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

Abstract

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 $4$ 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 byproduct 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.

arXiv comment29 pages; final version to appear in J. Math. Phys

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• New aspects of the Z${}_{\text{2}}$ $\times$ Z${}_{\text{2}}$-graded 1D superspace: induced strings and 2D relativistic models
• On classical $Z_2\times Z_2$-graded Lie algebras
• The $Z_2\times Z_2$-graded Lie superalgebras $pso(2n+1|2n)$ and $pso(\infty |\infty )$, and parastatistics Fock spaces
• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
• Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics
• Orthosymplectic $Z_2\times Z_2$-graded Lie superalgebras and parastatistics
• Integrable ${\mathbb{Z}}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace
• Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras
• Transmuted spectrum-generating algebras and detectable parastatistics of the Superconformal Quantum Mechanics
• On the detectability of paraparticles beyond bosons and fermions
• Universal weight systems from a minimal ${\mathbb{Z}}_2^2$-graded Lie algebra
• Braided quantum mechanics and Majorana qubits at third root of unity: a color Heisenberg-Lie (super)algebra framework