BV-operators and the secondary Hochschild complex

Abstract

We introduce the notion of a BV-operator) is determined by ∆ in a manner similar to the BV-formalism. As an application, we produce a BV-operator on the cochain complex defining the secondary Hochschild cohomology of a symmetric algebra A over a commutative algebra B . In this case, we also show that the operator ∆ • corresponds to Connes' operator.

Secondary Hochschild cohomology and derivations

“…Triples have now been studied quite broadly, and examples, extensions, and applications can be found in a number of places (see [1], [2], [3], [4], [5], [8], [9], [13], or [18], for example). When convenient and appropriate, we will denote a triple by T = (A, B, ε), so as to make it easier with notation.…”

Noncommutative Differential Calculus Structure on Secondary Hochschild (co)homology

“…It needs further investigation since 1) there is no derived functor description for the secondary Hochschild (co)homology, and 2)-even in the finite-dimensional symmetric algebra case, the most canonical choice for the BV-operator does not lift to the secondary Hochschild cohomology [1]. More precisely, in [1] the authors consider a finite-dimensional symmetric algebra A and define a BV-operator ∆ on the homotopy Gerstenhaber algebra C * ((A, B, ε), A), which due to proposition 9 and theorem 10 in [1], is the most canonical choice for a square zero BV-differential operator on H * ((A, B, ε), A). Though ∆ determines the graded Lie bracket on H * ((A, B, ε), A), it is not a cochain map in general (see theorem 7, [1]).…”

“…More precisely, in [1] the authors consider a finite-dimensional symmetric algebra A and define a BV-operator ∆ on the homotopy Gerstenhaber algebra C * ((A, B, ε), A), which due to proposition 9 and theorem 10 in [1], is the most canonical choice for a square zero BV-differential operator on H * ((A, B, ε), A). Though ∆ determines the graded Lie bracket on H * ((A, B, ε), A), it is not a cochain map in general (see theorem 7, [1]).…”

“…This has also prompted the authors to define the corresponding homology groups HH • (A, B, ε) and HC • (A, B, ε) associated to the triple (A, B, ε). We refer to [8] and [1] for more Hochschild-like results on the secondary Hochschild and secondary cyclic (co)homology groups, associated to the triple (A, B, ε).…”

Secondary Hochschild homology and differentials

“…The cohomology originally introduced in [15] by Staic, and the homology in [13], the theory involves a triple (A, B, ε) which consists of a commutative -algebra B inducing a B-algebra structure on an associative -algebra A by way of a morphism ε : B −→ A. The corresponding secondary cyclic (co)homology was also studied in [13], and properties, applications, and generalizations of the secondary Hochschild (co)homology were investigated in [1], [2], [7], [8], [11], [12], and [16], among others.…”