BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras

Giacomo Graziani (1) , Abdenacer Makhlouf (2, 3) , Claudia Menini (4) , Florin Panaite (5)

1 IF - Institut Fourier
2 Université de Haute-Alsace (UHA) - Université de Haute-Alsace (UHA) Mulhouse - Colmar
3 IRIMAS - Institut de Recherche en Informatique Mathématiques Automatique Signal - IRIMAS - UR 7499
4 Department of Mathematics [Ferrara]
5 IMAR - Institute of Mathematics of the Romanian Academy [Bucharest]

A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α,β:A→A such that α(a)(bc)=(ab)β(c), for all a,b,c∈A. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).

Domaines: Mathématiques [math]

Giacomo Graziani, Abdenacer Makhlouf, Claudia Menini, Florin Panaite. BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras. Symmetry, Integrability and Geometry : Methods and Applications, 2015, 11. ⟨hal-01990618⟩