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

量子群和量子代数的主题是在80年代的物理学思想中发展起来的。随后，在物理学中的应用范围，以及它们在数学的几个领域中的关键作用，导致这一主题成为数学中最活跃的学科之一。从代数的角度来看，最近已经变得很明显，这个问题应该作为非交换几何发展理论的一部分进行研究。在该理论中，通过代数方法但从几何角度研究了由经典交换情况的变形产生的非交换代数。这种发展有点类似于量子力学的物理学发展，作为经典牛顿物理学观点的非对易变形-非对易性反映了不确定性原理。从这个观点出发，经典几何中的“点”、“曲线”、“曲面”等在非交换几何中被素谱、本原谱和代数的表示论所取代。在本研究中出现的最重要的代数是量子坐标代数和量子包络代数产生的经典群和量子矩阵的代数。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。这些是这项建议所涉及的主要任务。这些任务是相互关联的。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素谱的相对较小的尺寸上。这反过来又限制了代数的可能自同构，并应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。

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.