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

𝑇 βˆ— πœƒ -Extensions of 𝑛 -Lie Algebras

Ruipu BaiΒ and Ying Li

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 𝑇 βˆ— πœƒ -extensions of 𝑛 -Lie algebras. The 𝑇 βˆ— πœƒ -extension 𝐿 πœƒ ( 𝐿 βˆ— ) 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 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 𝑇 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 𝑇 βˆ— -extension or a nondegenerate ideal of codimension 1 in a 𝑇 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 𝑇 πœƒ -extension and 𝑇 βˆ— -extension of metric 3-Lie algebras and provided a sufficient and necessary condition of a 𝑇 βˆ— -extension of 3-Lie algebra admitting a metric.

This paper defines the 𝑇 βˆ— πœƒ -extension of an 𝑛 -Lie algebra 𝐿 by the coadjoint module 𝐿 βˆ— and a cocycle πœƒ from 𝐿 ∧ 𝑛 on to the dual space 𝐿 βˆ— of 𝐿 . The main result of the paper is the complete classification of the 𝑇 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 𝑒 1 , β‹― , 𝑒 π‘š , then 𝑉 can be denoted by 𝑉 = 𝐹 𝑒 1 + β‹― + 𝐹 𝑒 π‘š .

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

To study the 𝑇 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 satisfying ξ€Ί π‘₯ 1 , … , π‘₯ 𝑛 ξ€» ξ€Ί π‘₯ = s g n ( 𝜎 ) 𝜎 ( 1 ) , … , π‘₯ 𝜎 ( 𝑛 ) ξ€» , ( 2 . 1 ) π‘₯ ξ€Ί ξ€Ί 1 , … , π‘₯ 𝑛 ξ€» , 𝑦 2 , … , 𝑦 𝑛 ξ€» = 𝑛  𝑖 = 1 ξ€Ί π‘₯ 1 ξ€Ί π‘₯ , … , 𝑖 , 𝑦 2 , … , 𝑦 𝑛 ξ€» , … , π‘₯ 𝑛 ξ€» ( 2 . 2 ) for every π‘₯ 1 , … , π‘₯ 𝑛 , 𝑦 2 , … , 𝑦 𝑛 ∈ 𝐿 and every permutation 𝜎 ∈ 𝑆 𝑛 . 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† 𝐡 ( [ 𝐡 , 𝐿 , … , 𝐿 ] βŠ† 𝐡 ). In particular, the subalgebra generated by [ π‘₯ 1 , … , π‘₯ 𝑛 ] for all π‘₯ 1 , … , π‘₯ 𝑛 ∈ 𝐿 is called the derived algebra of 𝐿 and is denoted by 𝐿 1 .

An 𝑛 -Lie algebra 𝐿 is called solvable if 𝐿 ( 𝑠 ) = 0 for some 𝑠 β‰₯ 0 , where 𝐿 ( 0 ) = 𝐿 and 𝐿 ( 𝑠 ) is defined as 𝐿 ( 𝑠 + 1 ) = [ 𝐿 ( 𝑠 ) , 𝐿 ( 𝑠 ) , 𝐿 , … , 𝐿 ] for 𝑠 β‰₯ 0 . An ideal 𝐿 is called nilpotent if 𝐿 𝑠 = 0 for some 𝑠 β‰₯ 0 , where 𝐿 0 = 𝐿 and 𝐿 𝑠 is defined as 𝐿 𝑠 = [ 𝐿 𝑠 βˆ’ 1 , 𝐿 , … , 𝐿 ] , for 𝑠 β‰₯ 1 . An 𝑛 -Lie algebra 𝐿 is called abelian if 𝐿 1 = 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 βˆ’ 1 ) β†’ 𝐸 𝑛 𝑑 ( 𝑉 ) satisfying 𝜌 π‘₯ ξ€· ξ€Ί 1 , … , π‘₯ 𝑛 ξ€» , 𝑦 2 , … , 𝑦 𝑛 βˆ’ 1 ξ€Έ = 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 𝜌 ξ€· π‘₯ 1 , … , Μ‚ π‘₯ 𝑖 , … , π‘₯ 𝑛 ξ€Έ 𝜌 ξ€· π‘₯ 𝑖 , 𝑦 2 , … , 𝑦 𝑛 βˆ’ 2 ξ€Έ ( 2 . 3 ) ξ€Ί 𝜌 ξ€· π‘₯ 1 , … , π‘₯ 𝑛 βˆ’ 1 ξ€Έ ξ€· 𝑦 , 𝜌 1 , … , 𝑦 𝑛 βˆ’ 1 ξ€· π‘₯ ξ€Έ ξ€» = 𝜌 1 , … , π‘₯ 𝑛 βˆ’ 1 ξ€Έ 𝜌 ξ€· 𝑦 1 , … , 𝑦 𝑛 βˆ’ 1 ξ€Έ ξ€· 𝑦 βˆ’ 𝜌 1 , … , 𝑦 𝑛 βˆ’ 1 ξ€Έ 𝜌 ξ€· π‘₯ 1 , … , π‘₯ 𝑛 βˆ’ 1 ξ€Έ = 𝑛  𝑖 = 1 𝜌 ξ€· 𝑦 1 ξ€Ί π‘₯ , … , 1 , … , π‘₯ 𝑛 βˆ’ 1 , 𝑦 𝑖 ξ€» , … , 𝑦 𝑛 βˆ’ 1 ξ€Έ ( 2 . 4 ) for all π‘₯ 𝑖 , 𝑦 𝑖 ∈ 𝐿 , 𝑖 = 1 , … , 𝑛 , then ( 𝑉 , 𝜌 ) is called a representation of 𝐿 or 𝑉 is an 𝐿 -module.

Let 𝜌 ( π‘₯ 1 , … , π‘₯ 𝑛 βˆ’ 1 ) = a d ( π‘₯ 1 , … , π‘₯ 𝑛 βˆ’ 1 ) for π‘₯ 1 , … , π‘₯ 𝑛 βˆ’ 1 ∈ 𝐿 . Then ( 𝐿 , a d ) is an 𝐿 -module and is called the adjoint module of 𝐿 . If ( 𝑉 , 𝜌 ) is an 𝐿 -module, then the dual space 𝑉 βˆ— of 𝑉 is an 𝐿 -module in the following way. For 𝑓 ∈ 𝑉 βˆ— , 𝑣 ∈ 𝑉 , π‘₯ 1 , … , π‘₯ 𝑛 βˆ’ 1 ∈ 𝐿 , defines 𝜌 βˆ— ∢ 𝐿 ∧ 𝑛 βˆ’ 1 β†’ 𝐸 𝑛 𝑑 ( 𝑉 βˆ— ) , 𝜌 βˆ— ξ€· π‘₯ 1 , … , π‘₯ 𝑛 βˆ’ 1 ξ€Έ ξ€· 𝜌 ξ€· π‘₯ ( 𝑓 ) ( 𝑣 ) = βˆ’ 𝑓 1 , … , π‘₯ 𝑛 βˆ’ 1 ξ€Έ ξ€Έ . ( 𝑣 ) ( 2 . 5 )   ( 𝑉 βˆ— , 𝜌 βˆ— ) is called the dual module of 𝑉 . If 𝑉 = 𝐿 and 𝜌 = ad, that is, a d βˆ— ( π‘₯ 1 , … , π‘₯ 𝑛 βˆ’ 1 ) ( 𝑓 ) ( π‘₯ ) = βˆ’ 𝑓 ( [ π‘₯ 1 , … , π‘₯ 𝑛 βˆ’ 1 , π‘₯ ] ) , ( 𝐿 βˆ— , a d βˆ— ) is called the coadjoint module of 𝐿 .

