Operations on the secondary Hochschild cohomology

Abstract

Abstract. We show that the secondary Hochschild cohomology associated to a triple (A, B, ε) has several of the properties of the usual Hochschild cohomology. Among others, we prove the existence of the cup and Lie products, discuss the connection with extensions of B-algebras, and give a Hodge type decomposition of the secondary Hochschild cohomology.

BV-operators and the secondary Hochschild complex

“…on the secondary cohomology. It follows (see [10,Corollary 3.2]) that the secondary cohomology H • (A, B, ε) carries the structure of a Gerstenhaber algebra in the sense of [3].…”

“…In [10], Staic and Stancu showed that the secondary Hochschild complex C • (A, B, ε) := C • ((A, B, ε); A) with coefficients in A is a non-symmetric operad with multiplication, giving it the structure of a homotopy G-algebra. Hence, the secondary cohomology H • (A, B, ε) is equipped with a graded commutative cup product and a Lie bracket which makes it a Gerstenhaber algebra.…”

Deformation theories controlled by Hochschild cohomologies

“…As introduced in [8], the secondary Hochschild cohomology concerns itself with a triple (A, B, ) , where (i), (ii), and (iii) are all satisfied from Definition 2.3. Beyond [8], more examples of triples can be found in [9]. Starting with a triple (A, B, ) , one can then consider several quintuples as consequence: for instance, (A, B, , , ) , (A, B, B, , id) , and (A, A, B, id, ) , where we require A to be commutative in the latter.…”

The Exterior Graded Swiss-Cheese Operad $Λ^{S^2}_V$ (with an appendix by Ana Lorena Gherman and Mihai D. Staic)

“…In this section we introduce GSD operads, which will provide the correct setting to generalize the exterior algebra of a vector space. The definition is inspired by Voronov's Swiss-Cheese operad ( [12]), results on higher Hochschild cohomology over S 2 ([2], [5]), and secondary Hochschild cohomology ( [10], [11]). First we need to define what we mean by a bioperad.…”

“…See [10] and [11] for similar situations where this notation is very useful to keep track of certain operations.…”