# quantum deformation

The standard quantum groups, say $GL_q(n)$ or $U_q(gl(n))$, depend on the parameter $q$, which in the classical limit tends toward 1. Let $t:=q-1$ be considered as a generic parameter then we can consider the ring of regular functions on $GL_q(n)$ or the algebra $U_q(gl(n))$ as algebras over the local ring $k[[t]]$. Are these algebras flat over $k[[t]]$? Does anyone have a reference for that?

• I think this is basically the PBW theorem, which is in most quantum groups books. – Peter Samuelson May 10 '15 at 21:16