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

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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.