# Schur indices for noncommutative reality-based algebras with two nonreal basis elements

## TL;DR

This paper explores the structure of noncommutative reality-based algebras with two nonreal basis elements, showing their simple components are generalized quaternion algebras and revealing implications for association schemes.

## Contribution

It provides a new characterization of noncommutative simple components as quaternion algebras in the specific case of two nonreal basis elements.

## Key findings

- Noncommutative simple components are generalized quaternion algebras over their field of definition.
- The real numbers always form a splitting field for these algebras.
- Certain rank 7 association schemes must have at least three symmetric relations.

## Abstract

This article discusses the representation theory of noncommutative algebras reality-based algebras with positive degree map over their field of definition. When the standard basis contains exactly two nonreal elements, the main result expresses the noncommutative simple component as a generalized quaternion algebra over its field of definition. The field of real numbers will always be a splitting field for this algebra, but there are noncommutative table algebras of dimension $6$ with rational field of definition for which it is a division algebra. The approach has other applications, one of which shows noncommutative association scheme of rank $7$ must have at least three symmetric relations.