Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality

Abstract

This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we associated a classical shadow to an irreducible representation $\rho$ of the skein algebra, which is a character $r_{\rho }\in {\mathcal{R}}_{{\mathrm{SL}}_2(\mathbb{C})}(S)$ represented by a group homomorphism ${\pi }_1(S)\to {\mathrm{SL}}_2(\mathbb{C})$. The main result of the current article is that, when the surface $S$ is closed, every character $r\in {\mathcal{R}}_{{\mathrm{SL}}_2(\mathbb{C})}(S)$ occurs as the classical shadow of an irreducible representation of the Kauffman bracket skein algebra. We also prove that the construction used in our proof is natural, and associates to each group homomorphism $r:{\pi }_1(S)\to {\mathrm{SL}}_2(\mathbb{C})$ a representation of the skein algebra ${\mathcal{S}}^A(S)$ that is uniquely determined up to isomorphism.

50 pages. Version 2: Takes into account the impact on uniqueness statements of sign-reversal symmetries of the data, which had been overlooked in the earlier versions of this article and of [BonWon4]; the manuscript is now ready for journal submission

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