Representations of the Kauffamn skein algebra of small surfaces

Abstract

We prove a uniqueness result for finite-dimensional representations of the Kauffman skein algebra ${\mathcal{S}}_A(S)$ of a surface $S$, when $A$ is a root of unity and when the surface $S$ is a sphere with at most four punctures or a torus with at most one puncture. We show that, if two irreducible representations of ${\mathcal{S}}_A(S)$ have the same classical shadow and the same puncture invariants, and if this classical shadow is sufficiently generic in the character variety ${\mathcal{X}}_{S{L}_2(\mathbb{C})}(S)$, then the two representations are isomorphic.

18 pages, 3 figures. arXiv admin note: text overlap with arXiv:1206.1638 by other authors

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