Sheaves of differential operators and D-modules over non-commutative projective spaces

Abstract

For a scheme, let D be the sheaf of differential operators, assigning to any open subscheme it’s ring of differential operators. The study of D-modules advances their theory independently, but pervades many other areas of modern mathematics as well. Most notably, the theory provided a framework to solve Hilbert’s 21-st problem, and to develop the Riemann-Hilbert correspondence, and eventually led to the resolution of the Kazhdan-Lustig conjecture in representation theory. For an affine patch of the scheme having dimension n, the sheaf will assign the n-th Weyl algebra. In [1], Hayashi develops the quantized Weyl algebra, a deformation of this algebra, and in [2] Lunts and Rosenberg develop versions of β and quantum differential operators for a graded non-commutative algebra. Iyer and McCune compute in [3] the ring of these quantum differential operators of Lunts and Rosenberg over the polynomial algebra in n-variables, or, over affine n-space. In [4], Bischof examines how a reconciliation of the β deformation in [2] and a 2-cocycle deformation of the graded algebra influence the category of these quantum D-modules, and considers some localizations. One naturally wonders about the category of modules for these quantum differential operators on a non-commutative space; about it’s objects and it’s structure. With the aim of future study in non-commutative grassmannians and flag varieties, of U[subscript]q(sl[subscript]n), for example, we consider a non-commutative projective space glued together from a covering of 2-cocycle deformed polynomial rings, as proposed in [5] and [4]. We determine when there exists a deformed polynomial ring from which we can obtain this covering, and the category of quasi-coherent sheaves can be realized via the categorical Proj construction. With a guiding hand from Rosenberg’s [5] we develop a general ring structure for containing these quantum differential operators on polynomial algebras. Finally, towards the goal of defining holonomic quantum D-modules, we consider the GK-dimension of the corresponding associated graded algebra for the purpose of determining the dimension of what might be considered the singular support for a quantum D-module.