Transparent Subalgebras and Local Module Categories

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.

Disclosure

“ivalent over E. It follows that B and BA Acknowledgements. HY was partly supported by a start-up grant from the University of Alberta and an NSERC Discovery Grant. KS was supported by JSPS KAKENHI Grant Number JP24K06676. The authors used ChatGPT while developing and drafting parts of this paper to explore possible formulations and proof strategies. All mathematical statements and proofs were independently checked by the authors, who take sole responsibility for the contents of the”

PDF page 2

Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-22T18:17:58.125110Z

Structural counts