Definition 2.1. Let 𝐿 be an 𝑛 -Lie algebra. If the 𝑛 -linear mapping πœƒ ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— satisfying for all π‘₯ 𝑖 , 𝑦 𝑗 ∈ 𝐿 , 1 ≀ 𝑖 ≀ 𝑛 , 2 ≀ 𝑗 ≀ 𝑛 , 𝑛  𝑖 = 1 πœƒ ξ€· π‘₯ 1 ξ€Ί π‘₯ , … , 𝑖 , 𝑦 2 , … , 𝑦 𝑛 ξ€» , … , π‘₯ 𝑛 ξ€Έ π‘₯ βˆ’ πœƒ ξ€· ξ€Ί 1 , … , π‘₯ 𝑛 ξ€» , 𝑦 2 , … , 𝑦 𝑛 ξ€Έ + 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 ξ€Ί π‘₯ 1 , … , Μ‚ π‘₯ 𝑖 , … , π‘₯ 𝑛 ξ€· π‘₯ , πœƒ 𝑖 , 𝑦 2 , … , 𝑦 𝑛 ξ€Έ ξ€» + ( βˆ’ 1 ) 𝑛 ξ€Ί 𝑦 2 , … , 𝑦 𝑛 ξ€· π‘₯ , πœƒ 1 , … , π‘₯ 𝑛 ξ€Έ ξ€» = 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œƒ ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— be a cocycle of 𝐿 . Then 𝐿 πœƒ ( 𝐿 βˆ— ) = 𝐿 βŠ• 𝐿 βˆ— is an 𝑛 -Lie algebra in the following multiplication: ξ€Ί 𝑦 1 + 𝑓 1 , … , 𝑦 𝑛 + 𝑓 𝑛 ξ€» πœƒ = ξ€Ί 𝑦 1 , … , 𝑦 𝑛 ξ€» 𝐿 ξ€· 𝑦 + πœƒ 1 , … , 𝑦 𝑛 ξ€Έ + 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 a d βˆ— ξ€· 𝑦 1 , … , Μ‚ 𝑦 𝑖 , … , 𝑦 𝑛 ξ€Έ 𝑓 𝑖 , ( 2 . 7 ) where 𝑦 𝑖 ∈ 𝐿 , 𝑓 𝑖 ∈ 𝐿 βˆ— , 1 ≀ 𝑖 ≀ 𝑛 .

Proof. It suffices to verify the Jacobi identity (2.2) for 𝐿 πœƒ ( 𝐿 βˆ— ) . For all 𝑦 𝑖 ∈ 𝐿 , 𝑓 𝑖 ∈ 𝐿 βˆ— , 1 ≀ 𝑖 ≀ 2 𝑛 βˆ’ 1 , set 𝑧 𝑖 = 𝑦 𝑖 + 𝑓 𝑖 , and by identity (2.7) we have 𝑧 ξ€Ί ξ€Ί 1 , … , 𝑧 𝑛 ξ€» πœƒ , 𝑧 𝑛 + 1 , … , 𝑧 2 𝑛 βˆ’ 1 ξ€» πœƒ = 𝑦 ξ€Ί ξ€Ί 1 + 𝑓 1 , … , 𝑦 𝑛 + 𝑓 𝑛 ξ€» πœƒ , 𝑦 𝑛 + 1 + 𝑓 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 + 𝑓 2 𝑛 βˆ’ 1 ξ€» πœƒ = 𝑦 ξ€Ί ξ€Ί 1 , … , 𝑦 𝑛 ξ€» 𝐿 , 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€» 𝐿 𝑦 + πœƒ ξ€· ξ€Ί 1 , … , 𝑦 𝑛 ξ€» 𝐿 , 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€Έ + ( βˆ’ 1 ) 𝑛 βˆ’ 1 a d βˆ— ξ€· 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€Έ πœƒ ξ€· 𝑦 1 , … , 𝑦 𝑛 ξ€Έ + a d βˆ— ξ€· 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€Έ 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑖 + 1 a d βˆ— ξ€· 𝑦 1 , … , Μ‚ 𝑦 𝑖 , … , 𝑦 𝑛 ξ€Έ 𝑓 𝑖 + 𝑛 βˆ’ 1  𝑗 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 βˆ’ 1 a d βˆ— 𝑦 ξ€· ξ€Ί 1 , … , 𝑦 𝑛 ξ€» 𝐿 , 𝑦 𝑛 + 1 , … , Μ‚ 𝑦 𝑛 + 𝑗 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€Έ 𝑓 𝑛 + 𝑗 ; ( 2 . 8 ) and for every 1 ≀ 𝑖 , π‘˜ ≀ 𝑛 , ξ€Ί 𝑧 1 , … , 𝑧 π‘˜ βˆ’ 1 , ξ€Ί 𝑧 π‘˜ , 𝑧 𝑛 + 1 , … , 𝑧 2 𝑛 βˆ’ 1 ξ€» πœƒ , 𝑧 π‘˜ + 1 , … , 𝑧 𝑛 ξ€» πœƒ = ξ€Ί 𝑦 1 , … , 𝑦 π‘˜ βˆ’ 1 , ξ€Ί 𝑦 π‘˜ , 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€» 𝐿 , 𝑦 π‘˜ + 1 , … , 𝑦 𝑛 ξ€» 𝐿 ξ€· 𝑦 + πœƒ 1 , … , 𝑦 π‘˜ βˆ’ 1 , ξ€Ί 𝑦 π‘˜ , 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€» 𝐿 , 𝑦 π‘˜ + 1 , … , 𝑦 𝑛 ξ€Έ + π‘˜ βˆ’ 1  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 𝜌 ξ€· 𝑦 1 , … , Μ‚ 𝑦 𝑖 , … , 𝑦 π‘˜ βˆ’ 1 , ξ€Ί 𝑦 π‘˜ , 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€» 𝐿 , 𝑦 π‘˜ + 1 , … , 𝑦 𝑛 ξ€Έ 𝑓 𝑖 + ( βˆ’ 1 ) 𝑛 βˆ’ π‘˜ a d βˆ— ξ€· 𝑦 1 , … , 𝑦 π‘˜ βˆ’ 1 , 𝑦 π‘˜ + 1 , … , 𝑦 𝑛 ξ€Έ πœƒ ξ€· 𝑦 π‘˜ , 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€Έ + ( βˆ’ 1 ) π‘˜ + 1 a d βˆ— ξ€· 𝑦 1 , … , 𝑦 π‘˜ βˆ’ 1 , 𝑦 π‘˜ + 1 , … , 𝑦 𝑛 ξ€Έ 𝜌 ξ€· 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€Έ 𝑓 π‘˜ + a d βˆ— ξ€· 𝑦 1 , … , 𝑦 π‘˜ βˆ’ 1 , 𝑦 π‘˜ + 1 , … , 𝑦 𝑛 ξ€Έ 𝑛 βˆ’ 1  𝑖 = 1 ( βˆ’ 1 ) π‘˜ + 𝑖 + 1 a d βˆ— ξ€· 𝑦 π‘˜ , 𝑦 𝑛 + 1 , … , Μ‚ 𝑦 𝑛 + 𝑖 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€Έ 𝑓 𝑛 + 𝑖 + 𝑛  𝑖 = π‘˜ + 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 a d βˆ— ξ€· 𝑦 1 , … , 𝑦 π‘˜ βˆ’ 1 , ξ€Ί 𝑦 π‘˜ , 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€» 𝐿 , 𝑦 π‘˜ + 1 , … , 𝑦 𝑛 ξ€Έ 𝑓 𝑖 . ( 2 . 9 ) Thanks for identity (2.5), for 1 ≀ π‘š ≀ 𝑛 , a d βˆ— ξ€· 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€Έ a d βˆ— ξ€· 𝑦 1 , … , Μ‚ 𝑦 π‘š , … , 𝑦 𝑛 ξ€Έ 𝑓 π‘š = ( βˆ’ 1 ) 𝑛 𝑛 βˆ’ 1  𝑗 β‰  π‘š , 𝑗 = 1 a d βˆ— ξ€· 𝑦 1 ξ€Ί 𝑦 , … , 𝑗 , 𝑦 𝑛 + 1 , … , 𝑦 2 𝑛 βˆ’ 1 ξ€» 𝐿 , … , Μ‚ 𝑦 π‘š , … , 𝑦 𝑛 ξ€Έ 𝑓 π‘š . ( 2 . 1 0 ) For 1 ≀ π‘š ≀ 𝑛 βˆ’ 1 , by identity (2.3), ( βˆ’ 1 ) 𝑛 βˆ’ π‘š βˆ’ 1 a d βˆ— 𝑦 ξ€· ξ€Ί 1 , … , 𝑦 𝑛 ξ€» 𝐿 , 𝑦 𝑛 + 1 , … , Μ‚ 𝑦 𝑛 + π‘š , … , 𝑦 2 𝑛 βˆ’ 1 ξ€Έ 𝑓 𝑛 + π‘š = 𝑛  𝑖 = 1 ( βˆ’ 1 ) βˆ’ π‘š βˆ’ 𝑖 βˆ’ 1 a d βˆ— ξ€· 𝑦 1 , … , Μ‚ 𝑦 𝑖 , … , 𝑦 𝑛 ξ€Έ a d βˆ— ξ€· 𝑦 𝑖 , 𝑦 𝑛 + 1 , … , Μ‚ 𝑦 𝑛 + π‘š , … , 𝑦 2 𝑛 βˆ’ 1 ξ€Έ 𝑓 𝑛 + π‘š . ( 2 . 1 1 ) Therefore, the multiplication of 𝐿 πœƒ ( 𝐿 βˆ— ) defined by identity (2.7) satisfies 𝑧 ξ€Ί ξ€Ί 1 , … , 𝑧 𝑛 ξ€» πœƒ , 𝑧 𝑛 + 1 , … , 𝑧 2 𝑛 βˆ’ 1 ξ€» πœƒ = 𝑛  π‘˜ = 1 ξ€Ί 𝑧 1 , … , 𝑧 π‘˜ βˆ’ 1 , ξ€Ί 𝑧 π‘˜ , 𝑧 𝑛 + 1 , … , 𝑧 2 𝑛 βˆ’ 1 ξ€» πœƒ , 𝑧 π‘˜ + 1 , … , 𝑧 𝑛 ξ€» πœƒ ( 2 . 1 2 ) for every 𝑧 𝑖 ∈ 𝐿 πœƒ ( 𝐿 βˆ— ) , 1 ≀ 𝑖 ≀ 2 𝑛 βˆ’ 1 .

