The Hom–Yang–Baxter equation and Hom–Lie algebras

Abstract

Abstract. Motivated by recent work on Hom-Lie algebras, a twisted version of the Yang-Baxter equation, called the Hom-Yang-Baxter equation (HYBE), was introduced by the author in [54]. In this paper, several more classes of solutions of the HYBE are constructed. Some of these solutions of the HYBE are closely related to the quantum enveloping algebra of sl (2), the Jones-Conway polynomial, and Yetter-Drinfel'd modules. Under some invertibility conditions, we construct a new infinite sequence of solutions of the HYBE from a given one.

Hom-Lie superalgebras and Hom-Lie admissible superalgebras

“…It is shown in [23] that the universal enveloping Hom-associative algebra carries a structure of Hom-bialgebra. See also [2,3,7,18,22,[24][25][26] for other works on twisted algebraic structures. This paper focusses on Z 2 -graded Hom-algebras.…”

Generalized Derivations of BiHom-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]. Recently, Hom-Lie algebras were studied in refs.…”

“…[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].…”

A classification of low dimensional multiplicative Hom-Lie superalgebras

“…For those algebras, the Jacobi condition is twisted [1]. Homalgebras were studied in [2][3][4][5][6][7][8][9][10]. Furthermore, Hom-Lie algebras on some q-deformations of Witt and Virasoro algebras were considered in [11][12][13].…”