$ \mathbb{Z}_2 \times \mathbb{Z}_2 $ generalizations of ${\cal N} = 1$ superconformal Galilei algebras and their representations

TL;DR

This paper introduces new $\mathbb{Z}_2 \times \mathbb{Z}_2$ graded superalgebras related to superconformal Galilei algebras, expanding their representations and providing explicit realizations.

Contribution

It constructs two classes of $\mathbb{Z}_2 \times \mathbb{Z}_2$ graded superalgebras from superconformal Galilei algebras and develops their representations and realizations.

Findings

Realized superalgebras within the universal enveloping algebra.

Extended superconformal Galilei algebra representations to $\mathbb{Z}_2 \times \mathbb{Z}_2$ graded algebras.

Provided a boson-fermion Fock space representation and vector field realizations.

Abstract

We introduce two classes of novel color superalgebras of $ \mathbb{Z}_2 \times \mathbb{Z}_2 $ grading. This is done by realizing members of each in the universal enveloping algebra of the ${\cal N}=1$ supersymmetric extension of the conformal Galilei algebra. This allows us to upgrade any representation of the super conformal Galilei algebras to a representation of the $ \mathbb{Z}_2 \times \mathbb{Z}_2 $ graded algebra. As an example, boson-fermion Fock space representation of one class is given. We also provide a vector field realization of members of the other class by using a generalization of the Grassmann calculus to $ \mathbb{Z}_2 \times \mathbb{Z}_2 $ graded setting.