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

Then the multiplication of 𝐿 0 ( 𝐿 βˆ— ) is as follows: ξ€Ί 𝑦 1 + 𝑓 1 , … , 𝑦 𝑛 + 𝑓 𝑛 ξ€» 0 = ξ€Ί 𝑦 1 , … , 𝑦 𝑛 ξ€» 𝐿 + 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 a d βˆ— ξ€· 𝑦 1 , … , Μ‚ 𝑦 𝑖 , … , 𝑦 𝑛 ξ€Έ 𝑓 𝑖 , ( 2 . 1 3 ) where 𝑦 𝑖 ∈ 𝐿 , 𝑓 𝑖 ∈ 𝑉 , 1 ≀ 𝑖 ≀ 𝑛 .

Theorem 2.4. Let 𝐿 be an 𝑛 -Lie algebra, and let πœƒ ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— be a cocycle of 𝐿 . Then one has the following results. (1) 𝐿 βˆ— is an abelian ideal of the 𝑇 βˆ— πœƒ -extension.(2)If 𝐿 is solvable, then the 𝑇 βˆ— πœƒ -extension 𝐿 πœƒ ( 𝐿 βˆ— ) is solvable.(3)If 𝐿 is a nilpotent 𝑛 -Lie algebra, then every 𝑇 βˆ— πœƒ -extension is nilpotent.(4)If πœƒ β‰  0 , then 𝐿 πœƒ ( 𝐿 βˆ— ) is an essential extension of 𝐿 by the module 𝐿 βˆ— . If πœƒ = 0 , 𝐿 0 ( 𝐿 βˆ— ) is a nonessential extension of 𝐿 .

Proof. From identity (2.7), 𝐿 βˆ— is an abelian ideal of 𝐿 πœƒ ( 𝐿 βˆ— ) since [ 𝐿 βˆ— , 𝐿 βˆ— , 𝐿 πœƒ ( 𝐿 βˆ— ) , … , 𝐿 πœƒ ( 𝐿 βˆ— ) ] πœƒ = 0 , and [ 𝐿 βˆ— , 𝐿 πœƒ ( 𝐿 βˆ— ) , … , 𝐿 πœƒ ( 𝐿 βˆ— ) ] πœƒ βŠ† 𝐿 βˆ— .
Now let 𝐿 be solvable and 𝐿 ( 𝑠 ) = 0 . By induction on π‘Ÿ , we have  𝐿 πœƒ ( π‘Ÿ + 1 ) ξ€· 𝐿 βˆ— ξ€Έ = [ 𝐿 πœƒ ( π‘Ÿ ) ξ€· 𝐿 βˆ— ξ€Έ , 𝐿 πœƒ ( π‘Ÿ ) ξ€· 𝐿 βˆ— ξ€Έ , 𝐿 πœƒ ξ€· 𝐿 βˆ— ξ€Έ , … , 𝐿 πœƒ ξ€· 𝐿 βˆ— ξ€Έ ξ‚„ πœƒ βŠ† 𝐿 ( π‘Ÿ + 1 ) ξ€· 𝐿 + πœƒ ( π‘Ÿ ) , 𝐿 ( π‘Ÿ ) ξ€Έ , 𝐿 , … , 𝐿 + 𝐿 βˆ— . ( 2 . 1 4 ) Then we have 𝐿 πœƒ ( 𝑠 + 1 ) ( 𝐿 βˆ— ) βŠ† 𝐿 βˆ— . Thanks to result (1), 𝐿 πœƒ ( 𝑠 + 2 ) ( 𝐿 βˆ— ) = 0 . Result (2) follows.
(3) Since 𝐿 is nilpotent, 𝐿 𝑠 = [ 𝐿 𝑠 βˆ’ 1 , 𝐿 , … , 𝐿 ] 𝐿 = 0 for some nonnegative integer 𝑠 . For every cocycle πœƒ ∢ 𝐿 𝑛 β†’ 𝐿 βˆ— , by identity (2.6), 𝐿 1 πœƒ ξ€· 𝐿 βˆ— ξ€Έ βŠ† 𝐿 1 + πœƒ ( 𝐿 , … , 𝐿 ) + a d βˆ— ξ€· 𝐿 ( 𝐿 , … , 𝐿 ) βˆ— ξ€Έ βŠ† 𝐿 1 + 𝐿 βˆ— . ( 2 . 1 5 )
Inductively, we have 𝐿 𝑠 πœƒ ( 𝐿 βˆ— ) βŠ† 𝐿 ( 𝑠 ) + 𝐿 βˆ— = 𝐿 βˆ— since 𝐿 𝑠 = 0 . Then we have 𝐿 πœƒ 2 𝑠 ( 𝐿 βˆ— ) βŠ† a d βˆ— 𝑠 ( 𝐿 , … , 𝐿 ) ( 𝐿 βˆ— ) . Note that for 𝑓 ∈ a d βˆ— 𝑠 ( 𝐿 , … , 𝐿 ) ( 𝐿 βˆ— ) , we have 𝑓 ( 𝐿 ) βŠ† 𝑓 ( 𝐿 𝑠 ) = 0 . Thus, 𝐿 πœƒ 2 𝑠 ( 𝐿 βˆ— ) = 0 , that is, 𝐿 πœƒ ( 𝐿 βˆ— ) is a nilpotent 𝑛 -Lie algebra.
It follows from result (4) that 𝐿 is a subalgebra of 𝐿 πœƒ ( 𝐿 βˆ— ) if πœƒ = 0 .

