Classification of minimal Z2×Z2-graded Lie (super)algebras and some applications

Abstract

This paper presents the classification over the fields of real and complex numbers, of the minimal Z2×Z2-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 Z2×Z2-graded Poincaré superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed Z2×Z2-graded (super)algebras. We mention, in particular, the notion of Z2×Z2-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 Z2×Z2-graded superalgebras, a theory invariant under a Z2×Z2-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

ℤ2 × ℤ2-graded Lie algebras and supseralgebras and related physical models

“…The Z 2 × Z 2 -graded algebras and related systems have been not investigated so much [19], [20]. In [19] a classifiation of low-dimensional Z 2 × Z 2 -graded Lie algebras and superalgebras is presented.…”

“…The Z 2 × Z 2 -graded algebras and related systems have been not investigated so much [19], [20]. In [19] a classifiation of low-dimensional Z 2 × Z 2 -graded Lie algebras and superalgebras is presented. In this Proceedings we describe a model which possesses a Z 2 × Z 2 -graded Lie algebra A1 from [19] as a symmetry algebra.…”

“…In [19] a classifiation of low-dimensional Z 2 × Z 2 -graded Lie algebras and superalgebras is presented. In this Proceedings we describe a model which possesses a Z 2 × Z 2 -graded Lie algebra A1 from [19] as a symmetry algebra. Some unusual properties of such a system are discussed.…”

“…The 1D Z 2 × Z 2 -graded supersymmetry induces 2D super-Poincaré model This section mostly based on [18]. The S10 Z 2 × Z 2 -graded abstract superalgebra is taken from [19] classification.…”

“…Consider Z 2 × Z 2 -graded color Lie superálgebra S = S10 from [19] represented by operators acting on graded superspace. The operators and the coordinates are associated with the graded parts as H, t ∈ S 00 , Q 01 , θ 01 ∈ S 01 , Q 10 , θ 10 ∈ S 10 , Z, z ∈ S 11 .…”