A classification of generalized quantum statistics associated with basic classical Lie superalgebras

TL;DR

This paper classifies all generalized quantum statistics linked to basic classical Lie superalgebras, extending previous work on Lie algebras to the superalgebra case using Z-gradings.

Contribution

It provides a complete classification of generalized quantum statistics associated with all basic classical Lie superalgebras, expanding the mathematical framework.

Findings

Classification of generalized quantum statistics for A(m|n), B(m|n), C(n), D(m|n) superalgebras

Extension of previous Lie algebra results to superalgebras

Connection between statistics and Z-gradings of superalgebras

Abstract

Generalized quantum statistics such as para-statistics is usually characterized by certain triple relations. In the case of para-Fermi statistics these relations can be associated with the orthogonal Lie algebra B_n=so(2n+1); in the case of para-Bose statistics they are associated with the Lie superalgebra B(0|n)=osp(1|2n). In a previous paper, a mathematical definition of ``a generalized quantum statistics associated with a classical Lie algebra G'' was given, and a complete classification was obtained. Here, we consider the definition of ``a generalized quantum statistics associated with a basic classical Lie superalgebra G''. Just as in the Lie algebra case, this definition is closely related to a certain Z-grading of G. We give in this paper a complete classification of all generalized quantum statistics associated with the basic classical Lie superalgebras A(m|n), B(m|n), C(n) and D(m|n).