For constructing 𝑇 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 𝜎 ∢ 𝐿 β†’ 𝐿 βˆ— , the skew-symmetric mapping πœƒ 𝜎 ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— given by, for all π‘₯ 1 , … , π‘₯ 𝑛 ∈ 𝐿 , πœƒ 𝜎 ξ€· π‘₯ 1 , … , π‘₯ 𝑛 ξ€Έ π‘₯ = 𝜎 ξ€· ξ€Ί 1 , … , π‘₯ 𝑛 ξ€» 𝐿 ξ€Έ βˆ’ 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 a d βˆ— ξ€· π‘₯ 1 , … , Μ‚ π‘₯ 𝑖 , … , π‘₯ 𝑛 ξ€Έ 𝜎 ξ€· π‘₯ 𝑖 ξ€Έ ( 2 . 1 6 ) is a cocycle.

Proof. A tedious calculation shows that, for every π‘₯ 𝑖 , 𝑦 𝑖 ∈ 𝐿 , 1 ≀ 𝑖 , π‘˜ ≀ 𝑛 , πœƒ 𝜎 ξ€· π‘₯ 1 , … , π‘₯ π‘˜ βˆ’ 1 , ξ€Ί 𝑦 2 , … , 𝑦 𝑛 , π‘₯ π‘˜ ξ€» 𝐿 , π‘₯ π‘˜ + 1 , … , π‘₯ 𝑛 ξ€Έ ξ‚€ ξ€Ί π‘₯ = 𝜎 1 , … , π‘₯ π‘˜ βˆ’ 1 , ξ€Ί 𝑦 2 , … , 𝑦 𝑛 , π‘₯ π‘˜ ξ€» 𝐿 , π‘₯ π‘˜ + 1 , … , π‘₯ 𝑛 ξ€» 𝐿  + π‘˜ βˆ’ 1  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 βˆ’ 1 a d βˆ— ξ€· π‘₯ 1 , … , Μ‚ π‘₯ 𝑖 , … , π‘₯ π‘˜ βˆ’ 1 , ξ€Ί 𝑦 2 , … , 𝑦 𝑛 , π‘₯ π‘˜ ξ€» 𝐿 , π‘₯ π‘˜ + 1 , … , π‘₯ 𝑛 ξ€Έ 𝜎 ξ€· π‘₯ 𝑖 ξ€Έ + ( βˆ’ 1 ) 𝑛 βˆ’ π‘˜ βˆ’ 1 a d βˆ— ξ€· π‘₯ 1 , … , π‘₯ π‘˜ βˆ’ 1 , π‘₯ π‘˜ + 1 , … , π‘₯ 𝑛 ξ€Έ 𝜎 ξ€· π‘₯ π‘˜ ξ€Έ + 𝑛  𝑗 = π‘˜ + 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 βˆ’ 1 a d βˆ— ξ€· π‘₯ 1 ξ€Ί 𝑦 , … , 2 , … , 𝑦 𝑛 , π‘₯ π‘˜ ξ€» 𝐿 , π‘₯ π‘˜ + 1 , … , Μ‚ π‘₯ 𝑗 , … , π‘₯ 𝑛 ξ€Έ 𝜎 ξ€· π‘₯ 𝑗 ξ€Έ ; πœƒ 𝜎 π‘₯ ξ€· ξ€Ί 1 , … , π‘₯ 𝑛 ξ€» 𝐿 , 𝑦 2 , … , 𝑦 𝑛 ξ€Έ ξ‚€ π‘₯ = 𝜎 ξ€Ί ξ€Ί 1 , … , π‘₯ 𝑛 ξ€» 𝐿 , 𝑦 2 , … , 𝑦 𝑛 ξ€» 𝐿  + 𝑛  𝑖 = 2 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 βˆ’ 1 a d βˆ— π‘₯ ξ€· ξ€Ί 1 , … , π‘₯ 𝑛 ξ€» 𝐿 , 𝑦 2 , … , Μ‚ 𝑦 𝑖 , … , 𝑦 𝑛 ξ€Έ 𝜎 ξ€· 𝑦 𝑖 ξ€Έ + ( βˆ’ 1 ) 𝑛 a d βˆ— ξ€· 𝑦 2 , … , 𝑦 𝑛 ξ€Έ 𝜎 π‘₯ ξ€· ξ€Ί 1 , … , π‘₯ 𝑛 ξ€» 𝐿 ξ€Έ ; a d βˆ— ξ€· π‘₯ 1 , … , Μ‚ π‘₯ π‘˜ , … , π‘₯ 𝑛 ξ€Έ πœƒ 𝜎 ξ€· 𝑦 2 , … , 𝑦 𝑛 , π‘₯ π‘˜ ξ€Έ = a d βˆ— ξ€· π‘₯ 1 , … , Μ‚ π‘₯ π‘˜ , … , π‘₯ 𝑛 ξ€Έ 𝜎 𝑦 ξ€· ξ€Ί 2 , … , 𝑦 𝑛 , π‘₯ π‘˜ ξ€» 𝐿 ξ€Έ + a d βˆ— ξ€· π‘₯ 1 , … , Μ‚ π‘₯ π‘˜ , … , π‘₯ 𝑛 ξ€Έ 𝑛  𝑖 = 2 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 a d βˆ— ξ€· 𝑦 2 , … , Μ‚ 𝑦 𝑖 , … , 𝑦 𝑛 , π‘₯ π‘˜ ξ€Έ 𝜎 ξ€· 𝑦 𝑖 ξ€Έ βˆ’ a d βˆ— ξ€· π‘₯ 1 , … , Μ‚ π‘₯ π‘˜ , … , π‘₯ 𝑛 ξ€Έ a d βˆ— ξ€· 𝑦 2 , … , 𝑦 𝑛 ξ€Έ 𝜎 ξ€· π‘₯ π‘˜ ξ€Έ ; a d βˆ— ξ€· 𝑦 2 , … , 𝑦 𝑛 ξ€Έ πœƒ 𝑓 ξ€· π‘₯ 1 , … , π‘₯ 𝑛 ξ€Έ = a d βˆ— ξ€· 𝑦 2 , … , 𝑦 𝑛 ξ€Έ 𝜎 π‘₯ ξ€· ξ€Ί 1 , … , π‘₯ 𝑛 ξ€» 𝐿 ξ€Έ βˆ’ a d βˆ— ξ€· 𝑦 2 , … , 𝑦 𝑛 ξ€Έ 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 a d βˆ— ξ€· π‘₯ 1 , … , Μ‚ π‘₯ 𝑖 , π‘₯ 𝑛 ξ€Έ 𝜎 ξ€· π‘₯ 𝑖 ξ€Έ . ( 2 . 1 7 ) Therefore, πœƒ 𝑓 satisfies identity (2.6). The proof is completed.

