The structure of the Kauffman bracket skein algebra at roots of unity

This paper examines the structure of the Kauffman bracket skein algebra of a punctured surface at roots of unity. A criterion that determines when a collection of skeins forms a basis of the skein algebra as a module over the SL(2,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL(2,{{\mathbb {C}}})$$\end{document} characters of the fundamental group of the surface, with appropriate localization is given. This is used to prove that when the algebra is localized so that every nonzero element of the center has a multiplicative inverse, it is a division algebra. Finally, it is proved that the localized skein algebra can be split as a module over its center as a tensor product of two commutative subalgebras.