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

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.