Center of stated $\mathrm{SL}(n)$-skein algebras: even roots of unity

Hiroaki Karuo; Zhihao Wang

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arXiv · 2509.18950

Center of stated $\mathrm{SL}(n)$-skein algebras: even roots of unity

Hiroaki Karuo·Zhihao Wang

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Abstract

We describe the center of quantum tori appearing in quantum higher Teichm\"uller theory when the quantum parameter is an even root of unity. This is a subsequent of the work in the odd roots of unity case. The centers of the quantum tori help to understand those of (reduced) stated $\mathrm{SL}(n)$-skein algebras via quantum trace maps. Moreover, we compute the PI-degree of the quantum tori, which are the same with those of (reduced) stated $\mathrm{SL}(n)$-skein algebras. As corollaries, we give matrix decompositions for anti-symmetric integer matrices for the quantum tori.

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Hiroaki Karuo, Zhihao Wang. 2025-09-23. Center of stated $\mathrm{SL}(n)$-skein algebras: even roots of unity. https://arxiv.org/abs/2509.18950

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