A classification of generalized quantum statistics associated with the exceptional Lie (super)algebras

TL;DR

This paper completes the classification of generalized quantum statistics linked to all exceptional Lie (super)algebras, expanding understanding of their algebraic structures and associated gradings.

Contribution

It finalizes the classification of GQS for all exceptional Lie algebras and superalgebras, including new Z-gradings for superalgebras.

Findings

Reproduces known Z-gradings of exceptional Lie algebras

Provides the first classification of Z-gradings for exceptional Lie superalgebras

Completes the classification of GQS for all exceptional Lie (super)algebras

Abstract

Generalized quantum statistics (GQS) associated to a Lie algebra or Lie superalgebra extends the notion of para-Bose or para-Fermi statistics. Such GQS have been classified for all classical simple Lie algebras and basic classical Lie superalgebras. In the current paper we finalize this classification for all exceptional Lie algebras and superalgebras. Since the definition of GQS is closely related to a certain Z-grading of the Lie (super)algebra G, our classification reproduces some known Z-gradings of exceptional Lie algebras. For exceptional Lie superalgebras such a classification of Z-gradings has not been given before.