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-Extensions of -Lie Algebras
The paper is mainly concerned with -extensions of -Lie algebras. The -extension of an -Lie algebra by a cocycle θ is defined, and a class of cocycles is constructed by means of linear mappings from an -Lie algebra on to its dual space. Finally all -extensions of -dimensional -Lie algebras are classified, and the explicit multiplications are given.
-Lie algebras (or Lie -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 -Lie algebras has been widely studied ([6–14]), it is quite necessary to get more examples of -Lie algebras and the method of constructing -Lie algebras. However it is not easy due to the -ary operation.
Bordemann in  introduced the notion of -extension of a Lie algebra and showed that each solvable quadratic Lie algebra over an algebraically closed field of characteristic zero is either a -extension or a nondegenerate ideal of codimension 1 in a -extension of some Lie algebra. In , 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  studied the -extension and -extension of metric 3-Lie algebras and provided a sufficient and necessary condition of a -extension of 3-Lie algebra admitting a metric.
This paper defines the -extension of an -Lie algebra by the coadjoint module and a cocycle from on to the dual space of . The main result of the paper is the complete classification of the -extensions of -dimensional -Lie algebras.
Throughout this paper, -Lie algebras are of finite dimensions and over an algebraically closed field of characteristic zero. Any multiplication of basis vectors which is not listed in the multiplication table of an -Lie algebra is assumed to be zero, and the symbol means that is omitted. If is a vector space over a field with a basis , then can be denoted by .
2. -Extensions of -Lie Algebras
To study the -extensions of -Lie algebras, we need some definitions and basic facts.
An -Lie algebra is a vector space with an -ary skew-symmetric operation satisfying for every and every permutation . Identity (2.2) is called the generalized Jacobi identity. A subspace of is referred to as a subalgebra (ideal) of if (). In particular, the subalgebra generated by for all is called the derived algebra of and is denoted by .
An -Lie algebra is called solvable if for some , where and is defined as for . An ideal is called nilpotent if for some , where and is defined as , for . An -Lie algebra is called abelian if .
Let be an -Lie algebra over the field and a vector space. If there exists a multilinear mapping satisfying for all , then is called a representation of or is an -module.
Let for . Then is an -module and is called the adjoint module of . If is an -module, then the dual space of is an -module in the following way. For , defines , is called the dual module of . If and ad, that is, ,, is called the coadjoint module of .
Definition 2.1. Let be an -Lie algebra. If the -linear mapping satisfying for all , then is called a cocycle of .
Theorem 2.2. Let be an -Lie algebra over , and let be a cocycle of . Then is an -Lie algebra in the following multiplication: where .
Proof. It suffices to verify the Jacobi identity (2.2) for . For all , set , and by identity (2.7) we have and for every , Thanks for identity (2.5), for , For , by identity (2.3), Therefore, the multiplication of defined by identity (2.7) satisfies for every .
Definition 2.3. The -Lie algebra with multiplication (2.7) is called the -extension of . In particular, the -extension corresponding to is called the trivial extension of and is denoted by .
Then the multiplication of is as follows: where .
Theorem 2.4. Let be an -Lie algebra, and let be a cocycle of . Then one has the following results. (1) is an abelian ideal of the -extension.(2)If is solvable, then the -extension is solvable.(3)If is a nilpotent -Lie algebra, then every -extension is nilpotent.(4)If , then is an essential extension of by the module . If , is a nonessential extension of .
Proof. From identity (2.7), is an abelian ideal of since , and .
Now let be solvable and . By induction on , we have Then we have . Thanks to result (1), . Result (2) follows.
(3) Since is nilpotent, for some nonnegative integer . For every cocycle , by identity (2.6),
Inductively, we have since . Then we have . Note that for , we have . Thus, , that is, is a nilpotent -Lie algebra.
It follows from result (4) that is a subalgebra of if .
For constructing -extensions of an -Lie algebra , we give the following method to get cocycles.
Theorem 2.5. Let be an -Lie algebra. Then for every linear mapping , the skew-symmetric mapping given by, for all , is a cocycle.
Proof. A tedious calculation shows that, for every , Therefore, satisfies identity (2.6). The proof is completed.
Theorem 2.6. Let be an -Lie algebra, and let be a cocycle. Then for every linear mapping , for all is an -Lie algebra isomorphism.
Proof. It is clear that is a linear isomorphism of the vector space to itself. Next, for every , the result follows.
Corollary 2.7. Let be an -Lie algebra, and let be cocycles. If there exists a linear mapping such that , then the -extension is isomorphic to the -extension of .
Proof. If there is a linear mapping such that , by Theorem 2.6, the -extension is isomorphic to the -extension .
3. The -Extension of -Dimensional -Lie Algebras
In this section, we study the -extension of -dimensional -Lie algebras over . First, we recall the classification theorem of -dimensional -Lie algebras.
Lemma 3.1 (see ). Let be an -dimensional -Lie algebra over and ,,, a basis of (). Then one and only one of the following possibilities hold up to isomorphisms. (a) If , then is an abelian -Lie algebra.(b)If and letting ,in the case that , ; if is not contained in , . (c) If and letting ,; ; . (d)If , , let . Then, where symbol means that is omitted.
We first introduce some notations. Let be an -dimensional -Lie algebra in the Lemma 3.1, and let , , be the basis of satisfying . For a cocycle
The -extensions of the classes , , and in Lemma 3.1 are denoted by , and , respectively.
Theorem 3.2. Let be an -dimensional -Lie algebra in the Lemma 3.1. Then up to isomorphisms the -extensions of are only of the following possibilities: is abelian . . . ,where ,,. . ,where .
Proof. Case is trivial. If is case , let be a basis of satisfying . By the direct computation, identity (2.6), and Lemma 3.1, for every cocycle , we have . The multiplication of in the basis is
By Theorem 2.5, omitting the computation process, for every linear mapping , the cocycle satisfies and ,. Then, define and . Follows Theorem 2.6 that is isomorphic to which with the multiplication .
In the case , let be a cocycle. Omitting the computation process, we have ,. The multiplication table of is as follows:
For every linear mapping , the cocycle : , by Theorem 2.5, omitting the computation process, ,,,,,. Then defining we have with the multiplication which is isomorphic to .
In case , for every cocycle , omitting the computation process, we have ,,,,. The multiplication table of is as follows:
Define the linear mapping , and others are zero. By the direct computation Then has the multiplication .
In the case , for every cocycle , we have ,,,,. The multiplication table of is as follows:
Define linear mapping ,, ,. Then we obtain ,, and others are zero. Therefore, has the multiplication in the basis ,,,.
In case , in similar discussions to above, for every cocycle , defining linear mapping , , we have and others are zero. Then has the multiplication in the basis ,,,.
Lastly, if is case , , for every cocycle , we have . By the direct computation, the multiplication of is as follows: Define linear mapping , and if . Then we obtain , for , and if . Therefore, with the multiplication in the basis ,, and is isomorphic to .
This project partially supported by NSF (10871192) of China, NSF (A2010000194) of Hebei Province, China.
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Copyright © 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.