On unitality conditions for Hom-associative algebras

Details On unitality conditions for Hom-associative algebras On unitality conditions for Hom-associative algebras

2009-04-30

arxiv.org/abs/0904.4874v2

Mathematics

Rings and Algebras

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

Scientific paper

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.

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