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dc.contributor.authorAzcárraga, José A. de-
dc.contributor.authorIzquierdo, José Manuel-
dc.identifier.citationJournal of Physics A - Mathematical and Theoretical 43 (29): 293001 (2010)en_US
dc.description.abstractThis 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 These generalizations are algebraic structures in which the two-entry Lie bracket has been replaced by a bracket with n entries.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 role 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, and to Filippov (or n-Lie) algebras, which are defined by fully antisymmetric n-brackets that satisfy the Filippov identity. 3-Lie algebras have surfaced recently in multi-brane theory in the context of the Bagger-Lambert-Gustavsson model. As a result, Filippov algebras will be discussed at length, including the cohomology complexes that govern their central extensions and their deformations ( it turns out that Whitehead's lemma extends to all semisimple n-Lie algebras). When the skewsymmetry of the Lie or n-Lie algebra bracket is relaxed, one is led to a more general type of n-algebras, 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 generalized Jacobi identity reproduces the pattern of the generalized Lie algebras, and the Nambu-Poisson structures, which satisfy the Filippov identity 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.en_US
dc.description.sponsorshipWe are grateful to I. Bandos, T. Curtright, J.M. Figueroa O’Farrill, J.-L. Loday, A.J. Macfarlane, A.J. Mountain, J. Navarro-Faus, A. Perelomov, J.C. P´erez-Bueno, M. Picón, D. Sorokin, J. Stasheff, P. K. Townsend and C. K. Zachos for their collaboration on past joint work and/or for helpful discussions, correspondence or references on parts of the material covered by this review. This work has been partially supported by the research grants FIS2008-01980 and FIS2009- 09002 from the Spanish MICINN and by VA-013-C05 from the Junta de Castilla y León.en_US
dc.format.extent1566860 bytes-
dc.publisherInstitute of Physics Publishingen_US
dc.titlen-ary algebras: a review with applicationsen_US
dc.typetrabajo de divulgaciónen_US
dc.description.peerreviewedPeer revieweden_US
item.openairetypetrabajo de divulgación-
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