Let's say I take the Lie group SO(2,3). It's maximal compact subgroup is SO(2)xSO(3). What I would like to know is what happens to this maximal compact subgroup if I deform to the quantum group SOq(2,3) [By this I mean a deformation of the universal enveloping algebra of SO(2,3) into the quantum group]. Is it simply SOq(2)xSOq(3) or is something more complicated going on?
Generally speaking few subgroups survives quantization. As an example the usual embedding o(n)->o(n+1) does not survive in quantization. A quantum group has few quantum subgroups (there are a couple of different notions of quantum subgroups though). In the strong sens of a Hopf subalgebra of the original Hopf algebra I have very strong doubts that U_q(so(2))xU_q(so(3) will sit inside U_q(so(2,3)). It has to be carefully checked with explicit formula of generators and relations.
Dear Jonas,
Thank you for your insight. I will take a look at your work and do some reading on quantum Schubert cells and coideal algebras.
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