Let G be a semisimple real Lie group with simply connected complexification G_c. Let U be the maximal compact subgroup of G_c. Then U is a compact semisimple Lie group which admits a q-deformation into a compact quantum group U_q. In this talk, we will explain how also the quotient space X = G\G_c can be q-deformed into a U_q-equivariant quantum space X_q with a classical space of orbits, denoted X_q/U_q. We further show how one particular orbit in X_q/U_q gives rise to a quantum symmetric space (in the sense of G. Letzter) through an adaptation of the universal K-matrix formalism (due to M. Balagovic and S. Kolb). This is joint work with Marco Matassa.
Quantum symmetric spaces and quantizations of semisimple real Lie groups