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Global dimension of generalized Weyl algebras. (English) Zbl 0857.16025

Bautista, Raymundo (ed.) et al., Representation theory of algebras. Seventh international conference, August 22-26, 1994, Cocoyoc, Mexico. Providence, RI: American Mathematical Society. CMS Conf. Proc. 18, 81-107 (1996).
As in many recent papers the author considers a class of algebras called generalized Weyl algebras \(A\). They are generated over a base ring \(D\) by \(2n\) generators subject to certain relations. As special cases one gets, for example, the first Weyl algebra and \(U_q({\mathfrak sl}_2)\). D. A. Jordan has studied some related algebras. The author shows that \(A\) has left global dimension equal to one of: \(\text{lgld }D\), \(\text{lgld }D+1\) or \(\infty\). When \(D\) is semiprime Noetherian the author obtains a criterion for the second of these three possibilities.
For the entire collection see [Zbl 0837.00015].

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16E10 Homological dimension in associative algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16S32 Rings of differential operators (associative algebraic aspects)
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