On Affine Version of Hom-Lie Algebras
Tarik Anowar, Ripan Saha
TL;DR
This work develops the theory of Hom-affgebras, uniting Brzeziński's affine algebras with Hom-type twisting. It defines and analyzes Hom-associative affgebras, Hom-pre-Lie affgebras, and Hom-Lie affgebras, and introduces generalized derivations for Hom-Lie algebras, establishing a fiber-based description: every Hom-Lie affgebra is determined by a Hom-Lie algebra together with a generalized derivation and a constant, and conversely such data yield a Hom-Lie affgebra. The authors prove a detailed correspondence between morphisms of Hom-Lie affgebras and morphisms of their Lie fibers augmented by compatibility data, and they show how Hom-affgebras retract to Hom-algebras on tangent fibers, providing a unified framework linking affine geometry and Hom-type algebras. These results lay groundwork for further exploration of cohomology, deformation, and non-associative structures in the affine-Hom setting, with a clear pathway to basepoint-free geometric interpretations. The work thus bridges affine geometry, heap-theoretic structures, and Hom-type algebras in a coherent, extensible framework.
Abstract
This paper introduces Hom-type analogues of affine algebraic structures, termed Hom-affgebras. Extending Brzeziński'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.