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

TL;DR

This paper explores the structure of $ ext{Z}_2 imes ext{Z}_2$-graded Lie (super)algebras and their connection to generalized quantum statistics, introducing new systems of parabosons and parafermions within this algebraic framework.

Contribution

It develops the structure theory of $ ext{Z}_2 imes ext{Z}_2$-graded Lie (super)algebras, identifying root vectors with parastatistics operators and constructing new quantum statistical systems.

Findings

Identified short root vectors with parastatistics operators.

Constructed systems of parabosons and parafermions satisfying specific relations.

Extended algebraic structures to include $ ext{Z}_2 imes ext{Z}_2$-graded Lie (super)algebras.

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.