‘›ξ€Έπœƒπ‘“ξ€·π‘₯1,…,π‘₯𝑛=adβˆ—ξ€·π‘¦2,…,π‘¦π‘›ξ€ΈπœŽπ‘₯ξ€·ξ€Ί1,…,π‘₯π‘›ξ€»πΏξ€Έβˆ’adβˆ—ξ€·π‘¦2,…,𝑦𝑛𝑛𝑖=1(βˆ’1)π‘›βˆ’π‘–adβˆ—ξ€·π‘₯1,…,Μ‚π‘₯𝑖,π‘₯π‘›ξ€ΈπœŽξ€·π‘₯𝑖.(2.17) Therefore, πœƒπ‘“ satisfies identity (2.6). The proof is completed.

Theorem 2.6. Let 𝐿 be an 𝑛-Lie algebra, and let πœƒβˆΆπΏβˆ§π‘›β†’πΏβˆ— be a cocycle. Then for every linear mapping πœŽβˆΆπΏβ†’πΏβˆ—, for all π‘¦βˆˆπΏ,π‘“βˆˆπΏβˆ—Ξ“βˆΆπΏπœƒξ€·πΏβˆ—ξ€ΈβŸΆπΏπœƒ+πœƒπœŽξ€·πΏβˆ—ξ€Έ,Ξ“(𝑦+𝑓)=𝑦+𝜎(𝑦)+𝑓,(2.18) is an 𝑛-Lie algebra isomorphism.

Proof. It is clear that Ξ“ is a linear isomorphism of the vector space πΏβŠ•πΏβˆ— to itself. Next, for every π‘“π‘–βˆˆπΏβˆ—,π‘¦π‘–βˆˆπΏ,1≀𝑖≀𝑛, Γ𝑦1+𝑓1,…,𝑦𝑛+π‘“π‘›ξ€»πœƒξ€Έξƒ©ξ€Ίπ‘¦=Ξ“1,…,𝑦𝑛𝐿𝑦+πœƒ1,…,𝑦𝑛+𝑛𝑖=1(βˆ’1)π‘›βˆ’π‘–adβˆ—ξ€·π‘¦1,…,𝑦𝑖,…,𝑦𝑛𝑓𝑖ξƒͺ=𝑦1,…,𝑦𝑛𝐿𝑦+πœƒ1,…,𝑦𝑛𝑦+πœŽξ€·ξ€Ί1,…,𝑦𝑛𝐿+𝑛𝑖=1(βˆ’1)π‘›βˆ’π‘–adβˆ—ξ€·π‘¦1,…,𝑦𝑖,…,𝑦𝑛𝑓𝑖.Γ𝑦1+𝑣1𝑦,…,Γ𝑛+π‘£π‘›ξ€Έξ€»πœƒ+πœƒπœŽ=𝑦1𝑦+𝜎1ξ€Έ+𝑓1,…,𝑦𝑛𝑦+πœŽπ‘›ξ€Έ+π‘“π‘›ξ€»πœƒ+πœƒπœŽ=𝑦1,…,𝑦𝑛𝐿+ξ€·πœƒ+πœƒπœŽπ‘¦ξ€Έξ€·1,…,𝑦𝑛+𝑛𝑖=1(βˆ’1)π‘›βˆ’π‘–adβˆ—ξ€·π‘¦1,…,𝑦𝑖,…,π‘¦π‘›πœŽξ€·π‘¦ξ€Έξ€·π‘–ξ€Έ+𝑓𝑖=𝑦1,…,𝑦𝑛𝐿𝑦+πœƒ1,…,𝑦𝑛𝑦+πœŽξ€·ξ€Ί1,…,𝑦𝑛𝐿+𝑛𝑖=1(βˆ’1)π‘›βˆ’π‘–adβˆ—ξ€·π‘¦1,…,𝑦𝑖,…,𝑦𝑛𝑓𝑖𝑦=Ξ“ξ€·ξ€Ί1+𝑓1,…,𝑦𝑛+π‘“π‘›ξ€»πœƒξ€Έ.(2.19) the result follows.

