On quantization of the Semenov–Tian–Shansky Poisson bracket on simple algebraic groups

Let $G$ be a simple complex factorizable Poisson algebraic group. Let $\mathcal U_\hbar(\mathfrak g)$ be the corresponding quantum group. We study the $\mathcal U_\hbar(\mathfrak g)$-equivariant quantization $\mathcal C_\hbar[G]$ of the affine coordinate ring $\mathcal C[G]$ along the Semenov–Tian–Shansky bracket. For a simply connected group $G$, we give an elementary proof for the analog of the Kostant–Richardson theorem stating that $\mathcal C_\hbar[G]$ is a free module over its center.