Differential operators and BV structures in noncommutative geometry

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 DerA, 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 grD(F(A)), the associated graded algebra, is commutative in some ‘twisted’ sense. The resulting double Poisson structure on grD(F(A)) is closely related to the one introduced by Van den Bergh. Specifically, we prove that grD(F(A)) ∼= F(TA(DerA)), provided the algebra 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 DerA gives rise to a second order (wheeled) differential operator, a noncommutative analogue of the BVoperator that makes F(TA(DerA)) a BV-wheelgebra.