Abstract
We review the basic definitions and properties of two types of n-ary structures, the Generalized Lie Algebras (GLA) and the Filippov (≡ n-Lie) algebras (FA), as well as those of their Poisson counterparts, the Generalized Poisson (GPS) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology complexes relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends Whitehead’s lemma to all n ≥ 2, n = 2 being the original Lie algebra case. Some comments onn-Leibniz algebras are also made.
Highlights
Introduction theFilippov identity (FI) [8]The Jacobi identity (JI) for Lie algebras g, [X, [Y, Z]]+ [Y, [Z, X]] + [Z, [X, Y ]] = 0, may be looked at in two ways
We look for structure constants Ωi1...i2p j that satisfy the generalized Jacobi identity (GJI) (1.3) i.e., such that
An important example of finite Filippov algebras is provided by the real euclidean simple n-Lie algebras An+1 defined on an euclidean (n + 1)-dimensional vector space
Summary
The definitions of ideals, solvable ideals and semisimple algebras can be extended to the n > 2 case as follows [9]. A FA is semisimple if it does not have solvable ideals, and simple if [G, . An important type of FAs, because of its relevance in physical applications where a scalar product is usually needed (as in the Bagger-Lambert-Gustavsson model in M-theory), is the class of metric Filippov algebras. These are endowed with a metric , on G, Y, Z = gabY a Zb, ∀ Y, Z ∈ G that is invariant i.e.,.
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