Giacomo Graziani, Abdenacer Makhlouf, Claudia Menini, Florin Panaite, “BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras”, SIGMA, 2015, Volume 11,086, 34 pp.

This article is cited in 62 papers

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

Giacomo Graziani^a, Abdenacer Makhlouf^b, Claudia Menini^c, Florin Panaite^d

^a Université Joseph Fourier Grenoble I Institut Fourier, 100, Rue des Maths BP74 38402 Saint-Martin-d’Hères, France
^b Université de Haute Alsace, Laboratoire de Mathématiques, Informatique et Applications, 4, Rue des frères Lumière, F-68093 Mulhouse, France
^c University of Ferrara, Department of Mathematics, Via Machiavelli 30, Ferrara, I-44121, Italy
^d Institute of Mathematics of the Romanian Academy, PO-Box 1-764, RO-014700 Bucharest, Romania

Abstract: A BiHom-associative algebra is a (nonassociative) algebra $A$ endowed with two commuting multiplicative linear maps $\alpha , \beta\colon A\rightarrow A$ such that $\alpha (a)(bc)=(ab)\beta (c)$, for all $a, b, c\in A$. 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).

Keywords: BiHom-associative algebra; BiHom-Lie algebra; BiHom-bialgebra; representation; twisting; smash product.

MSC: 17A99; 18D10; 16T99

Received: May 12, 2015; in final form October 13, 2015; Published online October 25, 2015

Language: English

DOI: 10.3842/SIGMA.2015.086