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

TL;DR

This work develops a framework connecting parabosons and parafermions to ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie (super)algebras. It identifies root-vector realizations of parastatistics operators and constructs two-ensemble systems in both the graded orthogonal algebra $so_q(2n{+}1)$ and the graded orthosymplectic superalgebra $\mathfrak{osp}(1,0|2n_1,2n_2)$, using relative relations that mix para- and nonpara-statistics. The authors provide structure theory for these graded algebras, including explicit matrix forms, Cartan subalgebras, roots, and short-root vectors corresponding to parafermons and parabosons, highlighting four-type root decompositions induced by the grading. They also show how the graded paraboson realization leads to two-ensemble parastatistics, suggesting natural extensions to larger graded superalgebras and outlining the need for representation theory to construct the associated Fock spaces. Overall, the work initiates a rich algebraic route to generalized quantum statistics with potential implications for higher-rank graded symmetries and their representations.

Abstract

We present systems of parabosons and parafermions in the context of Lie algebras, Lie superalgebras, $\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-graded Lie algebras and $\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-graded Lie superalgebras. For certain relevant $\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-graded Lie algebras and $\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-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 $\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-graded Lie algebra $so_q(2n+1)$, a system consisting of two ensembles of parafermions satisfying relative paraboson relations are introduced. For the $\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-graded Lie superalgebra $osp(1,0|2n_1,2n_2)$, a system consisting of two ensembles of parabosons satisfying relative parafermion relations are introduced.