Corollary 2.7. Let 𝐿 be an 𝑛-Lie algebra, and let πœƒ1,πœƒ2βˆΆπΏβˆ§π‘›β†’πΏβˆ— be cocycles. If there exists a linear mapping πœŽβˆΆπΏβ†’πΏβˆ— such that πœƒ1βˆ’πœƒ2=πœƒπœŽ, then the π‘‡βˆ—πœƒ1-extension πΏπœƒ1(πΏβˆ—) is isomorphic to the π‘‡βˆ—πœƒ2-extension πΏπœƒ2(πΏβˆ—) of 𝐿.

Proof. If there is a linear mapping πœŽβˆΆπΏβ†’πΏβˆ— such that πœƒ1=πœƒ2+πœƒπœŽ, by Theorem 2.6, the π‘‡βˆ—πœƒ1-extension πΏπœƒ1(πΏβˆ—)=πΏπœƒ2+πœƒπœŽ(πΏβˆ—) is isomorphic to the π‘‡βˆ—πœƒ2-extension πΏπœƒ2(πΏβˆ—).

3. The π‘‡βˆ—πœƒ-Extension of (𝑛+1)-Dimensional 𝑛-Lie Algebras

In this section, we study the π‘‡βˆ—πœƒ-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,…,𝑒𝑛+1 a basis of 𝐿 (𝑛β‰₯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 𝐿1βŠ†π‘(𝐿), (𝑏1)[𝑒2,…,𝑒𝑛+1]=𝑒1; if 𝐿1 is not contained in 𝑍(𝐿), (𝑏2)[𝑒1,…,𝑒𝑛]=𝑒1. (c) If dim𝐿1=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,π›ΌβˆˆπΉ,𝛼≠0. (d)If dim𝐿1=π‘Ÿ, 3β‰€π‘Ÿβ‰€π‘›+1, let 𝐿1=𝐹𝑒1+𝐹𝑒2+…+πΉπ‘’π‘Ÿ. Then(π‘‘π‘Ÿ)[𝑒1,…,̂𝑒𝑖,…,𝑒𝑛+1]=𝑒𝑖,1β‰€π‘–β‰€π‘Ÿ, where symbol ̂𝑒𝑖 means that 𝑒𝑖 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, …,𝑓𝑛+1 be the basis of πΏβˆ— satisfying 𝑓𝑖(𝑒𝑗)=𝛿𝑖𝑗,1≀𝑖,𝑗≀𝑛+1. For a cocycle πœƒβˆΆπΏβˆ§π‘›β†’πΏβˆ—πœƒξ€·π‘’1,…,̂𝑒𝑗,…,𝑒𝑛+1ξ€Έ=𝑛+1𝑠=1π‘Žπ‘—π‘ π‘“π‘ ,π‘Žπ‘—π‘ βˆˆπΉ,1≀𝑗≀𝑛+1.(3.1)

The π‘‡βˆ—πœƒ-extensions of the classes (𝑏𝑖), (𝑐𝑗), and (π‘‘π‘Ÿ) in Lemma 3.1 are denoted by (π‘βˆ—π‘–), (π‘βˆ—π‘—) and (π‘‘βˆ—π‘Ÿ), respectively.

