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Classification of minimal \(\mathbb{Z}_2 \times \mathbb{Z}_2\)-graded Lie (super)algebras and some applications. (English) Zbl 1467.81102

Summary: This paper presents the classification over the fields of real and complex numbers, of the minimal \(\mathbb{Z}_2 \times \mathbb{Z}_2\)-graded Lie algebras and Lie superalgebras spanned by four generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional \(\mathbb{Z}_2 \times \mathbb{Z}_2\)-graded Poincaré superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed \(\mathbb{Z}_2 \times \mathbb{Z}_2\)-graded (super)algebras. We mention, in particular, the notion of \(\mathbb{Z}_2 \times \mathbb{Z}_2\)-graded superspace and of invariant dynamical systems (both classical worldline sigma models and quantum Hamiltonians). As a further by-product, we point out that, contrary to \(\mathbb{Z}_2 \times \mathbb{Z}_2\)-graded superalgebras, a theory invariant under a \(\mathbb{Z}_2 \times \mathbb{Z}_2\)-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.
©2021 American Institute of Physics

MSC:

81V73 Bosonic systems in quantum theory
81T10 Model quantum field theories
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
17B70 Graded Lie (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
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