eBook ISBN:  9781470402396 
Product Code:  MEMO/136/650.E 
List Price:  $47.00 
MAA Member Price:  $42.30 
AMS Member Price:  $28.20 
eBook ISBN:  9781470402396 
Product Code:  MEMO/136/650.E 
List Price:  $47.00 
MAA Member Price:  $42.30 
AMS Member Price:  $28.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 136; 1998; 85 ppMSC: Primary 16; 32;
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.
ReadershipGraduate students and research mathematicians working in rings of differential operators; algebraic geometers and others interested in toric varieties.

Table of Contents

Chapters

1. Introduction

2. Notations and conventions

3. A certain class of rings

4. Some constructions

5. The algebras introduced by S.P. Smith

6. The Weyl algebras

7. Rings of differential operators for torus invariants

8. Dimension theory for $B^\chi $

9. Finite global dimension

10. Finite dimensional representations

11. An example


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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.
Graduate students and research mathematicians working in rings of differential operators; algebraic geometers and others interested in toric varieties.

Chapters

1. Introduction

2. Notations and conventions

3. A certain class of rings

4. Some constructions

5. The algebras introduced by S.P. Smith

6. The Weyl algebras

7. Rings of differential operators for torus invariants

8. Dimension theory for $B^\chi $

9. Finite global dimension

10. Finite dimensional representations

11. An example