Theorem 3.2. Let 𝐿 be an (𝑛+1)-dimensional 𝑛-Lie algebra in the Lemma 3.1. Then up to isomorphisms the π‘‡βˆ—πœƒ-extensions of 𝐿 are only of the following possibilities: (π‘Žβˆ—)πΏπœƒ(πΏβˆ—)is abelian (π‘βˆ—1)[𝑒2,…,𝑒𝑛+1]πœƒ=𝑒1,[𝑒1,𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1]πœƒ=βˆ‘π‘›+1𝑠=2π‘Žπ‘—π‘ π‘“π‘ ,[𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓1]πœƒ=(βˆ’1)𝑛+1+𝑗𝑓𝑗,π‘Žπ‘—π‘ βˆˆπΉ,2≀𝑗≀𝑛+1. (π‘βˆ—2)[𝑒1,…,𝑒𝑛]πœƒ=𝑒1,[𝑒1,…,̂𝑒𝑗,…,𝑒𝑛+1]πœƒ=βˆ‘π‘›+1𝑠=2π‘Žπ‘—π‘ π‘“π‘ ,[𝑒1,…,̂𝑒𝑗,…,𝑒𝑛,𝑓1]πœƒ=(βˆ’1)π‘›βˆ’π‘—+1𝑓𝑗,π‘Žπ‘—π‘ βˆˆπΉ,1≀𝑗≀𝑛. (π‘βˆ—1)[𝑒1,𝑒3,…,𝑒𝑛+1]πœƒ=𝑒2,[𝑒2,𝑒3,…,𝑒𝑛+1]πœƒ=𝑒1,[𝑒3,…,𝑒𝑛+1,𝑓2]πœƒ=(βˆ’1)𝑛𝑓1,[𝑒1,𝑒2,…,̂𝑒𝑗…,𝑒𝑛+1]πœƒ=βˆ‘π‘›+1𝑠=3π‘Žπ‘—π‘ π‘“π‘ ,[𝑒1,𝑒3,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓2]πœƒ=(βˆ’1)π‘›βˆ’π‘—π‘“π‘—,[𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓1]πœƒ=(βˆ’1)π‘›βˆ’π‘—π‘“π‘—,π‘Žπ‘—π‘ βˆˆπΉ,3≀𝑗≀𝑛+1.(π‘βˆ—2)[𝑒1,𝑒3,…,𝑒𝑛+1]πœƒ=𝑒2,[𝑒2,𝑒3,…,𝑒𝑛+1]πœƒ=𝛼𝑒1+𝑒2,[𝑒2,𝑒3,…,̂𝑒𝑖,…,𝑒𝑛+1,𝑓1]πœƒβ€‰ =(βˆ’1)π‘›βˆ’π‘–π›Όπ‘“π‘–,[𝑒2,𝑒3,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓2]πœƒβ€‰ =(βˆ’1)π‘›βˆ’π‘–π‘“π‘—,[𝑒1,𝑒3,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓2]πœƒβ€‰ =(βˆ’1)π‘›βˆ’π‘—π‘“π‘—,[𝑒3,…,…,𝑒𝑛+1,𝑓2]πœƒβ€‰ =(βˆ’1)𝑛(𝑓2+𝑓1),where π›ΌβˆˆπΉ,𝛼≠0,2≀𝑖≀𝑛+1,3≀𝑗≀𝑛+1.(π‘βˆ—3)[𝑒1,𝑒3,…,𝑒𝑛+1]πœƒ=𝑒1,[𝑒2,𝑒3,…,𝑒𝑛+1]πœƒ=𝑒2,[𝑒3,…,𝑒𝑛+1,𝑓1]πœƒ=(βˆ’1)𝑛𝑓1,[𝑒1,𝑒3,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓1]πœƒβ€‰ =(βˆ’1)π‘›βˆ’π‘—π‘“π‘—,3≀𝑗≀𝑛+1,[𝑒2,…,̂𝑒𝑖,…,𝑒𝑛+1,𝑓2]πœƒ=(βˆ’1)π‘›βˆ’π‘–π‘“π‘–,2≀𝑖≀𝑛+1.(π‘‘βˆ—π‘Ÿ)[𝑒1,…,̂𝑒𝑗,…,𝑒𝑛+1]πœƒ=𝑒𝑗,1β‰€π‘—β‰€π‘Ÿ,[𝑒1,𝑒2,…,π‘’π‘Ÿ,…,̂𝑒𝑗,…,𝑒𝑛+1]πœƒβ€‰ =βˆ‘π‘›+1𝑠=π‘Ÿ+1π‘Žπ‘—π‘ π‘“π‘ ,π‘Žπ‘—π‘ βˆˆπΉ,π‘Žπ‘—π‘ βˆˆπΉ,π‘Ÿ<𝑗,[𝑒1,…,̂𝑒𝑗,…,̂𝑒𝑖,…,𝑒𝑛+1,𝑓𝑖]πœƒβ€‰ =(βˆ’1)π‘›βˆ’π‘—+1𝑓𝑗,1≀𝑗<π‘–β‰€π‘Ÿ,[𝑒1,…,̂𝑒𝑖,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓𝑖]πœƒβ€‰ =(βˆ’1)π‘›βˆ’π‘—π‘“π‘—,1≀𝑖<π‘—β‰€π‘Ÿ,where 3β‰€π‘Ÿβ‰€π‘›+1.

