Arithmetic Witt-hom-Lie algebras

Abstract

This paper is concerned with explaining and further developing the rather technical definition of a hom-Lie algebra given in a previous paper which was an adaption of the ordinary definition to the language of number theory and arithmetic geometry. To do this we here introduce the notion of Witt-hom-Lie algebras and give interesting arithmetic applications, both in the Lie algebra case and in the hom-Lie algebra case. The paper ends with a discussion of a few possible applications of the developed hom-Lie language.

Hom-quantum groups III: Representations and module Hom-algebras

“…Hom-type algebras first appeared in the form of Hom-Lie algebras [15], which satisfy an α-twisted version of the Jacobi identity. Hom-Lie algebras are closely related to deformed vector fields [1,15,27,28,29,39,42] and number theory [26]. Homassociative algebras were introduced in [33] to construct Hom-Lie algebras using the commutator bracket.…”

Hom-quantum groups: I. Quasi-triangular Hom-bialgebras

“…Earlier precursors of Hom-Lie algebras can be found in [24,38]. Hom-Lie algebras are closely related to deformed vector fields [2,22,34,35,36,50,53] and number theory [33].…”

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

“…Hom-Lie algebras (without multiplicativity) were introduced in [14] to describe the structures on some q-deformations of the Witt and the Virasoro algebras. They are also closely related to discrete and deformed vector fields, differential calculus [14,36,37,46,48], and number theory [35]. Earlier precursors of Hom-Lie algebras can be found in [1,17,38].…”