Classification of Homogeneous Fourier Matrices

TL;DR

This paper establishes a correspondence between Fourier matrices linked to modular data and self-dual C-algebras, revealing that homogeneous C-algebras from these matrices have all degrees equal to one.

Contribution

It introduces a novel correspondence between Fourier matrices and self-dual C-algebras, and characterizes homogeneous C-algebras derived from these matrices.

Findings

One-to-one correspondence between Fourier matrices and self-dual C-algebras.

Homogeneous C-algebras from Fourier matrices have all degrees equal to one.

Provides new insights into the structure of modular data and related algebraic objects.

Abstract

Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group $SL_2(\mathbb{Z})$. In this paper, we show that there is a one-to-one correspondence between Fourier matrices associated to modular data and self-dual $C$-algebras that satisfy a certain condition. Also, we prove that a homogenous $C$-algebra arising from a Fourier matrix has all the degrees equal to $1$.