On unitality conditions for Hom-associative algebras

Abstract

In hom-associative structures, the associativity condition $(xy)z=x(yz)$ is twisted to $\alpha (x)(yz)=(xy)\alpha (z)$, with $\alpha$ a map in the appropriate category. In the present paper, we consider two different unitality conditions for hom-associative algebras. The first one, existence of a unit in the classical sense, is stronger than the second one, which we call weak unitality. We show associativity conditions connected to the size of the image of the twisting map for unital hom-associative algebras. Also the problem of embedding arbitrary hom-associative algebras into unital or weakly unital ones is investigated. Finally, we show that weakly unital hom-associative algebras with bijective twisting map are twisted versions of associative algebras.

arXiv commentContains some results in arxiv:0903.3393v1, plus subsequent additions. Final version submitted for publication

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• Hom-quantum groups II: cobraided Hom-bialgebras and Hom-quantum geometry
• Hom-quantum groups III: Representations and module Hom-algebras
• Paradigm of Nonassociative Hom-algebras and Hom-superalgebras
• Hom-Maltsev, Hom-alternative, and Hom-Jordan algebras
• Cyclic homology for Hom-associative algebras
• On Hom type algebras
• Deformations of Hom-Alternative and Hom-Malcev algebras
• On hom-algebras with surjective twisting
• Infinitesimal Hom-bialgebras and Hom-Lie bialgebras
• On the hom-associative Weyl algebras
• Hilbert's basis theorem for non-associative and hom-associative Ore extensions
• Hom-Akivis algebras
• The hom-associative Weyl algebras in prime characteristic