Deformations of differential operators

Abstract

The Weyl algebra is the algebra of differential operators on a commutative ring of polynomials in finitely many variables. In Hayashi1990, Hayashi defines an algebra which he refers to as the quantized n-th Weyl algebra given by a deformation of the classical Weyl algebra. In luntsdifferential, Lunts and Rosenberg define [beta] and quantum differential operators for localization of quantum groups by deforming the relations that algebras of differential operators satisfy. In Iyer2007, Iyer and Mccune compute the quantum differential operators on the polynomial algebra with n variables. One naturally wonders ``What is the relationship between the quantized Weyl algebra and the quantum differential operators on the polynomial algebra with n variables?" In this thesis we answer this question by comparing the natural representations of U[subscript]q(sl[subscript]2) emerging from each algebra. Additionally, we connect the differential operators on the big cell of the flag variety of U[subscript]q(sl[subscript]n) with our deformed algebras. We also show the relationship between these algebras of differential operators and those appearing in the quantum Beilinson-Bernstein equivalence. Next we discuss analogous results in the case of [beta]-differential operators, as introduced in luntsdifferential. We consider both deformations on the underlying coordinate rings, and of the algebra of differential operators. We relate these results to the gluing problem for differential operators on noncommutative coordinate rings. We collect some of the different deformations of the usual Weyl algebra, and compare them based on a common bicharacter [beta]. Finally, we show a geometric result need in order to be able to glue deformed spaces and have their algebras of deformed differential operators cohere.