Deformations of Lie Algebras using σ-Derivations

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

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In this article we develop an approach to deformations of the Witt and Virasoro algebras based on sigma-derivations. We show that sigma-twisted Jacobi type identity holds for generators of such deformations. For the sigma-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the q-deformations of the Witt and Virasoro algebras associated to q-difference operators, providing also corresponding q-deformed Jacobi identities. (c) 2005 Elsevier Inc. All rights reserved.

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

View Paper: https://www.semanticscholar.org/paper/d1b03dba549fef0cf88950d320c0a44843c48733

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Citations in the corpus

Hom-bialgebras and comodule Hom-algebras (2008, methodology): “Hom-Lie algebras were introduced in [8] to describe the structures on certain deformations of the Witt algebra and the Virasoro algebra.”
On universal central extensions of Hom-Lie algebras (2012, methodology): “The Hom-Lie algebra structure was initially introduced in [4] motivated by examples of deformed Lie algebras coming from twisted discretizations of vector fields.”
Characterization of Higher Derivations on Algebras (2010, background): “(For other approaches to generalized derivations and their applications see [1, 2, 4, 10, 11] and references therein.”
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 ”
(Co)Homology and universal central extension of Hom-Leibniz algebras (2011, methodology): “The following theorem is similar to [ 2 ] and we overleap its proof.”
On hom-algebras with surjective twisting (2009, background): “The study of hom-algebras originates with [3], who introduced a notion of hom-Lie algebra in the context of deformation theory of Witt and Virasoro algebras.”
Hom-algebras and homology (2007, methodology): “Hom-Lie algebras are closely related to discrete and deformed vector fields and differential calculus [14, 18, 19].”
Automatic continuity of -derivations on *-algebras (2005, background): “and for all t ∈ [1, 2], d(fg)(t) = f(t)g(t)h(t)”
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.”
On Hom-algebra structures (2006, methodology): “The class of Quasi-Lie algebras incorporates as special cases Hom-Lie algebras and more general quasi-Hom-Lie algebras (qhl-algebras) which appear naturally in the algebraic study of σ-derivations (see [4]) and related deformations of infinite-dimensional and finite-dimensional Lie algebras.”
Monoidal Hom–Hopf Algebras (2009, background): “ategory such that Homalgebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras and Lie algebras. Introduction Hom-Lie algebras were introduced in [3]. They have been investigated by Silvestrov and his collaborators, see [9] for a survey. The idea”
Unital algebras of Hom-associative type and surjective or injective twistings (2009, methodology): “At the same time, on the side of algebra structures, formal deformations and elements of co-homology for Hom-associative and Hom-Lie algebras have been considered by Makhlouf and Silvestrov in [18], and elements of homology for Hom-algebras have been considered by Yau in [20].”
Enveloping algebras of Hom-Lie algebras (2007, methodology): “Hom-Lie algebras and related algebras were introduced in [1] to construct deformations of the Witt algebra, which is the Lie algebra of derivations on the Laurent polynomial algebra C[z±1].”
Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras (2010, methodology): “In 1999 in [20], this example (up to minor change of deformation parameter and generators) of q-Witt and q-Virasoro algebras associated to Jackson q-derivative was presented in a way close to the way it appeared in [18], namely using Jackson q-derivative by stressing the fact that it is a σ-derivati”
Rota–Baxter Hom-Lie–Admissible Algebras (2011, methodology): “This family of Hom–Lie algebras was constructed in [45, 52], and it carries a Hom–Lie algebra structure but not a Lie algebra structure when q = 1.”