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

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 $\mathcal{N}=1$ 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.

17 pages, no figure