Proof. Case (π‘Žβˆ—) is trivial. If 𝐿 is case (𝑏1), let 𝑓1,…,𝑓𝑛+1 be a basis of πΏβˆ— satisfying 𝑓𝑖(𝑒𝑗)=𝛿𝑖𝑗,1≀𝑖,𝑗≀𝑛+1. By the direct computation, identity (2.6), and Lemma 3.1, for every cocycle πœƒ0βˆΆπΏβˆ§π‘›β†’πΏβˆ—, we have πœƒ0(𝑒2,𝑒3,…,𝑒𝑛+1βˆ‘)=𝑛+1𝑠=1π‘Ž1𝑠𝑓𝑠,πœƒ0(𝑒1,𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1βˆ‘)=𝑠𝑗=2π‘Žπ‘—π‘ π‘“π‘ ,π‘Žπ‘—π‘ βˆˆπΉ,2≀𝑗≀𝑛+1. The multiplication of πΏπœƒ0(πΏβˆ—) in the basis 𝑒1,…,𝑒𝑛+1,𝑓1,…,𝑓𝑛+1 is 𝑒2,…,𝑒𝑛+1ξ€»πœƒ0=𝑒1+𝑛+1βˆ‘π‘ =1π‘Ž1𝑠𝑓𝑠,π‘Ž1π‘ ξ€Ίπ‘’βˆˆπΉ,1,𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1ξ€»πœƒ0=π‘ βˆ‘π‘—=2π‘Žπ‘—π‘ π‘“π‘ ,π‘Žπ‘—π‘ ξ€Ίπ‘’βˆˆπΉ,2≀𝑗≀𝑛+1,2,,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓1ξ€»πœƒ0=(βˆ’1)𝑛+𝑗+1𝑓𝑗,2≀𝑗≀𝑛+1.(3.2)
By Theorem 2.5, omitting the computation process, for every linear mapping πœŽβˆΆπΏβ†’πΏβˆ—, the cocycle πœƒπœŽβˆΆπΏβˆ§π‘›β†’πΏβˆ— satisfies πœƒπœŽ(𝑒2,𝑒3,…,𝑒𝑛+1)=(𝑛+1)𝜎(𝑒1) and πœƒπœ‚(𝑒1,𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1)=0,2≀𝑗≀𝑛+1. Then, define πœŽξ€·π‘’1ξ€Έ=βˆ’1πœƒπ‘›+10𝑒2,𝑒3,…,𝑒𝑛+1ξ€Έ=βˆ’π‘›+1𝑠=1π‘Ž1𝑠𝑓𝑠,(3.3) and 𝜎(𝑒𝑖)=0,2≀𝑖≀𝑛+1. Follows Theorem 2.6 that πΏπœƒ0(πΏβˆ—) is isomorphic to πΏπœƒ0+πœƒπœŽ(πΏβˆ—) which with the multiplication (π‘βˆ—1).
