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

• URL: http://arxiv.org/abs/2501.15541v1
• Date: Sun, 26 Jan 2025 14:27:07 GMT
• Title: $\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-graded Lie (super)algebras and generalized quantum statistics
• Authors: N. I. Stoilova, J. Van der Jeugt,
• Abstract summary: We present systems of parabosons and parafermions in the context of Lie algebras, Lie superalgebras, and $mathbfmathbb Z times mathbb Z$-graded Lie algebras.
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• 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.

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