Representations of the Kauffman bracket skein algebra of the punctured torus

We describe the action of the Kauffman bracket skein algebra on some vector spaces that arise as relative Kauffman bracket skein modules of tangles in the punc- tured torus. We show how this action determines the Reshetikhin-Turaev representation of the punctured torus. We rephrase our results to describe the quantum group quantiza- tion of the moduli space of at SU(2)-connections on the punctured torus with fixed trace of the holonomy around the boundary.