The Structure of the Kauffman Bracket Skein Algebra at Roots of Unity

Abstract: This paper is focused on 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 an extension over the $SL(2,{\mathbb C})$ 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, that it is a division algebra. Finally, it is proved that the localized skein algebra can be split over its center as a tensor product of two commutative subalgebras.