Invariants Under Tori of Rings of Differential Operators and Related Topics
Seiten
1998
American Mathematical Society (Verlag)
978-0-8218-0885-6 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-0885-6 (ISBN)
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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. This book shows that this is indeed the case when $G$ is a torus and $X=k^r/times (k^*)^s$.
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.
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.
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.
| Erscheint lt. Verlag | 30.10.1998 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Gewicht | 198 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Analysis | |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| ISBN-10 | 0-8218-0885-0 / 0821808850 |
| ISBN-13 | 978-0-8218-0885-6 / 9780821808856 |
| Zustand | Neuware |
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