Theorem 2.6. Let 𝐿 be an 𝑛 -Lie algebra, and let πœƒ ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— be a cocycle. Then for every linear mapping 𝜎 ∢ 𝐿 β†’ 𝐿 βˆ— , for all 𝑦 ∈ 𝐿 , 𝑓 ∈ 𝐿 βˆ— Ξ“ ∢ 𝐿 πœƒ ξ€· 𝐿 βˆ— ξ€Έ ⟢ 𝐿 πœƒ + πœƒ 𝜎 ξ€· 𝐿 βˆ— ξ€Έ , Ξ“ ( 𝑦 + 𝑓 ) = 𝑦 + 𝜎 ( 𝑦 ) + 𝑓 , ( 2 . 1 8 ) is an 𝑛 -Lie algebra isomorphism.

Proof. It is clear that Ξ“ is a linear isomorphism of the vector space 𝐿 βŠ• 𝐿 βˆ— to itself. Next, for every 𝑓 𝑖 ∈ 𝐿 βˆ— , 𝑦 𝑖 ∈ 𝐿 , 1 ≀ 𝑖 ≀ 𝑛 , Ξ“ 𝑦 ξ€· ξ€Ί 1 + 𝑓 1 , … , 𝑦 𝑛 + 𝑓 𝑛 ξ€» πœƒ ξ€Έ  ξ€Ί 𝑦 = Ξ“ 1 , … , 𝑦 𝑛 ξ€» 𝐿 ξ€· 𝑦 + πœƒ 1 , … , 𝑦 𝑛 ξ€Έ + 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 a d βˆ— ξ€· 𝑦 1 , … ,  𝑦 𝑖 , … , 𝑦 𝑛 ξ€Έ 𝑓 𝑖 ξƒͺ = ξ€Ί 𝑦 1 , … , 𝑦 𝑛 ξ€» 𝐿 ξ€· 𝑦 + πœƒ 1 , … , 𝑦 𝑛 ξ€Έ 𝑦 + 𝜎 ξ€· ξ€Ί 1 , … , 𝑦 𝑛 ξ€» 𝐿 ξ€Έ + 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 a d βˆ— ξ€· 𝑦 1 , … ,  𝑦 𝑖 , … , 𝑦 𝑛 ξ€Έ 𝑓 𝑖 . ξ€Ί Ξ“ ξ€· 𝑦 1 + 𝑣 1 ξ€Έ ξ€· 𝑦 , … , Ξ“ 𝑛 + 𝑣 𝑛 ξ€Έ ξ€» πœƒ + πœƒ 𝜎 = ξ€Ί 𝑦 1 ξ€· 𝑦 + 𝜎 1 ξ€Έ + 𝑓 1 , … , 𝑦 𝑛 ξ€· 𝑦 + 𝜎 𝑛 ξ€Έ + 𝑓 𝑛 ξ€» πœƒ + πœƒ 𝜎 = ξ€Ί 𝑦 1 , … , 𝑦 𝑛 ξ€» 𝐿 + ξ€· πœƒ + πœƒ 𝜎 𝑦 ξ€Έ ξ€· 1 , … , 𝑦 𝑛 ξ€Έ + 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 a d βˆ— ξ€· 𝑦 1 , … ,  𝑦 𝑖 , … , 𝑦 𝑛 𝜎 ξ€· 𝑦 ξ€Έ ξ€· 𝑖 ξ€Έ + 𝑓 𝑖 ξ€Έ = ξ€Ί 𝑦 1 , … , 𝑦 𝑛 ξ€» 𝐿 ξ€· 𝑦 + πœƒ 1 , … , 𝑦 𝑛 ξ€Έ 𝑦 + 𝜎 ξ€· ξ€Ί 1 , … , 𝑦 𝑛 ξ€» 𝐿 ξ€Έ + 𝑛  𝑖 = 1 ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 a d βˆ— ξ€· 𝑦 1 , … ,  𝑦 𝑖 , … , 𝑦 𝑛 ξ€Έ 𝑓 𝑖 𝑦 = Ξ“ ξ€· ξ€Ί 1 + 𝑓 1 , … , 𝑦 𝑛 + 𝑓 𝑛 ξ€» πœƒ ξ€Έ . ( 2 . 1 9 ) the result follows.

Corollary 2.7. Let 𝐿 be an 𝑛 -Lie algebra, and let πœƒ 1 , πœƒ 2 ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— be cocycles. If there exists a linear mapping 𝜎 ∢ 𝐿 β†’ 𝐿 βˆ— such that πœƒ 1 βˆ’ πœƒ 2 = πœƒ 𝜎 , then the 𝑇 βˆ— πœƒ 1 -extension 𝐿 πœƒ 1 ( 𝐿 βˆ— ) is isomorphic to the 𝑇 βˆ— πœƒ 2 -extension 𝐿 πœƒ 2 ( 𝐿 βˆ— ) of 𝐿 .

Proof. If there is a linear mapping 𝜎 ∢ 𝐿 β†’ 𝐿 βˆ— such that πœƒ 1 = πœƒ 2 + πœƒ 𝜎 , by Theorem 2.6, the 𝑇 βˆ— πœƒ 1 -extension 𝐿 πœƒ 1 ( 𝐿 βˆ— ) = 𝐿 πœƒ 2 + πœƒ 𝜎 ( 𝐿 βˆ— ) is isomorphic to the 𝑇 βˆ— πœƒ 2 -extension 𝐿 πœƒ 2 ( 𝐿 βˆ— ) .

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

In this section, we study the 𝑇 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 𝑒 1 , 𝑒 2 , … , 𝑒 𝑛 + 1 a basis of 𝐿 ( 𝑛 β‰₯ 3 ). Then one and only one of the following possibilities hold up to isomorphisms. (a) If d i m 𝐿 1 = 0 , then 𝐿 is an abelian 𝑛 -Lie algebra.(b)If d i m 𝐿 1 = 1 and letting 𝐿 1 = 𝐹 𝑒 1 ,in the case that 𝐿 1 βŠ† 𝑍 ( 𝐿 ) , ( 𝑏 1 ) [ 𝑒 2 , … , 𝑒 𝑛 + 1 ] = 𝑒 1 ; if 𝐿 1 is not contained in 𝑍 ( 𝐿 ) , ( 𝑏 2 ) [ 𝑒 1 , … , 𝑒 𝑛 ] = 𝑒 1 . (c) If d i m 𝐿 1 = 2 and letting 𝐿 1 = 𝐹 𝑒 1 + 𝐹 𝑒 2 , ( 𝑐 1 ) [ 𝑒 2 , … , 𝑒 𝑛 + 1 ] = 𝑒 1 , [ 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ] = 𝑒 2 ; ( 𝑐 2 ) [ 𝑒 2 , … , 𝑒 𝑛 + 1 ] = 𝛼 𝑒 1 + 𝑒 2 , [ 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ] = 𝑒 2 ; ( 𝑐 3 ) [ 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ] = 𝑒 1 , [ 𝑒 2 , … , 𝑒 𝑛 + 1 ] = 𝑒 2 , 𝛼 ∈ 𝐹 , 𝛼 β‰  0 . (d)If d i m 𝐿 1 = π‘Ÿ , 3 ≀ π‘Ÿ ≀ 𝑛 + 1 , let 𝐿 1 = 𝐹 𝑒 1 + 𝐹 𝑒 2 + … + 𝐹 𝑒 π‘Ÿ . Then ( 𝑑 π‘Ÿ ) [ 𝑒 1 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 ] = 𝑒 𝑖 , 1 ≀ 𝑖 ≀ π‘Ÿ , where symbol Μ‚ 𝑒 𝑖 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 𝑓 1 , … , 𝑓 𝑛 + 1 be the basis of 𝐿 βˆ— satisfying 𝑓 𝑖 ( 𝑒 𝑗 ) = 𝛿 𝑖 𝑗 , 1 ≀ 𝑖 , 𝑗 ≀ 𝑛 + 1 . For a cocycle πœƒ ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— πœƒ ξ€· 𝑒 1 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ξ€Έ = 𝑛 + 1  𝑠 = 1 π‘Ž 𝑗 𝑠 𝑓 𝑠 , π‘Ž 𝑗 𝑠 ∈ 𝐹 , 1 ≀ 𝑗 ≀ 𝑛 + 1 . ( 3 . 1 )

