### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 086, 34 pages      arXiv:1505.00469      https://doi.org/10.3842/SIGMA.2015.086

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

Giacomo Graziani a, Abdenacer Makhlouf b, Claudia Menini c and 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

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

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).

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

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