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Prime spectra, automorphism groups and poisson structures associated with quantum algebras.

与量子代数相关的素谱、自同构群和泊松结构。

基本信息

批准号：
EP/D034167/1
负责人：
Tom Lenagan
金额：
$ 15.42万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2006
资助国家：
英国
起止时间：
2006 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FD034167%2F1
关键词：
Prime spectra automorphism groups poisson

项目摘要

The subject of Quantum Groups and Quantum Algebras developed out of ideas in physics in the 80s. Subsequently, the range of applications in physics, and their pivotal role in several areas of mathematics has lead to this subject being one of the most active in mathematics. From an algebraic point of view, it has recently become apparent that the subject should be studied as part of the developing theory of Noncommutative Geometry. In this theory, the noncommutative algebras arising from deformations of the classical commutative case are studied by algebraic means, but from a geometrical perspective. This development is somewhat akin to the development in physics of Quantum Mechanics as a noncommutative deformation of the classical Newtonian view of physics - the noncommutativity reflecting the uncertainty principle. From this point of view, the 'points', 'curves', 'surfaces', etc. in classical geometry are replaced in noncommutative geometry by the prime and primitive spectra and the representation theory of the algebras. The most important algebras that arise in this study are the quantum coordinate algebras and quantum enveloping algebras arising from the classical groups and the algebra of quantum matrices. Important tasks are to understand the prime spectra, to calculate automorphism groups and to understand the poisson structure that the classical world inherits from the quantum world. These are the main tasks involved in this proposal. The tasks are interlinked. In contrast with their classical counterparts, the quantum deformations are much more rigid objects (at least in the generic case) and this is reflected by the relatively small size of the so-called H-prime spectrum of these algebras. This in turn puts restrictions on the possible automorphism of the algebras and should lead to much smaller automorphism groups. The poisson structure in the classical case should then be linked in a natural way to the corresponding quantum features.

量子群和量子代数这门学科是从80年代的物理学思想发展而来的。随后，物理学的广泛应用，以及它们在数学几个领域的关键作用，使这门学科成为数学中最活跃的学科之一。从代数的角度来看，最近很明显，这个主题应该作为发展中的非交换几何理论的一部分来研究。在这个理论中，由经典交换情况的变形引起的非交换代数用代数的方法，但从几何的角度进行了研究。这种发展在某种程度上类似于量子力学在物理学中的发展，作为经典牛顿物理学观点的非对易变形——反映不确定性原理的非对易性。从这个角度来看，经典几何中的“点”、“曲线”、“曲面”等在非交换几何中被素数谱和原始谱以及代数的表示理论所取代。在本研究中出现的最重要的代数是由经典群和量子矩阵代数产生的量子坐标代数和量子包络代数。重要的任务是理解素谱，计算自同构群和理解经典世界从量子世界继承的泊松结构。这些是本提案所涉及的主要任务。任务是相互关联的。与它们的经典对应物相比，量子变形是更加刚性的对象（至少在一般情况下），这反映在这些代数的所谓h -素数谱的相对较小的尺寸上。这反过来又限制了代数可能的自同构，并应该导致更小的自同构群。经典情况下的泊松结构应该以一种自然的方式与相应的量子特征联系起来。