The 𝑇 βˆ— πœƒ -extensions of the classes ( 𝑏 𝑖 ) , ( 𝑐 𝑗 ) , and ( 𝑑 π‘Ÿ ) in Lemma 3.1 are denoted by ( 𝑏 βˆ— 𝑖 ) , ( 𝑐 βˆ— 𝑗 ) and ( 𝑑 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 𝑇 βˆ— πœƒ -extensions of 𝐿 are only of the following possibilities: ( π‘Ž βˆ— ) 𝐿 πœƒ ( 𝐿 βˆ— ) is abelian ( 𝑏 βˆ— 1 ) [ 𝑒 2 , … , 𝑒 𝑛 + 1 ] πœƒ = 𝑒 1 , [ 𝑒 1 , 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ] πœƒ = βˆ‘ 𝑛 + 1 𝑠 = 2 π‘Ž 𝑗 𝑠 𝑓 𝑠 , [ 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 1 ] πœƒ = ( βˆ’ 1 ) 𝑛 + 1 + 𝑗 𝑓 𝑗 , π‘Ž 𝑗 𝑠 ∈ 𝐹 , 2 ≀ 𝑗 ≀ 𝑛 + 1 . ( 𝑏 βˆ— 2 ) [ 𝑒 1 , … , 𝑒 𝑛 ] πœƒ = 𝑒 1 , [ 𝑒 1 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ] πœƒ = βˆ‘ 𝑛 + 1 𝑠 = 2 π‘Ž 𝑗 𝑠 𝑓 𝑠 , [ 𝑒 1 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 , 𝑓 1 ] πœƒ = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 + 1 𝑓 𝑗 , π‘Ž 𝑗 𝑠 ∈ 𝐹 , 1 ≀ 𝑗 ≀ 𝑛 . ( 𝑐 βˆ— 1 ) [ 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ] πœƒ = 𝑒 2 , [ 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ] πœƒ = 𝑒 1 , [ 𝑒 3 , … , 𝑒 𝑛 + 1 , 𝑓 2 ] πœƒ = ( βˆ’ 1 ) 𝑛 𝑓 1 , [ 𝑒 1 , 𝑒 2 , … , Μ‚ 𝑒 𝑗 … , 𝑒 𝑛 + 1 ] πœƒ = βˆ‘ 𝑛 + 1 𝑠 = 3 π‘Ž 𝑗 𝑠 𝑓 𝑠 , [ 𝑒 1 , 𝑒 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 2 ] πœƒ = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 𝑓 𝑗 , [ 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 1 ] πœƒ = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 𝑓 𝑗 , π‘Ž 𝑗 𝑠 ∈ 𝐹 , 3 ≀ 𝑗 ≀ 𝑛 + 1 . ( 𝑐 βˆ— 2 ) [ 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ] πœƒ = 𝑒 2 , [ 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ] πœƒ = 𝛼 𝑒 1 + 𝑒 2 , [ 𝑒 2 , 𝑒 3 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 , 𝑓 1 ] πœƒ   = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 𝛼 𝑓 𝑖 , [ 𝑒 2 , 𝑒 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 2 ] πœƒ   = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 𝑓 𝑗 , [ 𝑒 1 , 𝑒 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 2 ] πœƒ   = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 𝑓 𝑗 , [ 𝑒 3 , … , … , 𝑒 𝑛 + 1 , 𝑓 2 ] πœƒ   = ( βˆ’ 1 ) 𝑛 ( 𝑓 2 + 𝑓 1 ) ,where 𝛼 ∈ 𝐹 , 𝛼 β‰  0 , 2 ≀ 𝑖 ≀ 𝑛 + 1 , 3 ≀ 𝑗 ≀ 𝑛 + 1 . ( 𝑐 βˆ— 3 ) [ 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ] πœƒ = 𝑒 1 , [ 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ] πœƒ = 𝑒 2 , [ 𝑒 3 , … , 𝑒 𝑛 + 1 , 𝑓 1 ] πœƒ = ( βˆ’ 1 ) 𝑛 𝑓 1 , [ 𝑒 1 , 𝑒 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 1 ] πœƒ   = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 𝑓 𝑗 , 3 ≀ 𝑗 ≀ 𝑛 + 1 , [ 𝑒 2 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 , 𝑓 2 ] πœƒ = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑖 𝑓 𝑖 , 2 ≀ 𝑖 ≀ 𝑛 + 1 . ( 𝑑 βˆ— π‘Ÿ ) [ 𝑒 1 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ] πœƒ = 𝑒 𝑗 , 1 ≀ 𝑗 ≀ π‘Ÿ , [ 𝑒 1 , 𝑒 2 , … , 𝑒 π‘Ÿ , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ] πœƒ   = βˆ‘ 𝑛 + 1 𝑠 = π‘Ÿ + 1 π‘Ž 𝑗 𝑠 𝑓 𝑠 , π‘Ž 𝑗 𝑠 ∈ 𝐹 , π‘Ž 𝑗 𝑠 ∈ 𝐹 , π‘Ÿ < 𝑗 , [ 𝑒 1 , … , Μ‚ 𝑒 𝑗 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 , 𝑓 𝑖 ] πœƒ   = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 + 1 𝑓 𝑗 , 1 ≀ 𝑗 < 𝑖 ≀ π‘Ÿ , [ 𝑒 1 , … , Μ‚ 𝑒 𝑖 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 𝑖 ] πœƒ   = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 𝑓 𝑗 , 1 ≀ 𝑖 < 𝑗 ≀ π‘Ÿ ,where 3 ≀ π‘Ÿ ≀ 𝑛 + 1 .

