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ISRN Algebra
VolumeΒ 2011Β (2011), Article IDΒ 381875, 11 pages
http://dx.doi.org/10.5402/2011/381875
Research Article

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

College of Mathematics and Computer Science, Hebei University, Baoding 071002, China

Received 28 May 2011; Accepted 6 July 2011

Academic Editors: W.Β de Graaf and A.Β Zimmermann

Copyright Β© 2011 Ruipu Bai and Ying Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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