# ${\mathbb Z}_2 \times {\mathbb Z}_2 $ generalizations of ${\cal N} = 2$ super Schr\"odinger algebras and their representations

## TL;DR

This paper extends ${
m N}=2$ super Schr"odinger algebras to ${
m Z}_2 imes {
m Z}_2$-graded superalgebras, providing new algebraic structures, calculus, and vector field realizations, including a generalization of the ${
m N}=1$ case.

## Contribution

It introduces ${
m Z}_2 imes {
m Z}_2$-graded superalgebras as generalizations of super Schr"odinger algebras, with explicit realizations and calculus adaptations.

## Key findings

- Constructed ${
m Z}_2 imes {
m Z}_2$-graded superalgebras from ${
m N}=2$ super Schr"odinger algebras.
- Developed a ${
m Z}_2 imes {
m Z}_2$ calculus extending Grassmann numbers.
- Provided vector field realizations of the generalized superalgebras.

## Abstract

We generalize the real and chiral $ {\cal N} =2 $ super Schr\"odinger algebras to ${\mathbb Z}_2 \times {\mathbb Z}_2$-graded Lie superalgebras. This is done by $D$-module presentation and as a consequence, the $D$-module presentations of ${\mathbb Z}_2 \times {\mathbb Z}_2$-graded superalgebras are identical to the ones of super Schr\"odinger algebras. We then generalize the calculus over Grassmann number to ${\mathbb Z}_2 \times {\mathbb Z}_2 $ setting. Using it and the standard technique of Lie theory, we obtain a vector field realization of ${\mathbb Z}_2 \times {\mathbb Z}_2$-graded superalgebras. A vector field realization of the ${\mathbb Z}_2 \times {\mathbb Z}_2 $ generalization of ${\cal N} = 1 $ super Schr\"odinger algebra is also presented.