Memoirs of the American Mathematical Society 1998; 85 pp; softcover Volume: 136 ISBN10: 0821808850 ISBN13: 9780821808856 List Price: US$47 Individual Members: US$28.20 Institutional Members: US$37.60 Order Code: MEMO/136/650
 If \(G\) is a reductive algebraic group acting rationally on a smooth affine variety \(X\), then it is generally believed that \(D(X)^G\) has properties very similar to those of enveloping algebras of semisimple Lie algebras. In this book, the authors show that this is indeed the case when \(G\) is a torus and \(X=k^r\times (k^*)^s\). They give a precise description of the primitive ideals in \(D(X)^G\) and study in detail the ring theoretical and homological properties of the minimal primitive quotients of \(D(X)^G\). The latter are of the form \(B^x=D(X)^G/({\mathfrak g}\chi({\mathfrak g}))\) where \({\mathfrak g}= \mathrm{Lie}(G)\), \(\chi\in {\mathfrak g}^\ast\) and \({\mathfrak g}\chi({\mathfrak g})\) is the set of all \(v\chi(v)\) with \(v\in {\mathfrak g}\). They occur as rings of twisted differential operators on toric varieties. It is also proven that if \(G\) is a torus acting rationally on a smooth affine variety, then \(D(X/\!/G)\) is a simple ring. Readership Graduate students and research mathematicians working in rings of differential operators; algebraic geometers and others interested in toric varieties. Table of Contents  Introduction
 Notations and conventions
 A certain class of rings
 Some constructions
 The algebras introduced by S. P. Smith
 The Weyl algebras
 Rings of differential operators for torus invariants
 Dimension theory for \(B^\chi\)
 Finite global dimension
 Finite dimensional representations
 An example
 References
