Dyson's Classification And Real Division Superalgebras

TL;DR

This paper proves the equivalence between Dyson's 10-fold classification of group representations involving antiunitary operators and the classification involving superdivision algebras, unifying two important frameworks in representation theory.

Contribution

It establishes a rigorous proof demonstrating the equivalence of Dyson's 10-fold classification with the superdivision algebra classification scheme.

Findings

Confirmed the equivalence of the two 10-fold classification schemes

Provided a detailed mathematical proof of the correspondence

Clarified the relationship between antiunitary symmetries and superdivision algebras

Abstract

It is well-known that unitary irreducible representations of groups can be usefully classified in a 3-fold classification scheme: Real, Complex, Quaternionic. In 1962 Freeman Dyson pointed out that there is an analogous 10-fold classification of irreducible representations of groups involving both unitary and antiunitary operators. More recently it was realized that there is also a 10-fold classification scheme involving superdivision algebras. Here we give a careful proof of the equivalence of these two 10-fold ways.