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Inverse and direct images for quantum Weyl algebras. (English) Zbl 0999.16035
This paper is concerned with quantum analogues of ideas (inverse and direct images, Kashiwara’s theorem, and preservation of holonomicity) that are well established for modules over Weyl algebras and are explained in Chapters 14 to 18 of ‘A primer of algebraic $$\mathcal D$$-modules’ by S. C. Coutinho [Lond. Math. Soc. Student Texts 33, Cambridge University Press (1995; Zbl 0848.16019)].
The quantum Weyl algebra $$A_n(q,P)$$ is generated by $$n$$ variables and $$n$$ partial differential operators, or skew partial differential operators, on a quantum affine space. It arises from work of J. Wess and B. Zumino [Nucl. Phys. B, Proc. Suppl. 18B, 302-312 (1990; Zbl 0957.46514)], and is associated with a differential graded algebra generated by the variables and corresponding skew differentials and called a Wess-Zumino differential calculus. By a result of the reviewer, [J. Algebra 174, No. 1, 267-281 (1995; Zbl 0833.16025)], under suitable conditions on the parameters, $$A_n(q,P)$$ has a simple localization $$B_n(q,P)$$ which is Noetherian of Krull and global dimension $$n$$ and may be regarded as a better analogue of the Weyl algebra $$A_n$$ than $$A_n(q,P)$$ itself. L. Rigal, [Bull. Sci. Math. 121, No. 6, 477-505 (1997; Zbl 0895.17007)], introduced a notion of holonomicity for $$B_n(q,P)$$-modules and proved an analogue of Bernstein’s inequality. In the present paper, given a differential graded algebra morphism between two Wess-Zumino differential calculi, the author constructs, under appropriate conditions on the parameters, inverse and direct image functors between categories of left modules over the corresponding quantum Weyl algebras. The restrictions on the parameters are stronger for direct images than for inverse images because of the need for an involution to switch from right modules to left modules. Holonomicity is defined for modules over $$A_n(q,P)$$ and results giving sufficient conditions for its preservation by inverse and direct images are given. An analogue of Kashiwara’s theorem on the equivalence of categories arising from the direct image for an embedding is given for modules over the simple localized quantum Weyl algebras.

MSC:
 16W35 Ring-theoretic aspects of quantum groups (MSC2000) 16S36 Ordinary and skew polynomial rings and semigroup rings 16D30 Infinite-dimensional simple rings (except as in 16Kxx) 16D90 Module categories in associative algebras 16E45 Differential graded algebras and applications (associative algebraic aspects) 16S32 Rings of differential operators (associative algebraic aspects) 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory
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