On Integrality of Non-Semisimple Quantum Representations of Mapping Class Groups

Marco De Renzi, Jules Martel
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Abstract

For a root of unity $\zeta$ of odd prime order, we restrict coefficients of non-semisimple quantum representations of mapping class groups associated with the small quantum group $\mathfrak{u}_\zeta \mathfrak{sl}_2$ from $\mathbb{Q}(\zeta)$ to $\mathbb{Z}[\zeta]$. We do this by exhibiting explicit bases of states spaces that span $\mathbb{Z}[\zeta]$-lattices that are invariant under projective actions of mapping class groups.
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论映射类群的非半简单量子表示的积分性
对于奇素数阶的合根 $\zeta$,我们将与小量子群 $\mathfrak{u}_\zeta \mathfrak{sl}_2$ 相关联的映射类群的非半纯量子表示的系数从 $\mathbb{Q}(\zeta)$ 限制到 $\mathbb{Z}[\zeta]$ 。我们通过展示跨 $\mathbb{Z}[\zeta]$ 格的状态空间的显式基,这些状态空间在映射类群的投影作用下是不变的。
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