On Affine Version of Hom-Lie Algebras

TL;DR

This paper develops Hom-affgebras, a Hom-type extension of affine algebraic structures, exploring their properties, relationships, and derivations, with a focus on Hom-Lie affgebras and their fibers.

Contribution

It introduces Hom-affgebras, studies their structure, and establishes connections with generalized derivations and Lie fibers, extending classical affine algebra concepts.

Findings

Hom-affgebras generalize classical affine structures with a twisting map.

Hom-Lie affgebras are characterized via generalized derivations.

Hom-Lie affgebras correspond to Lie algebras with derivations and constants.

Abstract

This paper introduces Hom-type analogues of affine algebraic structures, termed Hom-affgebras. Extending Brzezi\'nski's theory of affgebras and the Hom-algebra framework developed by Hartwig-Larsson-Silvestrov, we define and study Hom-associative, Hom-pre-Lie, and Hom-Lie affgebras, where the classical identities are twisted by an affine self-map. We show how Hom-associative, Hom-pre-Lie, and Hom-Lie affgebras are related to one another. The main focus of this paper is on Hom-Lie affgebras and their fibers. We study the concept of generalized derivations for Hom-Lie algebras, extending the notion of generalized derivations for Lie algebras. We explore the close relationship between Hom-Lie affgebras and such derivations. We show that every Hom-Lie affgebra both determines and is determined by a Hom-Lie algebra together with such a generalized derivation and a constant. Furthermore, we establish that a homomorphism between Lie affgebras corresponds to a homomorphism between their associated Lie fibers along with a constant, and vice versa.