Proof. Case ( π‘Ž βˆ— ) is trivial. If 𝐿 is case ( 𝑏 1 ) , let 𝑓 1 , … , 𝑓 𝑛 + 1 be a basis of 𝐿 βˆ— satisfying 𝑓 𝑖 ( 𝑒 𝑗 ) = 𝛿 𝑖 𝑗 , 1 ≀ 𝑖 , 𝑗 ≀ 𝑛 + 1 . By the direct computation, identity (2.6), and Lemma 3.1, for every cocycle πœƒ 0 ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— , we have πœƒ 0 ( 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 βˆ‘ ) = 𝑛 + 1 𝑠 = 1 π‘Ž 1 𝑠 𝑓 𝑠 , πœƒ 0 ( 𝑒 1 , 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 βˆ‘ ) = 𝑠 𝑗 = 2 π‘Ž 𝑗 𝑠 𝑓 𝑠 , π‘Ž 𝑗 𝑠 ∈ 𝐹 , 2 ≀ 𝑗 ≀ 𝑛 + 1 . The multiplication of 𝐿 πœƒ 0 ( 𝐿 βˆ— ) in the basis 𝑒 1 , … , 𝑒 𝑛 + 1 , 𝑓 1 , … , 𝑓 𝑛 + 1 is ξ€Ί 𝑒 2 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑒 1 + 𝑛 + 1 βˆ‘ 𝑠 = 1 π‘Ž 1 𝑠 𝑓 𝑠 , π‘Ž 1 𝑠 ξ€Ί 𝑒 ∈ 𝐹 , 1 , 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑠 βˆ‘ 𝑗 = 2 π‘Ž 𝑗 𝑠 𝑓 𝑠 , π‘Ž 𝑗 𝑠 ξ€Ί 𝑒 ∈ 𝐹 , 2 ≀ 𝑗 ≀ 𝑛 + 1 , 2 , , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 1 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 + 𝑗 + 1 𝑓 𝑗 , 2 ≀ 𝑗 ≀ 𝑛 + 1 . ( 3 . 2 )
By Theorem 2.5, omitting the computation process, for every linear mapping 𝜎 ∢ 𝐿 β†’ 𝐿 βˆ— , the cocycle πœƒ 𝜎 ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— satisfies πœƒ 𝜎 ( 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) = ( 𝑛 + 1 ) 𝜎 ( 𝑒 1 ) and πœƒ πœ‚ ( 𝑒 1 , 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ) = 0 , 2 ≀ 𝑗 ≀ 𝑛 + 1 . Then, define 𝜎 ξ€· 𝑒 1 ξ€Έ = βˆ’ 1 πœƒ 𝑛 + 1 0 ξ€· 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ξ€Έ = βˆ’ 𝑛 + 1  𝑠 = 1 π‘Ž 1 𝑠 𝑓 𝑠 , ( 3 . 3 ) and 𝜎 ( 𝑒 𝑖 ) = 0 , 2 ≀ 𝑖 ≀ 𝑛 + 1 . Follows Theorem 2.6 that 𝐿 πœƒ 0 ( 𝐿 βˆ— ) is isomorphic to 𝐿 πœƒ 0 + πœƒ 𝜎 ( 𝐿 βˆ— ) which with the multiplication ( 𝑏 βˆ— 1 ) .
In the case ( 𝑏 2 ) , let πœƒ 0 ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— be a cocycle. Omitting the computation process, we have πœƒ 0 ( 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) = π‘Ž 1 1 𝑓 1 + … + π‘Ž 1 𝑛 + 1 𝑓 𝑛 + 1 , πœƒ 0 ( 𝑒 1 , 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ) = π‘Ž 𝑗 2 𝑓 2 + … + π‘Ž 𝑗 𝑛 + 1 𝑓 𝑛 + 1 , 𝑗 ∈ 2 … , 𝑛 + 1 . The multiplication table of 𝐿 πœƒ 0 ( 𝐿 βˆ— ) is as follows: ξ€Ί 𝑒 2 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑒 1 + 𝑛 + 1  𝑠 = 1 π‘Ž 1 𝑠 𝑓 𝑠 , ξ€Ί 𝑒 1 , 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑛 + 1  𝑠 = 2 π‘Ž 𝑗 𝑠 𝑓 𝑠 , ξ€Ί 𝑒 2 , 𝑒 3 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 , 𝑓 1 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 + 𝑖 + 1 𝑓 𝑖 , 2 ≀ 𝑖 ≀ 𝑛 + 1 . ( 3 . 4 )
For every linear mapping 𝜎 ∢ 𝐿 β†’ 𝐿 βˆ— , the cocycle πœƒ 𝜎 : 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— , by Theorem 2.5, omitting the computation process, πœƒ πœ‚ ( 𝑒 1 , 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ) = 0 , 2 ≀ 𝑗 ≀ 𝑛 + 1 , πœƒ πœ‚ ( 𝑒 2 , … , 𝑒 𝑛 + 1 ) = ( 𝑛 + 1 ) πœ‚ ( 𝑒 1 ) . Then defining 𝜎 ξ€· 𝑒 1 ξ€Έ 1 = βˆ’ πœƒ ξ€· 𝑒 𝑛 + 1 2 , … , 𝑒 𝑛 + 1 ξ€Έ = π‘Ž 1 1 𝑓 1 + β‹― + π‘Ž 1 𝑛 + 1 𝑓 𝑛 + 1 ξ€· 𝑒 , πœ‚ 𝑗 ξ€Έ = 0 , 2 ≀ 𝑗 ≀ 𝑛 + 1 , ( 3 . 5 ) we have 𝐿 πœƒ 0 + πœƒ 𝜎 ( 𝐿 βˆ— ) with the multiplication ( 𝑏 βˆ— 2 ) which is isomorphic to 𝐿 πœƒ 0 ( 𝐿 βˆ— ) .
In case ( 𝑐 1 ) , for every cocycle πœƒ 0 ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— , omitting the computation process, we have πœƒ 0 ( 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 βˆ‘ ) = 𝑛 + 1 𝑠 = 1 π‘Ž 2 𝑠 𝑓 𝑠 , πœƒ 0 ( 𝑒 2 , … , 𝑒 𝑛 + 1 βˆ‘ ) = 𝑛 + 1 𝑠 = 1 π‘Ž 1 𝑠 𝑓 𝑠 , πœƒ 0 ( 𝑒 1 , 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 βˆ‘ ) = 𝑛 + 1 𝑠 = 3 π‘Ž 𝑗 𝑠 𝑓 𝑠 , 𝑗 = 3 , … , 𝑛 + 1 . The multiplication table of 𝐿 πœƒ 0 ( 𝐿 βˆ— ) is as follows: ξ€Ί 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑒 2 + 𝑛 + 1  𝑠 = 1 π‘Ž 2 𝑠 𝑓 𝑠 , ξ€Ί 𝑒 2 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑒 1 + 𝑛 + 1  𝑠 = 1 π‘Ž 1 𝑠 𝑓 𝑠 , ξ€Ί 𝑒 1 , 𝑒 2 , 𝑒 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑛 + 1  𝑠 = 3 π‘Ž 𝑗 𝑠 𝑓 𝑠 ξ€Ί 𝑒 , 3 ≀ 𝑗 ≀ 𝑛 + 1 , 1 , 𝑒 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 2 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 𝑓 𝑗 ξ€Ί 𝑒 , 3 ≀ 𝑗 ≀ 𝑛 + 1 , 2 , 𝑒 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 1 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 𝑓 𝑗 ξ€Ί 𝑒 , 3 ≀ 𝑗 ≀ 𝑛 + 1 , 3 , … , 𝑒 𝑛 + 1 , 𝑓 2 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 𝑓 1 . ( 3 . 6 )
Define the linear mapping 𝜎 ∢ 𝐿 β†’ 𝐿 βˆ— ∢ 𝜎 ( 𝑒 2 ) = βˆ’ ( 1 / ( 𝑛 + 1 ) ) πœƒ 0 ( 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) , 𝜎 ( 𝑒 1 ) = βˆ’ ( 1 / ( 𝑛 + 1 ) ) πœƒ 0 ( 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) and others are zero. By the direct computation πœƒ 𝜎 ξ€· 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ξ€Έ = ( 𝑛 + 1 ) πœ‚ 0 ξ€· 𝑒 2 ξ€Έ , πœƒ 𝜎 ξ€· 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ξ€Έ = ( 𝑛 + 1 ) πœ‚ 0 ξ€· 𝑒 1 ξ€Έ . ( 3 . 7 ) Then 𝐿 πœƒ 0 + πœƒ 𝜎 ( 𝐿 βˆ— ) has the multiplication ( 𝑐 βˆ— 1 ) .
