Abstract. 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).
We introduce and investigate the properties of Hochschild cohomology of algebras in an abelian monoidal category M. We show that the second Hochschild cohomology group of an algebra in M classifies extensions of A up to an equivalence. We characterize algebras of Hochschild dimension 0 (separable algebras), and of Hochschild dimension ≤1 (formally smooth algebras). Several particular cases and applications are included in the last section of the paper.Let k be a field and let A be a k-algebra. Recall that an A-bimodule is, by definition, a left module over the enveloping algebra A e := A ⊗ A op . Hochschild cohomology H * (A, M) of A with coefficients in M was introduced in [8] in order to classify, up to equivalence, all extensions of A with kernel M. Many other applications of this cohomology have been discovered since then. Let us mention here a few of them.The algebra A is called separable if A is projective as an A-bimodule. Separable algebras are characterized by the fact that their Hochschild dimension is zero, that is H 1 (A, M) = 0, for every bimodule M. Other homological characterizations of separable algebras can be found for example in [17] and [5].The set of equivalence classes of extensions of A with kernel M is in one-to-one correspondence with H 2 (A, M). In particular, an algebra A has no non-trivial extensions if and only if H 2 (A, M) = 0, for any bimodule M, i.e. its Hochschild dimension is less than or equal to 1. These algebras were introduced by Cuntz and Quillen in [5], where they are called quasi-free and play the role of "functions algebras" of a "noncommutative smooth affine variety". One can prove that an algebra A is quasi-free if and only if it has the following "lifting property": for any algebra E and
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