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

TL;DR

This paper classifies minimal ${ m Z}_2 imes { m Z}_2$-graded Lie (super)algebras over real and complex fields, and explores their applications in invariant dynamical systems and superspaces, extending previous special cases.

Contribution

It provides a complete classification of minimal ${ m Z}_2 imes { m Z}_2$-graded Lie (super)algebras with four generators, and extends their applications to dynamical systems and superspaces.

Findings

Classified all minimal ${ m Z}_2 imes { m Z}_2$-graded Lie (super)algebras with four generators.

Extended constructions from the ${ m Z}_2 imes { m Z}_2$-graded Poincaré superalgebra to other graded superalgebras.

Identified that such algebras imply the presence of exotic bosons with specific anticommutation properties.

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 Poincar\'e 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.