In the case (𝑏2), let πœƒ0βˆΆπΏβˆ§π‘›β†’πΏβˆ— be a cocycle. Omitting the computation process, we have πœƒ0(𝑒2,𝑒3,…,𝑒𝑛+1)=π‘Ž11𝑓1+…+π‘Ž1𝑛+1𝑓𝑛+1,πœƒ0(𝑒1,𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1)=π‘Žπ‘—2𝑓2+…+π‘Žπ‘—π‘›+1𝑓𝑛+1,π‘—βˆˆ2…,𝑛+1. The multiplication table of πΏπœƒ0(πΏβˆ—) is as follows: 𝑒2,…,𝑒𝑛+1ξ€»πœƒ0=𝑒1+𝑛+1𝑠=1π‘Ž1𝑠𝑓𝑠,𝑒1,𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1ξ€»πœƒ0=𝑛+1𝑠=2π‘Žπ‘—π‘ π‘“π‘ ,𝑒2,𝑒3,…,̂𝑒𝑖,…,𝑒𝑛+1,𝑓1ξ€»πœƒ0=(βˆ’1)𝑛+𝑖+1𝑓𝑖,2≀𝑖≀𝑛+1.(3.4)
For every linear mapping πœŽβˆΆπΏβ†’πΏβˆ—, the cocycle πœƒπœŽ: πΏβˆ§π‘›β†’πΏβˆ—, by Theorem 2.5, omitting the computation process, πœƒπœ‚(𝑒1,𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1)=0,2≀𝑗≀𝑛+1,πœƒπœ‚(𝑒2,…,𝑒𝑛+1)=(𝑛+1)πœ‚(𝑒1). Then defining πœŽξ€·π‘’1ξ€Έ1=βˆ’πœƒξ€·π‘’π‘›+12,…,𝑒𝑛+1ξ€Έ=π‘Ž11𝑓1+β‹―+π‘Ž1𝑛+1𝑓𝑛+1𝑒,πœ‚π‘—ξ€Έ=0,2≀𝑗≀𝑛+1,(3.5) we have πΏπœƒ0+πœƒπœŽ(πΏβˆ—) with the multiplication (π‘βˆ—2) which is isomorphic to πΏπœƒ0(πΏβˆ—).
In case (𝑐1), for every cocycle πœƒ0βˆΆπΏβˆ§π‘›β†’πΏβˆ—, omitting the computation process, we have πœƒ0(𝑒1,𝑒3,…,𝑒𝑛+1βˆ‘)=𝑛+1𝑠=1π‘Ž2𝑠𝑓𝑠,πœƒ0(𝑒2,…,𝑒𝑛+1βˆ‘)=𝑛+1𝑠=1π‘Ž1𝑠𝑓𝑠,πœƒ0(𝑒1,𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1βˆ‘)=𝑛+1𝑠=3π‘Žπ‘—π‘ π‘“π‘ ,𝑗=3,…,𝑛+1. The multiplication table of πΏπœƒ0(πΏβˆ—) is as follows: 𝑒1,𝑒3,…,𝑒𝑛+1ξ€»πœƒ0=𝑒2+𝑛+1𝑠=1π‘Ž2𝑠𝑓𝑠,𝑒2,…,𝑒𝑛+1ξ€»πœƒ0=𝑒1+𝑛+1𝑠=1π‘Ž1𝑠𝑓𝑠,𝑒1,𝑒2,𝑒3,…,̂𝑒𝑗,…,𝑒𝑛+1ξ€»πœƒ0=𝑛+1𝑠=3π‘Žπ‘—π‘ π‘“π‘ ξ€Ίπ‘’,3≀𝑗≀𝑛+1,1,𝑒3,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓2ξ€»πœƒ0=(βˆ’1)π‘›βˆ’π‘—π‘“π‘—ξ€Ίπ‘’,3≀𝑗≀𝑛+1,2,𝑒3,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓1ξ€»πœƒ0=(βˆ’1)π‘›βˆ’π‘—π‘“π‘—ξ€Ίπ‘’,3≀𝑗≀𝑛+1,3,…,𝑒𝑛+1,𝑓2ξ€»πœƒ0=(βˆ’1)𝑛𝑓1.(3.6)
Define the linear mapping πœŽβˆΆπΏβ†’πΏβˆ—βˆΆπœŽ(𝑒2)=βˆ’(1/(𝑛+1))πœƒ0(𝑒1,𝑒3,…,𝑒𝑛+1), 𝜎(𝑒1)=βˆ’(1/(𝑛+1))πœƒ0(𝑒2,𝑒3,…,𝑒𝑛+1) and others are zero. By the direct computation πœƒπœŽξ€·π‘’1,𝑒3,…,𝑒𝑛+1ξ€Έ=(𝑛+1)πœ‚0𝑒2ξ€Έ,πœƒπœŽξ€·π‘’2,𝑒3,…,𝑒𝑛+1ξ€Έ=(𝑛+1)πœ‚0𝑒1ξ€Έ.(3.7) Then πΏπœƒ0+πœƒπœŽ(πΏβˆ—) has the multiplication (π‘βˆ—1).
