Limit properties of quantum twists at q->1, from semi-classical twists to noncommutative geometry

Abstract

A problem of defining the quantum analogues for semi-classical twists in $U(\mathfrak{g})[[t]]$ is considered. First, we study specialization at $q=1$ of singular coboundary twists defined in $U_q(\mathfrak{g})[[t]]$ for $\mathfrak{g}$ being a nonexceptional Lie algebra, then we consider specialization of noncoboundary twists when $\mathfrak{g}={\mathfrak{sl}}_3$ and obtain $q-$deformation of the semi-classical twist introduced by Connes and Moscovici in noncommutative geometry.