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arXiv · 2608.19153

Transparent Subalgebras and Local Module Categories

Kenichi Shimizu·Harshit Yadav
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Abstract

Let $A$ be a commutative simple algebra in a braided finite tensor category $\mathcal{B}$. We identify the largest transparent subalgebra of $A$ as the algebra induced by a central lift of the free-module functor. This identification gives formulas for the Frobenius-Perron dimension and the Müger center of the category of local $A$-modules. These formulas give criteria for nondegeneracy, symmetry, and modularity, together with sharp bounds on $\mathrm{FPdim}_{\mathcal{B}}(A)$. We also realize the Müger center of $\mathcal{B}$ as a category of local modules over an adjoint algebra. Finally, we prove a relative-center factorization and deduce that taking the category of local modules preserves the relative Witt class.

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Kenichi Shimizu, Harshit Yadav. 2026-08-19. Transparent Subalgebras and Local Module Categories. https://arxiv.org/abs/2608.19153

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