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

TL;DR

This paper demonstrates the integrality of non-semisimple quantum representations of mapping class groups at roots of unity by constructing explicit bases that form invariant lattices over cyclotomic integers.

Contribution

It provides explicit bases for state spaces that ensure the integrality of quantum representations associated with the small quantum group at roots of unity.

Findings

Constructed explicit bases spanning Z[ζ]-lattices

Proved invariance under mapping class group actions

Extended integrality results to non-semisimple quantum representations

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.