Differential operators and BV structures in noncommutative geometry

TL;DR

This paper develops a new formalism for differential operators on associative algebras, introducing a noncommutative BV structure and a functorial Fock space framework that generalizes classical concepts.

Contribution

It introduces a novel formalism of differential operators for noncommutative algebras, including the concept of wheelgebras and a noncommutative BV operator, extending classical geometric ideas.

Findings

The algebra of differential operators is filtered with a commutative associated graded in a twisted sense.

A new notion of wheelgebra and wheelgebras is introduced, related to wheeled PROPs.

A noncommutative BV operator is constructed from Ricci-flat, torsion-free bimodule connections.

Abstract

We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck's notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DDer(A), the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A), a certain `Fock space' associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)), of differential operators, is filtered and gr D(F(A)), the associated graded algebra, is commutative in some `twisted' sense. The resulting double Poisson structure on gr D(F(A)) is closely related to the one introduced by Van den Bergh. Specifically, we prove that gr D(F(A))=F(T_A(DDer(A)), provided A is smooth. It is crucial for our construction that the Fock space F(A) carries an extra-structure of a wheelgebra, a new notion closely related to the notion of a wheeled PROP. There are also notions of Lie wheelgebras, and so on. In that language, D(F(A)) becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on DDer(A) gives rise to a second order (wheeled) differential operator, a noncommutative analogue of the BV-operator.