Prime Ideals in Skew and Q-Skew Polynomial Rings

by Goodearl, K. R.; Letzter, E. S.

• ISBN13: 9780821825839
• ISBN10: 0821825836
• Format: Paperback
• Copyright: 1994-07-01
• Publisher: Amer Mathematical Society

Summary

There has been continued interest in skew polynomial rings and related constructions since Ore's initial studies in the 1930s. New examples not covered by previous analyses have arisen in the current study of quantum groups. The aim of this work is to introduce and develop new techniques for understanding the prime ideals in skew polynomial rings $S=R[y;\tau, \delta]$, for automorphisms $\tau$ and $\tau$-derivations $\delta$ of a noetherian coefficient ring $R$. Goodearl and Letzter give particular emphasis to the use of recently developed techniques from the theory of noncommutative noetherian rings. When $R$ is an algebra over a field $k$ on which $\tau$ and $\delta$ act trivially, a complete description of the prime ideals of $S$ is given under the additional assumption that $\tau {-1 \delta \tau = q\delta$ for some nonzero $q\in k$. This last hypothesis is an abstraction of behavior found in many quantum algebras, including $q$-Weyl algebras and coordinate rings of quantum matrices, and specific examples along these lines are considered in detail.

Table of Contents

Introduction

Preliminaries for $S=R[y;\tau,\delta ]$

Tau-delta-prime coefficient rings

Each prime ideal of $S$ is associated to a unique $\tau$-orbit in $\operatorname{spec}R$

Annihilator primes and induced bimodules

Prime ideals in quadratic $(-1)$-skew extensions

Prime ideals in $S$ associated to infinite orbits

The general case Prime ideals in $S$ associated to infinite orbits

The $q$-skew case Prime ideals in $S$ associated to finite orbits

The general case Prime ideals in $S$ associated to finite orbits

The $q$-skew case Classification of prime ideals in $q$-skew extensions

Irreducible finite dimensional representations of $q$-skew extensions

Quantized Weyl algebras

Prime factors of coordinate rings of quantum matrices

Chains of prime ideals in iterated Ore extensions

References

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