In the case (𝑐2), for every cocycle πœƒ0βˆΆπΏβˆ§π‘›β†’πΏβˆ—, we have πœƒ0(𝑒1,𝑒3,…,𝑒𝑛+1βˆ‘)=𝑛+1𝑠=1π‘Ž2𝑠𝑓𝑠,πœƒ0(𝑒2,…,𝑒𝑛+1βˆ‘)=𝑛+1𝑠=1π‘Ž1𝑠𝑓𝑠,πœƒ0(𝑒1,𝑒2,…,̂𝑒𝑗,…,𝑒𝑛+1βˆ‘)=𝑛+1𝑠=3π‘Žπ‘—π‘ π‘“π‘ ,𝑗=3,…,𝑛+1. The multiplication table of πΏπœƒ0(πΏβˆ—) is as follows: 𝑒1,𝑒3,…,𝑒𝑛+1ξ€»πœƒ0=𝑒2+𝑛+1𝑠=1π‘Ž2𝑠𝑓𝑠,𝑒2,…,𝑒𝑛+1ξ€»πœƒ0=𝛼𝑒1+𝑒2+𝑛+1𝑠=1π‘Ž1𝑠𝑓𝑠,𝑒1,𝑒2,𝑒3,…,̂𝑒𝑗,…,𝑒𝑛+1ξ€»πœƒ0=𝑛+1𝑠=3π‘Žπ‘—π‘ π‘“π‘ ξ€Ίπ‘’,3≀𝑗≀𝑛+1,2,…,̂𝑒𝑖,…,𝑒𝑛+1,𝑓1ξ€»πœƒ0=(βˆ’1)𝑛+𝑖𝛼𝑓𝑖𝑒,2≀𝑖≀𝑛+1,2,𝑒3,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓2ξ€»πœƒ0=(βˆ’1)π‘›βˆ’π‘—π‘“π‘—ξ€Ίπ‘’,3≀𝑗≀𝑛+1,1,𝑒3,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓2ξ€»πœƒ0=(βˆ’1)π‘›βˆ’π‘—π‘“π‘—ξ€Ίπ‘’,3≀𝑗≀𝑛+1,3,…,̂𝑒𝑗,…,𝑒𝑛+1,𝑓2ξ€»πœƒ0=(βˆ’1)𝑛𝑓2+𝑓1ξ€Έ.(3.8)
Define linear mapping πœŽβˆΆπΏβ†’πΏβˆ—βˆΆπœŽ(𝑒2)=βˆ’(1/(𝑛+1))πœƒ0(𝑒1,𝑒3,…,𝑒𝑛+1), 𝜎(𝑒1)=(1/𝛼(𝑛+1))(πœƒ0(𝑒1,𝑒3,…,𝑒𝑛+1)βˆ’πœƒ0(𝑒2,𝑒3,…,𝑒𝑛+1)). Then we obtain πœƒπœŽ(𝑒1,𝑒3,…,𝑒𝑛+1)=(𝑛+1)𝜎(𝑒2)=βˆ’πœƒ0(𝑒1,𝑒3,…,𝑒𝑛+1),πœƒπœŽ(𝑒2,𝑒3,…,𝑒𝑛+1)=(𝑛+1)𝜎(𝛼𝑒1+𝑒2)=βˆ’πœƒ0(𝑒2,𝑒3,…,𝑒𝑛+1) and others are zero. Therefore, πΏπœƒ0+πœƒπœŽ(πΏβˆ—) has the multiplication (π‘βˆ—2) in the basis 𝑒1,…,𝑒𝑛+1,𝑓1,…,𝑓𝑛+1.
In case (𝑐3), in similar discussions to above, for every cocycle πœƒ0βˆΆπΏβˆ§π‘›β†’πΏβˆ—, defining linear mapping πœŽβˆΆπΏβ†’πΏβˆ—βˆΆπœŽ(𝑒1)=βˆ’(1/(𝑛+1))πœƒ0(𝑒1,𝑒3,…,𝑒𝑛+1), πœ‚0(𝑒2)=βˆ’(1/(𝑛+1))πœƒ0(𝑒2,𝑒3,…,𝑒𝑛+1), we have πœƒπœŽξ€·π‘’1,𝑒3,…,𝑒𝑛+1ξ€Έ=𝑒(𝑛+1)𝜎1ξ€Έ=βˆ’πœƒ0𝑒1,𝑒3,…,𝑒𝑛+1ξ€Έ,(3.9) β€‰πœƒπœŽ(𝑒2,𝑒3,…,𝑒𝑛+1)=(𝑛+1)𝜎(𝑒2)=βˆ’πœƒ0(𝑒2,𝑒3,…,𝑒𝑛+1) and others are zero. Then πΏπœƒ0+πœƒπœŽ(πΏβˆ—) has the multiplication (π‘βˆ—3) in the basis 𝑒1,…,𝑒𝑛+1,𝑓1,…,𝑓𝑛+1.
