Gerstenhaber and Batalin-Vilkovisky Structures on Modules over Operads

Abstract

ABSTRACT. In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin-Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalised Lie derivative and a generalised (cyclic) cap product that obey a Cartan-Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Hochschild theory of associative algebras, for Poisson structures, but above all the calculus for general left Hopf algebroids with respect to general coefficients (in which the classical calculus of vector fields and differential forms is contained).

Higher Brackets on Cyclic and Negative Cyclic (Co)Homology

“…Graphically, the condition (6.7) can be understood by a picture similar to those that depict cyclic operads: In particular [Ko1,Prop. 3.5], a cyclic unital opposite module N over an operad with multiplication pO, µ, eq carries the structure of a cyclic K-module with cyclic operator t : N pnq Ñ N pnq, along with faces d i : N pnq Ñ N pn´1q and degeneracies s j : N pnq Ñ N pn`1q of the underlying simplicial object given by…”

“…The operator T was implicitly introduced in [GDTs] in the context of associative algebras. The obvious advantage of the explicit expression (6.17) is that it not only applies to the endomorphism operad of an associative algebra acting on chains that governs Hochschild theory, but more general to any operad with multiplication, such as for example those arising in the context of Hopf algebras, differential operators, or more general bialgebroids [Ko1], and moreover also to cyclic operads as we will see in §7.…”

“…Not only the preceding Examples 6.16 & 6.17 by setting pU, Aq :" pA e , Aq are included in this theory, but also the duality Lie-Rinehart algebras in [Hue] (geometrically, Lie algebroids that lead to the space of differential operators on a smooth manifold); in particular, this also comprises Poisson algebras. As described in detail in [Ko1,§6.3], the operad needed here is not precisely an endomorphism operad, but rather…”

Noncommutative differential calculus on (co)homology of hom-associative algebras

“…For a smooth manifold M , the pair (X • (M ), Ω • (M )) of multivector fields and differential forms; for an associative algebra A, the pair (H • (A), H • (A)) of Hochschild cohomology and homology yields differential calculus structure. In [9] Kowalzig extends the result of Gerstenhaber and Voronov by introducing a cyclic comp module over an operad (with multiplication). Such a structure induces a simplicial homology on the underlying graded space of the comp module and certain action maps.…”

“…We mention how a cyclic comp module induces a noncommutative differential calculus. Our main references are [6,9]. We mainly follow the sign conventions of [9].…”