Research in Ring Theory and Quantum Groups

提案编号：dms -9970159提案题目：环理论与量子群的研究首席研究员建议继续他在环理论（代数/数学）方面的研究，集中在量子坐标代数（量化代数函数）的结构，诺etherian环和相关环。主要的焦点将放在量子代数的素数谱和原始谱的结构上。目标包括素理想的分类，素理想在自同构环下不变的有限条件，验证支持性质，如auslander -正则性，Cohen-Macaulayness和正态分离（对证明链性和研究链接至关重要），以及经典光谱或谱的量化素数或原始谱的拓扑商的表示。在理论物理学发展的刺激下，包括李代数包络代数的量子类似物和代数变体的协调，量子代数理论得到了迅速的发展，许多数学家都参与其中。尽管量子代数与它们的经典前辈之间的某些相似之处是显而易见的，同时也有其他相似之处浮出水面，但需要找到新的视角，开发新的工具来更全面地理解这些代数。这个建议集中在量子坐标环（相对于量子化包络代数）和类量子代数表现出相关的行为。在上述类型的广泛代数中已经发现了足够多的常见现象，使人们推测这类代数中一般的、公理化的基础是它们行为相似的原因。这项研究的主要长期目标是揭示和描述这样的一般结构。从中期来看，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.