Quantum symmetric spaces (Q1896836)

Let \(G\) be a semisimple Lie group with Lie algebra \({\mathfrak g}\), and let \(M\) be a symmetric space of \(G\). The Drinfeld-Jimbo classical \(r\)-matrix \(r \in \Lambda^2 {\mathfrak g}\) gives rise to a skew-bilinear bracket \(\{ , \}\) on \(C^\infty (M)\). The authors show that \(\{ , \}\) is a Poisson bracket, and construct an algebra deformation \(\mu_h = m + h\{ , \} + o(h)\) of the multiplication \(m\) on \(C^\infty (M)\) which is invariant under the quantum group \(U_h({\mathfrak g})\). If \(M\) is, in addition, equipped with a \(G\)-invariant Poisson bracket \(\{ , \}_{\text{inv}}\), the authors construct a 2-parameter deformation \[ \mu_{\nu,h} = m + \nu \{ , \}_{\text{inv}} + h\{ , \} + o(\nu,h) \] which is also \(U_h ({\mathfrak g})\)-invariant.