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 $r\colon \pi_1(S) \to \mathrm{SL}_2(\mathbb C)$ a representation of the skein algebra $\mathcal S^A(S)$ that is uniquely determined up to isomorphism.

中文翻译：

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考夫曼括号绞线代数 III 的表示：封闭曲面和自然性

这是从 [BonWon3, BonWon4] 开始的系列中的第三篇文章，专门讨论定向表面 $S$ 的 Kauffman 支架绞线代数的有限维表示。在 [BonWon3] 中，我们将经典阴影与绞线代数的不可约表示 $\rho$ 相关联，这是一个字符 $r_\rho \in \mathcal R_{\mathrm{SL}_2(\mathbb C)}(S )$ 由群同态 $\pi_1(S) \to \mathrm{SL}_2(\mathbb C)$ 表示。当前文章的主要结果是，当表面 $S$ 闭合时，每个字符 $r\in \mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$ 作为经典阴影出现考夫曼括号绞线代数的不可约表示。我们还证明了我们证明中使用的构造是自然的，