Universal algebra of a Hom-Lie algebra and group-like elements

Abstract

Abstract. We construct the universal enveloping algebra of a Hom-Lie algebra and endow it with a Hom-Hopf algebra structure. We discuss group-like elements that we see as a Hom-group integrating the initial Hom-Lie algebra.

Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

“…Throughout this paper, we consider regular Hom-groups that is the case when the structure map is invertible, and this notion can be traced back to Caenepeel and Goyvaerts's pioneering work [3]. The axioms in the following definition of Hom-group is different from the one in [7,8,13]. However, we show that if the structure map is invertible, then some axioms in the original definition are redundant and can be obtained from the Hom-associativity condition.…”

“…In Proposition 2.13, we show that this axiom is redundant in the regular case. Let us recall the Hom-invertibility condition in the definition of a Hom-group (G, Φ) in [13]: for each x ∈ G, there exists a positive integer k such that…”

“…He also proved Lagrange's theorem for finite Hom-groups in [8]. The recent developments on Hom-groups (see [7,8,13]) make it natural to study Hom-Lie groups and to explore the relationship between Hom-Lie groups and Hom-Lie algebras.…”

“…There is a growing interest in Hom-algebraic structures because of their close relationship with the discrete and deformed vector fields, and differential calculus [6,10,11]. In particular, representations and deformations of Hom-Lie algebras were studied in [1,15,17]; a categorical interpretation of Hom-Lie algebras was described in [3]; the categorification of Hom-Lie algebras was given in [18]; geometric and algebraic generalizations of Hom-Lie algebras were given in [4,14,16]; quantization of Hom-Lie algebras was studied in [21]; and the universal enveloping algebra of Hom-Lie algebras was studied in [13,20].…”

“…The notion of Hom-groups was initially introduced by Caenepeel and Goyvaerts in [3]. In [13], Laurent-Gengoux, Makhlouf and Teles first gave a new construction of the universal enveloping algebra that is different from the one in [20]. This new construction leads to a Hom-Hopf algebra structure on the universal enveloping algebra of a Hom-Lie algebra.…”