This paper reviews the properties and applications of certain n-ary
generalizations of Lie algebras in a self-contained and unified way. These
generalizations are algebraic structures in which the two entries Lie bracket
has been replaced by a bracket with n entries. Each type of n-ary bracket
satisfies a specific characteristic identity which plays the r\^ole of the
Jacobi identity for Lie algebras. Particular attention will be paid to
generalized Lie algebras, which are defined by even multibrackets obtained by
antisymmetrizing the associative products of its n components and that satisfy
the generalized Jacobi identity (GJI), and to Filippov (or n-Lie) algebras,
which are defined by fully antisymmetric n-brackets that satisfy the Filippov
identity (FI). Three-Lie algebras have surfaced recently in multi-brane theory
in the context of the Bagger-Lambert-Gustavsson model. Because of this,
Filippov algebras will be discussed at length, including the cohomology
complexes that govern their central extensions and their deformations
(Whitehead's lemma extends to all semisimple n-Lie algebras). When the
skewsymmetry of the n-Lie algebra is relaxed, one is led the n-Leibniz
algebras. These will be discussed as well, since they underlie the
cohomological properties of n-Lie algebras.
The standard Poisson structure may also be extended to the n-ary case. We
shall review here the even generalized Poisson structures, whose GJI reproduces
the pattern of the generalized Lie algebras, and the Nambu-Poisson structures,
which satisfy the FI and determine Filippov algebras. Finally, the recent work
of Bagger-Lambert and Gustavsson on superconformal Chern-Simons theory will be
briefly discussed. Emphasis will be made on the appearance of the 3-Lie algebra
structure and on why the A_4 model may be formulated in terms of an ordinary
Lie algebra, and on its Nambu bracket generalization.Comment: Invited topical review for JPA Math.Theor. v2: minor changes,
references added. 120 pages, 318 reference
