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

By Daniel S. D. Larsson, Sergei Silvestrov. 2004

345 citations recorded by the source corpus.

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 sigma-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 delta 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. (Less)

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c1d34ea7dca90769a44f39ab5b101d52324638bb

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Citations in the corpus

Hom-algebras and Hom-coalgebras (2008, methodology): “Motivated by these and the new examples arising as application of the general quasi-deformation construction of [17, 26 , 27] on the one hand, and the desire to be able to treat within the same framework such well-known generalizations of Lie algebras as the color and super Lie algebras on the othe”
Rota–Baxter Hom-Lie–Admissible Algebras (2011, methodology): “The notion of Hom–Lie algebra was introduced by Hartwig, Larsson, and Silvestrov in [45, 50, 51], motivated initially by examples of deformed Lie algebras coming from twisted discretizations of vector fields.”
Lie Color and Hom-Lie Algebras of Witt Type and Their Central Extensions (2009, background): “The charge c is assumed to be homogeneous of degree zero. For further reference see [5, 6 , 9].”
Hom-Lie Superalgebras and Hom-Lie admissible Superalgebras (2009, background): “l deforming twisting maps and also in quasi-Lie and quasi-hom-Lie algebras the Jacobi identity in general contains 6 twisted triple bracket terms. 1 The Hom-Lie algebras were discussed intensively in [8, 10, 11, 12]. The graded case was mentioned in [13]. Hom-associative algebras were introduced in ”
Yetter-Drinfeld modules for Hom-bialgebras (2013, background): “as observed in the pioneering works [1, 7, 8, 9, 10, 11, 13, 18, 19, 25] that they are no longer Lie algebras. Motivated by these examples and their generalization, Hartwig, Larsson and Silvestrov in [17, 21, 22, 23] introduced the notion of Hom-Lie algebra as a deformation of Lie algebras in which ”
Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras (2010, methodology): “This quasi-deformation method was devised in [18, 22, 23, 24].”
Hom-bialgebras and comodule Hom-algebras (2008, background): “tures on certain deformations of the Witt algebra and the Virasoro algebra. Earlier precursors of Hom-Lie algebras can be found in [1,9,16]. Hom-Lie algebras are also related to deformed vector ﬁelds [8,13,14,15,22,23], number theory [12], the various versionsof the Yang-Baxter equations, braid grou”
BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras (2015, background): “Hartwig, Larsson and Silvestrov studied this kind of algebras in [15, 18] and called them Hom-Lie algebras because they involve a homomorphism in the defining identity.”
(Co)Homology and universal central extension of Hom-Leibniz algebras (2011, methodology): “Similarly to the theory of group and Lie algebras [13–15], we define the cohomology theory of Hom-Leibniz algebras.”
Hom-Lie Algebras with Symmetric Invariant NonDegenerate Bilinear Forms (2010, methodology): “The notion of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov in [20, 25, 26] motivated initially by examples of deformed Lie algebras coming from twisted discretizations of vector fields.”
Notes on Formal Deformations of Hom-associative and Hom-Lie Algebras (2007, neutral): “This quasi-deformation method was devised in [12, 15, 16].”
On Hom-algebra structures (2006, neutral): “2000 MSC: 17A30, 16Y99, 17A01, 17A20, 17D25.”
HOM-ALTERNATIVE ALGEBRAS AND HOM-JORDAN ALGEBRAS (2009, methodology): “The Hom-Lie algebras were introduced and discussed in [12, 15, 16, 17], motivated by quasi-deformations of Lie algebras of vector fields, in particular q-deformations of Witt and Virasoro algebras.”