Research in Ring Theory and Quantum Groups

提案编号：DMS-9970159提案标题：环论和量子群研究主要研究员：K.R.Goodearl技术描述。首席研究员建议继续他在环论(代数/数学)方面的研究，集中在量子坐标代数(函数的量子化代数)、诺瑟环和相关环的结构上。主要集中在量子代数的素谱和本原谱的结构上。目标包括素理想的分类，在自同构环下不变的素理想的有限条件，支持性质的验证，如Auslander正则性，Cohen-Macaulay和正规分离(对证明链性和研究联系至关重要)，以及经典谱或簇的量子素谱或本原谱的拓扑商的表示。在李代数的包络代数的量子模拟和代数簇的坐标的理论物理的发展的推动下，量子代数的理论迅速扩展，许多数学家参与其中。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，同时也出现了其他的相似之处，但需要找到新的视角，并开发新的工具来更全面地理解这些代数。这一建议集中在量子坐标环(相对于量子化的包络代数)和表现出相关行为的一类量子代数上。在上述类型的代数的广泛范围内已经发现了许多共同的现象，这使得人们猜测这类代数中的一般的、可公理的基础可能是它们行为上的类似之处。这项研究的主要长期主旨是发现和描述这种一般结构。从中期来看，该提议旨在扩大这类代数中已知的共同现象的范围，以便更好地了解它们的共同点。

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.