A classification of generalized quantum statistics associated with classical Lie algebras

TL;DR

This paper provides a comprehensive classification of generalized quantum statistics linked to classical Lie algebras, introducing new classes and establishing a general framework based on Z-gradings and root vectors.

Contribution

It introduces a general definition of quantum statistics associated with classical Lie algebras and classifies all such statistics for A_n, B_n, C_n, D_n, including new classes.

Findings

Complete classification of generalized quantum statistics for classical Lie algebras

Introduction of a Z-grading framework for these statistics

Discovery of several new classes of quantum statistics

Abstract

Generalized quantum statistics such as para-Fermi statistics is characterized by certain triple relations which, in the case of para-Fermi statistics, are related to the orthogonal Lie algebra B_n=so(2n+1). In this paper, we give a quite general definition of ``a generalized quantum statistics associated to a classical Lie algebra G''. This definition is closely related to a certain Z-grading of G. The generalized quantum statistics is then determined by a set of root vectors (the creation and annihilation operators of the statistics) and the set of algebraic relations for these operators. Then we give a complete classification of all generalized quantum statistics associated to the classical Lie algebras A_n, B_n, C_n and D_n. In the classification, several new classes of generalized quantum statistics are described.