Abstract:
Motivated by Van den Bergh's recent `double Poisson'
algebras, we introduce a new formalism of differential operators for a
general associative algebra A. It replaces Grothendieck's notion for
commutative algebras in such a way that derivations of a commutative
algebra are replaced by Der(A, A ⊗ A), the A-bimodule of
`double derivations' of A. The algebra D(A) of differential operators
is filtered, and its associated graded is commutative in some
`twisted' sense. Parallel to the commutative case, there is an
induced double Poisson structure on gr(A), which identifies gr(A) with
Van den Bergh's `double Schouten-Nijenhuis algebra',
T
A Der(A, A ⊗ A), which is a noncommutative version of
polyvector fields on a variety.
A crucial step of our construction is replacing A with a certain `Fock
space' F(A), which carries the structure of a wheeled PROP.
We then consider the case where A is the path algebra of a quiver,
which may be viewed as a `noncommutative' version of affine space.
The original motivation for the above constructions was to construct a
`wheeled BV' structure on the double Schouten-Nijenhuis algebra of A,
which justifies calling A a `wheeled Calabi-Yau' algebra. In this
case, D(A) generalizes the quiver version of the Moyal star product
on Rn, and is also a quiver version of Turaev's skein algebra of
links on surfaces.