Hom–Lie algebras with symmetric invariant nondegenerate bilinear forms

Abstract

The aim of this paper is to introduce and study quadratic Hom-Lie algebras, which are Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the double extension theory to Hom-Lie algebras. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. We establish a correspondence between the class of involutive quadratic Hom-Lie algebras and quadratic simple Lie algebras with symmetric involution. Centerless involutive quadratic Hom-Lie algebras are characterized. Also elements of a representation theory for Hom-Lie algebras, including adjoint and coadjoint representations are supplied with application to quadratic Hom-Lie algebras.2000 Mathematics Subject Classification. 17B10,17A30,17A45.

Constructions and Cohomology of Hom–Lie Color Algebras

“…We extend to Hom-Lie color algebras, the concept of -module introduced in [3,10,32], and then define a family of cohomology complexes for Hom-Lie color algebras.…”

“…In the particular case of Hom-Lie superalgebras, a cohomology theory was provided in [3], see also [28]. Notice that for Hom-Lie algebras, cohomology was described in [2,20,32] and representations also in [10].…”

Cyclic homology for Hom-associative algebras

“…For a not necessarily multiplicative Hom-associative algebra (A, µ, α), we say α ∈ End(A) is an element of the centroid if 6) for any x, y ∈ A. We refer the reader to [19] for more information on the centroid of a ring, and [5] for the centroid of a Lie algebra in the concept of Hom-Lie algebras. We note that for any unital Hom-associative algebra (A, α), α ∈ End(A) is an element of the centroid.…”

Another approach to Hom-Lie bialgebras via Manin triples

“…With this generalization of the Lie algebra, some q-deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra [8]. Due to their close relationship with discrete and deformed vector fields and differential calculus [8,9,10], Hom-Lie algebras have been studied in broad areas [1,2,3,4,5,11,12,14,15,16,17,18,20].…”