${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie (super)algebras and generalized quantum statistics

Abstract

We present systems of parabosons and parafermions in the context of Lie algebras, Lie superalgebras, ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie algebras and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie superalgebras. For certain relevant ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie algebras and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie superalgebras, some structure theory in terms of roots and root vectors is developed. The short root vectors of these algebras are identified with parastatistics operators. For the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie algebra $s{o}_q(2n+1)$, a system consisting of two ensembles of parafermions satisfying relative paraboson relations are introduced. For the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie superalgebra $osp(1,0|2n_1,2n_2)$, a system consisting of two ensembles of parabosons satisfying relative parafermion relations are introduced.