Lastly, if 𝐿 is case (π‘‘π‘Ÿ), 3β‰€π‘Ÿβ‰€π‘›+1, for every cocycle πœƒ0βˆΆπΏβˆ§π‘›β†’πΏβˆ—, we have πœƒ0(𝑒1,,…,̂𝑒𝑖,…,𝑒𝑛+1βˆ‘)=𝑛+1𝑠=1π‘Žπ‘–π‘ π‘“π‘ ,1β‰€π‘–β‰€π‘Ÿ,πœƒ0(𝑒1,…,π‘’π‘Ÿ,…,̂𝑒𝑗,…,𝑒𝑛+1βˆ‘)=𝑛+1𝑗=π‘Ÿ+1π‘Žπ‘—π‘ π‘“π‘ ,π‘Ÿ<𝑗≀𝑛+1. By the direct computation, the multiplication of πΏπœƒ0(πΏβˆ—) is as follows: 𝑒1,…,̂𝑒𝑖,…,𝑒𝑛+1ξ€»πœƒ0=𝑒𝑖+𝑛+1𝑠=1π‘Žπ‘–π‘ π‘“π‘ ξ€Ίπ‘’,1β‰€π‘–β‰€π‘Ÿ,1,…,π‘’π‘Ÿ,…,̂𝑒𝑗,…,𝑒𝑛+1ξ€»πœƒ0=𝑛+1𝑠=π‘Ÿ+1π‘Žπ‘—π‘ π‘“π‘—ξ€Ίπ‘’,π‘Ÿ<𝑗≀𝑛+1,1,…,̂𝑒𝑗,…,̂𝑒𝑖,…,𝑒𝑛+1,π‘“π‘–ξ€»πœƒ0=(βˆ’1)π‘›βˆ’π‘—+1𝑓𝑗𝑒,1≀𝑗<π‘–β‰€π‘Ÿ,1,…,̂𝑒𝑖,…,̂𝑒𝑗,…,𝑒𝑛+1,π‘“π‘–ξ€»πœƒ0=(βˆ’1)π‘›βˆ’π‘—π‘“π‘—,1≀𝑖<π‘—β‰€π‘Ÿ.(3.10) Define linear mapping πœŽβˆΆπΏβ†’πΏβˆ—βˆΆπœŽ(𝑒𝑖)=βˆ’(1/(𝑛+1))πœƒ0(𝑒1,…,̂𝑒𝑖,…,𝑒𝑛+1),1β‰€π‘–β‰€π‘Ÿ, and 𝜎(𝑒𝑖)=0 if π‘Ÿ<𝑖. Then we obtain πœƒπœŽ(𝑒1,…,̂𝑒𝑖,…,𝑒𝑛+1)=(𝑛+1)𝜎(𝑒𝑖)=βˆ’πœƒ0(𝑒1,…,̂𝑒𝑖,…,𝑒𝑛+1) for 1β‰€π‘–β‰€π‘Ÿ, and πœƒπœŽ(𝑒1,…,̂𝑒𝑖,…,𝑒𝑛+1)=0 if 𝑖>π‘Ÿ. Therefore, πΏπœƒ0+πœƒπœŽ(πΏβˆ—) with the multiplication (π‘‘βˆ—π‘Ÿ) in the basis 𝑒1,…,𝑒𝑛+1,𝑓1,…,𝑓𝑛+1 and πΏπœƒ0(πΏβˆ—) is isomorphic to πΏπœƒ0+πœƒπœŽ(πΏβˆ—).

Acknowledgments

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