Invariants under Tori of Rings of Differential Operators and Related Topics (Memoirs of the AMS) (Memoirs of the American Mathematical Society) - Softcover

Ian M. Musson; Michel Van Den Bergh

Softcover

ISBN 10: 0821808850 ISBN 13: 9780821808856

Publisher: American Mathematical Society, 1999

Synopsis

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.

Edition details

• Publisher: American Mathematical Society
• Publication date: 1999
• Language: English
• ISBN 10: 0821808850
• ISBN 13: 9780821808856
• Binding: Paperback
• Number of pages: 85