Elsevier

Journal of Algebra

Volume 288, Issue 2, 15 June 2005, Pages 321-344
Journal of Algebra

Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities

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Abstract

This paper introduces the notion of a quasi-hom-Lie algebra, or simply, a qhl-algebra, which is a natural generalization of hom-Lie algebras introduced in a previous paper [J.T. Hartwig, D. Larsson, S.D. Silvestrov, Deformations of Lie algebras using σ-derivations, math.QA/0408064]. Quasi-hom-Lie algebras include also as special cases (color) Lie algebras and superalgebras, and can be seen as deformations of these by maps, twisting the Jacobi identity and skew-symmetry. The natural realm for these quasi-hom-Lie algebras is generalizations-deformations of the Witt algebra d of derivations on the Laurent polynomials C[t,t−1]. We also develop a theory of central extensions for qhl-algebras which can be used to deform and generalize the Virasoro algebra by centrally extending the deformed Witt type algebras constructed here. In addition, we give a number of other interesting examples of quasi-hom-Lie algebras, among them a deformation of the loop algebra.

Keywords

Deformations
Central extensions
Quasi-hom-Lie algebras
(Color) Lie algebras
Witt algebras
Virasoro algebras
Loop algebras

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