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

李群或代数群的球面子群由于其有趣的几何、代数、组合和表示理论性质，自20世纪70年代以来一直受到研究。如今，像球形簇这样的推广是人们感兴趣的焦点。对于量子群的另一个推广，首先必须找到适当的设置：由于缺乏足够多的Hopf子代数，通过Hopf代数的Hopf子代数的明显描述失败了。基于20世纪90年代量子对称空间的逐例构造和量子化包络代数的右共理想子代数的最新发展，现在我们可以利用右共理想子代数发展量子化包络代数的球面子代数的实质结构理论，并开始分类工程. (In经典的、余交换的设置右余理想子代数自动是Hopf子代数。这是一个非常有趣的问题，在何种程度上经典和量子理论和例子是相似的，以及人们是否可以观察到重要的新方面。

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.