The recognition problem for table algebras and reality-based algebras

TL;DR

This paper investigates the recognition problem for real basis algebras (RBAs), establishing existence results and exploring structural properties like integrality, positivity, and generalized table algebra structures for specific classes.

Contribution

It proves that any finite-dimensional noncommutative semisimple algebra with involution has an RBA-basis and analyzes conditions for integral, rational, and positive structure constants.

Findings

Every such algebra has an RBA-basis.

Conditions for positive degree maps and nonnegative structure constants are characterized.

Results are explicitly settled for algebras of the form C ⊕ M_n(C), n ≥ 2.

Abstract

Given a finite-dimensional noncommutative semisimple algebra $A$ with involution, we show that $A$ always has an RBA-basis. We look for an RBA-basis that has integral or rational structure constants, and ask if the RBA admits a positive degree map. For RBAs that have a positive degree map, we try to find an RBA-basis with nonnegative structure constants to determine if there is a generalized table algebra structure. We settle these questions for the algebras $\mathbb{C} \oplus M_n(\mathbb{C})$, $n \ge 2$.