Spherical subalgebras of quantized enveloping algebras - structure theory and classification problems

李群或代数群的球子群自20世纪70年代以来一直被研究，因为它们具有有趣的几何、代数、组合和表示理论性质。如今，像球形变体这样的泛化是人们感兴趣的焦点。对于量子群的另一个推广，首先必须找到合适的设置：由于缺乏足够多的Hopf子代数，用Hopf子代数来描述Hopf代数的明显方法是失败的。在90年代量子对称空间的逐个构造和量子化包络代数的右上理想子代数的最新发展的基础上，我们现在能够利用右上理想子代数建立量子化包络代数的球子代数的实质结构理论，并启动分类方案。(在经典的余交换环境下，右上理想子代数自动成为Hopf子代数。)这是一个非常有趣的问题，经典理论和量子理论和例子在多大程度上是相似的，一个人是否可以观察到有意义的新方面。

Spherical subgroups of Lie or algebraic groups have been investigated since the 1970ies because of their interesting geometric, algebraic, combinatorial and representation theoretical properties. Nowadays generalizations like spherical varieties are in the focus of interest. For another generalization towards quantum groups one first has to find the proper setting: an obvious description via Hopf subalgebras of Hopf algebras fails due to the lack of sufficiently many Hopf subalgebras. Based on the case-by-case construction of quantum symmetric spaces in the 1990ies and on recent developments on right coideal subalgebras of quantized enveloping algebras, now we are in the position to develop a substantial structure theory of spherical subalgebras of quantized enveloping algebras using right coideal subalgebras and to initiate classification projects. (In the classical, cocommutative setting right coideal subalgebras are automatically Hopf subalgebras.) It is a very interesting question, to which extent the classical and the quantum theories and examples are analogous and whether one can observe significant new aspects.