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 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 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 byproduct we point out that, contrary to Z2 × Z2graded 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.