Noncommutative reality-based algebras of rank 6

TL;DR

This paper classifies 6-dimensional noncommutative semisimple reality-based algebras with positive degree maps, providing parametrizations, formulas, and a list of integral table algebras up to order 150, with applications to association schemes.

Contribution

It offers a complete classification of rank 6 noncommutative RBAs with positive degree maps, including explicit formulas and enumeration of integral table algebras.

Findings

Parametrization of RBAs by seven real numbers.

Explicit formulas for bases and structure constants.

List of all integral table algebras of rank 6 up to order 150.

Abstract

We classify the RBA-bases of $6$-dimensional noncommutative semisimple algebras for which the algebra has a positive degree map. We show that these RBAs are parametrized by seven real numbers, the first four of which are positive and the remaining three arbitrary. Our classification gives formulas for their standard bases and structure constants. Using these we give a list of all noncommutative integral table algebras of rank 6 with order up to 150. Four in the list are primitive, but we show these cannot be realized as adjacency algebras of association schemes. In the last section of the paper we apply our methods to give a precise description of the noncommutative integral table algebras of rank 6 for which the multiplicity of both linear characters is 1.