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:
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:
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.