2019/12/04 by Creamer, Daniel
#Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1912.02269
This paper introduces a computational approach to classifying low rank modular categories up to their modular data. The modular data of a modular category is a pair of matrices, (S,T). Virtually all the numerical information of the category is contained within or derived from the modular data. The modular data satisfy a variety of criteria that Bruillard, Ng, Rowell, and Wang call the admissibility criteria. Of note is the Galois group of the S matrix is an abelian group that acts faithfully on the columns of the eigenvalue matrix, s = (\fracSijS0j). This gives an injection from Gal(ℚ(S),ℚ) → Symr, where r is the rank of the category. Our approach begins by listing all the possible abelian subgroups of Sym6 and building all the possible modular data for each group. We run each set of modular data through a series of Gröbner basis calculations until we either find a contradiction or solve for the modular data.