# $\cal N$-Extension of duble-graded supersymmetric and superconformal quantum mechanics

## TL;DR

This paper extends double-graded supersymmetric quantum mechanics to higher N values and demonstrates a method to convert Lie superalgebras into $Z_2^2$-graded superalgebras, linking superconformal mechanics with double grading.

## Contribution

It introduces a higher N extension of Bruce-Duplij's double-graded SQM and presents a general method to derive double-graded superalgebras from Lie superalgebras.

## Key findings

- Constructed higher N double-graded SQM models.
- Showed the method converts superalgebras to double-graded superalgebras.
- Analyzed the simplest N=1 double-graded superconformal mechanics.

## Abstract

In the recent paper, Bruce and Duplij introduced a double-graded version of supersymmetric quantum mechanics (SQM). It is an extension of Lie superalgebraic nature of ${\cal N}=1$ SQM to a $\mathbb{Z}_2^2$-graded superalgebra. In this work, we propose an extension of Bruce-Duplij model to higher values of $\cal N.$ Furthermore, it is shown that our construction of double-graded SQM is a special case of the method which converts a given Lie superalgebra to a $\mathbb{Z}_2^2$-graded superalgebra. By employing this method one may convert a model of superconformal mechanics to its double-graded version. The simplest example of ${\cal N}=1$ double-graded superconformal mechanics is studied in some detail.