# Invariants Under Tori of Rings of Differential Operators and Related Topics (Memoirs of the American Mathematical Society No. 650)

By: Ian M. Musson (author), Michel van den Bergh (author)Paperback

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### Description

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 ${\mathfrak g}=\textnormal{Lie}(G)$, $\chi\in {\mathfrak g}^\ast$ 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 $D(X[LAMBDA]!/G)$ is a simple ring.

### 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.

### Product Details

• publication date: 15/10/1998
• ISBN13: 9780821808856
• Format: Paperback
• Number Of Pages: 85
• ID: 9780821808856
• weight: 198
• ISBN10: 0821808850

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