Research in Ring Theory and Quantum Groups

项目编号：DMS-9970159项目名称：环理论与量子群研究项目负责人：K.R. Goodear技术说明。主要研究者建议继续他的调查在环理论（代数/数学），集中onthe结构的量子坐标代数（量子代数函数），noether环，和相关的环。主要研究量子代数的素谱和本原谱的结构，目标包括素理想的分类，素理想在自同构环面下不变的有限性条件，支持性质如Auslander正则性，Cohen-Macaulay性和正规分离性的证明（对于证明连锁性和重复链接至关重要），以及量子化素谱或本原谱与经典谱或簇的拓扑关系的表示。受李代数包络代数的量子类似物和代数簇的坐标化等理论物理学发展的推动，量子代数理论迅速发展，许多数学家也参与其中。尽管量子代数与其经典前辈之间的某些相似之处是显而易见的，同时也出现了其他相似之处，但需要找到新的观点，and new新tools工具developed发达towunderstand理解these algebras代数more fully充分.这一建议集中于量子坐标环（而不是量子化包络代数）和表现出相关行为的量子代数类。在上述类型的广泛代数中已经发现了足够多的共同现象，从而使人们猜想这类代数中的普遍的、可公理化的基础是它们行为的相似之处。这项研究的主要长期目标是揭示和描述这种一般结构。在中期，该提案旨在扩大这类代数中已知的共享现象的范围，以便更好地了解它们的共同基础。

Proposal No: DMS-9970159Proposal Title: Research in Ring Theory and Quantum GroupsPrincipal Investigator: K.R. GoodearlTechnical Description. The Principal Investigator proposes to continuehis investigations in ring theory (algebra/mathematics), concentrating onthe structure of quantum coordinate algebras (quantized algebras offunctions), noetherian rings, and related rings. The main focus will beon the structure of the prime and primitive spectra of quantum algebras.The goals include classification of prime ideals, finiteness conditionsfor prime ideals invariant under tori of automorphisms, verification ofsupporting properties such as Auslander-regularity, Cohen-Macaulayness,and normal separation (crucial to proving catenarity and investigatinglinks), and presentations of quantized prime or primitive spectra astopological quotients of classical spectra or varieties.Abstract. Spurred by developments in theoretical physics involvingquantum analogs of enveloping algebras of Lie algebras and coordinaterings of algebraic varieties, the theory of quantum algebras has rapidlyexpanded, and numerous mathematicians have become involved in it.Although certain similarities between quantum algebras and theirclassical predecessors are evident and others have surfaced in themeantime, new perspectives need to be found, and new tools developed tounderstand these algebras more fully. This proposal concentrates onquantum coordinate rings (as opposed to quantized enveloping algebras)and classes of quantum algebras exhibiting related behavior.Sufficiently many common phenomena have been discovered in a wide rangeof algebras of the above type to lead one to conjecture that general,axiomatizable underpinnings within this class of algebras areresponsible for the parallels in their behavior. The main long-termthrust of the research is to uncover and describe such generalstructures. In the medium term, the proposal aims to extend the range ofknown shared phenomena within this class of algebras, in order to gainbetter insight into their common base.