Inverse and Direct Images for Quantum Weyl Algebras

Abstract

In [18] Wess and Zumino gave a method for constructing noncommutative differential calculus (or de Rham complex) on the quantum affine space associated to a Hecke symmetry R. Also, they constructed the corresponding algebra of linear differential operators. Since the algebra of linear differential operators on the n-dimensional affine space is the n-th Weyl algebra, this algebra is regarded as a quantum analogue of the Weyl algebra, and called the quantum Weyl algebra (associated to R). Let Rq,P be the multiparameter R-matrix of the quantum deformation of GLn parameterized by a scalar q and an n× n matrix P = (pij) in [3]. For the quantum Weyl algebra An(q, P ) associated to Rq,P , Demidov [6] and Rigal [15] consider quantum versions of classical theory of the Weyl algebras including Bernstein’s inequality. And, some ring-theoretic properties of An(q, P ) have been studied in [1, 2, 9, 10, 11] etc. In [11] Jordan constructed a simple localization Bn(q, P ) of An(q, P ), which is a better analogue of the Weyl algebra An from the point of view of noncommutative ring theory. The purpose of this paper is to define an analogue of the inverse and direct images for the quantum Weyl algebra An(q, P ), and to investigate their properties. In particular, we prove a quantum analogue of Kashiwara’s theorem (Section 4), and consider preservation of holonomicity under inverse and direct images (Section 5). Throughout this paper we fix a ground field K and let q be a nonzero element of K such that q is not a root of unity, and we use the following q-integer notation: