Central extensions of mapping class groups of surfaces from stated skein algebras

Let $\mathrm{\Sigma }$ be a surface of genus $g$ with zero or one boundary component and $n$ marked points, and $H$ a finite-dimensional factorizable ribbon Hopf algebra. We compute the central extension of the mapping class group of $\mathrm{\Sigma }$, associated to the projective representation defined from the stated skein algebra of $\mathrm{\Sigma }$ and $H$. Our proof is purely two-dimensional, and makes no use of TQFT arguments.