Homotopy G-algebra structure on the cochain complex of hom-type algebras

Abstract

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. We show that the Hochschild type cochain complex of a hom-associative algebra carries a homotopy G-algebra structure. As a consequence, we get a Gerstenhaber algebra structure on the cohomology of a hom-associative algebra. We also find similar results for hom-dialgebras.

Cohomology of BiHom-associative algebras

“…Remark. When α = β, one recovers the Gerstenhaber algebra structure on the Hochschild cohomology of a hom-associative algebra [6] (see also [5]). In particular, if α = β = id, one gets the classical result that the Hochschild cohomology of an associative algebra inherits a Gerstenhaber algebra structure [8,11].…”

“…Please see subsection 4.2 for details clarification. (A similar description for the cohomology of hom-associative algebras can be found in [6].) The cohomology of this complex is called the Hochschild cohomology of the bihom-associative algebra (A, µ, α, β) and denoted by H • Hoch (A, A).…”

“…Remark. When α = β, one recovers the operad considered in [6] associated to any vector space A with a linear map α. In particular, when α and β are both identity map, one gets our favourite endomorphism operad.…”

“…A hom-associative algebra is an algebra (A, µ) whose associativity is twisted by an algebra homomorphism α in the sense that: µ(α(a), µ(b, c)) = µ(µ(a, b), α(c)), for all a, b, c ∈ A. See [1,5,6,18,19] (and references there in) for more details.…”

On equivariant dendriform algebras

“…Some generalizations of dendriform structures twisted by linear maps were studied in [8,25,19]. See [6,24] for more details about algebraic structures twisted by linear maps.…”