A classification of generalized quantum statistics associated with basic classical Lie 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 Bn=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).