In the case ( 𝑐 2 ) , for every cocycle πœƒ 0 ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— , we have πœƒ 0 ( 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 βˆ‘ ) = 𝑛 + 1 𝑠 = 1 π‘Ž 2 𝑠 𝑓 𝑠 , πœƒ 0 ( 𝑒 2 , … , 𝑒 𝑛 + 1 βˆ‘ ) = 𝑛 + 1 𝑠 = 1 π‘Ž 1 𝑠 𝑓 𝑠 , πœƒ 0 ( 𝑒 1 , 𝑒 2 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 βˆ‘ ) = 𝑛 + 1 𝑠 = 3 π‘Ž 𝑗 𝑠 𝑓 𝑠 , 𝑗 = 3 , … , 𝑛 + 1 . The multiplication table of 𝐿 πœƒ 0 ( 𝐿 βˆ— ) is as follows: ξ€Ί 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑒 2 + 𝑛 + 1  𝑠 = 1 π‘Ž 2 𝑠 𝑓 𝑠 , ξ€Ί 𝑒 2 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝛼 𝑒 1 + 𝑒 2 + 𝑛 + 1  𝑠 = 1 π‘Ž 1 𝑠 𝑓 𝑠 , ξ€Ί 𝑒 1 , 𝑒 2 , 𝑒 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑛 + 1  𝑠 = 3 π‘Ž 𝑗 𝑠 𝑓 𝑠 ξ€Ί 𝑒 , 3 ≀ 𝑗 ≀ 𝑛 + 1 , 2 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 , 𝑓 1 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 + 𝑖 𝛼 𝑓 𝑖 ξ€Ί 𝑒 , 2 ≀ 𝑖 ≀ 𝑛 + 1 , 2 , 𝑒 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 2 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 𝑓 𝑗 ξ€Ί 𝑒 , 3 ≀ 𝑗 ≀ 𝑛 + 1 , 1 , 𝑒 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 2 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 𝑓 𝑗 ξ€Ί 𝑒 , 3 ≀ 𝑗 ≀ 𝑛 + 1 , 3 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 2 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 ξ€· 𝑓 2 + 𝑓 1 ξ€Έ . ( 3 . 8 )
Define linear mapping 𝜎 ∢ 𝐿 β†’ 𝐿 βˆ— ∢ 𝜎 ( 𝑒 2 ) = βˆ’ ( 1 / ( 𝑛 + 1 ) ) πœƒ 0 ( 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) , 𝜎 ( 𝑒 1 ) = ( 1 / 𝛼 ( 𝑛 + 1 ) ) ( πœƒ 0 ( 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) βˆ’ πœƒ 0 ( 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) ) . Then we obtain πœƒ 𝜎 ( 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) = ( 𝑛 + 1 ) 𝜎 ( 𝑒 2 ) = βˆ’ πœƒ 0 ( 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) , πœƒ 𝜎 ( 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) = ( 𝑛 + 1 ) 𝜎 ( 𝛼 𝑒 1 + 𝑒 2 ) = βˆ’ πœƒ 0 ( 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) and others are zero. Therefore, 𝐿 πœƒ 0 + πœƒ 𝜎 ( 𝐿 βˆ— ) has the multiplication ( 𝑐 βˆ— 2 ) in the basis 𝑒 1 , … , 𝑒 𝑛 + 1 , 𝑓 1 , … , 𝑓 𝑛 + 1 .
In case ( 𝑐 3 ) , in similar discussions to above, for every cocycle πœƒ 0 ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— , defining linear mapping 𝜎 ∢ 𝐿 β†’ 𝐿 βˆ— ∢ 𝜎 ( 𝑒 1 ) = βˆ’ ( 1 / ( 𝑛 + 1 ) ) πœƒ 0 ( 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) , πœ‚ 0 ( 𝑒 2 ) = βˆ’ ( 1 / ( 𝑛 + 1 ) ) πœƒ 0 ( 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) , we have πœƒ 𝜎 ξ€· 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ξ€Έ = ξ€· 𝑒 ( 𝑛 + 1 ) 𝜎 1 ξ€Έ = βˆ’ πœƒ 0 ξ€· 𝑒 1 , 𝑒 3 , … , 𝑒 𝑛 + 1 ξ€Έ , ( 3 . 9 )   πœƒ 𝜎 ( 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) = ( 𝑛 + 1 ) 𝜎 ( 𝑒 2 ) = βˆ’ πœƒ 0 ( 𝑒 2 , 𝑒 3 , … , 𝑒 𝑛 + 1 ) and others are zero. Then 𝐿 πœƒ 0 + πœƒ 𝜎 ( 𝐿 βˆ— ) has the multiplication ( 𝑐 βˆ— 3 ) in the basis 𝑒 1 , … , 𝑒 𝑛 + 1 , 𝑓 1 , … , 𝑓 𝑛 + 1 .
Lastly, if 𝐿 is case ( 𝑑 π‘Ÿ ) , 3 ≀ π‘Ÿ ≀ 𝑛 + 1 , for every cocycle πœƒ 0 ∢ 𝐿 ∧ 𝑛 β†’ 𝐿 βˆ— , we have πœƒ 0 ( 𝑒 1 , , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 βˆ‘ ) = 𝑛 + 1 𝑠 = 1 π‘Ž 𝑖 𝑠 𝑓 𝑠 , 1 ≀ 𝑖 ≀ π‘Ÿ , πœƒ 0 ( 𝑒 1 , … , 𝑒 π‘Ÿ , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 βˆ‘ ) = 𝑛 + 1 𝑗 = π‘Ÿ + 1 π‘Ž 𝑗 𝑠 𝑓 𝑠 , π‘Ÿ < 𝑗 ≀ 𝑛 + 1 . By the direct computation, the multiplication of 𝐿 πœƒ 0 ( 𝐿 βˆ— ) is as follows: ξ€Ί 𝑒 1 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑒 𝑖 + 𝑛 + 1  𝑠 = 1 π‘Ž 𝑖 𝑠 𝑓 𝑠 ξ€Ί 𝑒 , 1 ≀ 𝑖 ≀ π‘Ÿ , 1 , … , 𝑒 π‘Ÿ , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 ξ€» πœƒ 0 = 𝑛 + 1  𝑠 = π‘Ÿ + 1 π‘Ž 𝑗 𝑠 𝑓 𝑗 ξ€Ί 𝑒 , π‘Ÿ < 𝑗 ≀ 𝑛 + 1 , 1 , … , Μ‚ 𝑒 𝑗 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 , 𝑓 𝑖 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 + 1 𝑓 𝑗 ξ€Ί 𝑒 , 1 ≀ 𝑗 < 𝑖 ≀ π‘Ÿ , 1 , … , Μ‚ 𝑒 𝑖 , … , Μ‚ 𝑒 𝑗 , … , 𝑒 𝑛 + 1 , 𝑓 𝑖 ξ€» πœƒ 0 = ( βˆ’ 1 ) 𝑛 βˆ’ 𝑗 𝑓 𝑗 , 1 ≀ 𝑖 < 𝑗 ≀ π‘Ÿ . ( 3 . 1 0 ) Define linear mapping 𝜎 ∢ 𝐿 β†’ 𝐿 βˆ— ∢ 𝜎 ( 𝑒 𝑖 ) = βˆ’ ( 1 / ( 𝑛 + 1 ) ) πœƒ 0 ( 𝑒 1 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 ) , 1 ≀ 𝑖 ≀ π‘Ÿ , and 𝜎 ( 𝑒 𝑖 ) = 0 if π‘Ÿ < 𝑖 . Then we obtain πœƒ 𝜎 ( 𝑒 1 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 ) = ( 𝑛 + 1 ) 𝜎 ( 𝑒 𝑖 ) = βˆ’ πœƒ 0 ( 𝑒 1 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 ) for 1 ≀ 𝑖 ≀ π‘Ÿ , and πœƒ 𝜎 ( 𝑒 1 , … , Μ‚ 𝑒 𝑖 , … , 𝑒 𝑛 + 1 ) = 0 if 𝑖 > π‘Ÿ . Therefore, 𝐿 πœƒ 0 + πœƒ 𝜎 ( 𝐿 βˆ— ) with the multiplication ( 𝑑 βˆ— π‘Ÿ ) in the basis 𝑒 1 , … , 𝑒 𝑛 + 1 , 𝑓 1 , … , 𝑓 𝑛 + 1 and 𝐿 πœƒ 0 ( 𝐿 βˆ— ) is isomorphic to 𝐿 πœƒ 0 + πœƒ 𝜎 ( 𝐿 βˆ— ) .

Acknowledgments

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

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