Deformations of Lie algebras using σ-derivations

Abstract

In this article we develop an approach to deformations of the Witt and Virasoro algebras based on σ -derivations. We show that σ -twisted Jacobi type identity holds for generators of such deformations. For the σ -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.

On Identities in Hom-Malcev Algebras

“…Hom-Lie algebras were introduced in [3] as a tool in understanding the structure and constructions of q-deformations of the Witt and the Virasoro algebras within the general framework of quasi-Lie algebras and quasi-Hom-Lie algebras introduced in [6,7,13]. Hom-associative algebras, as an analogue and generalization of associative algebras for Hom-Lie algebras, have been introduced in [9].…”

“…(ii) (A, ·, α) is called a Hom-Lie algebra ( [3]) if the Hom-Jacobi identity such that the Hom-Malcev identity (see (3))…”

Generalized Derivations of BiHom-Lie Algebras

“…Hom-Lie algebra, introduced by Hartwig, Larson and Silvestrov in ref. [1], is a triple (L, The main feature of these algebras is that the identities defining the structures are twisted by homomorphisms. The paradigmatic examples are q-deformations of Witt and Virasoro algebras, Heisenberg-Virasoro algebra and other algebraic structure constructed in pioneering works [2][3][4][5].…”

“…In the particular case that the twisted map is the identity map, Hom-Lie algebras become Lie algebras. The notion of Hom-Lie algebras was introduced by Hartwig, Larsson and Silvestrov to describe the structures on certain deformations of the Witt algebras and the Virasoro algebras [1]. HomLie algebras are also related to deformed vector fields, the various versions of the Yang-Baxter equations, braid group representations, and quantum groups [1,5,7].…”

A classification of low dimensional multiplicative Hom-Lie superalgebras

“…The Hom-Lie and quasi-Hom-Lie structures are obtained from twisted derivations of discrete modifications of vector fields. For those algebras, the Jacobi condition is twisted [1]. Homalgebras were studied in [2][3][4][5][6][7][8][9][10].…”