A classification of lowest weight irreducible modules over $\mathbb{Z}_2^2$-graded extension of $osp(1|2)$

TL;DR

This paper classifies the irreducible lowest weight modules of a $Z_2^2$-graded extension of $osp(1|2)$, relevant for $Z_2^2$-graded superconformal mechanics, by constructing Verma modules and identifying singular vectors.

Contribution

It provides the first classification of irreducible lowest weight modules for this graded extension of $osp(1|2)$, including explicit formulas for singular vectors.

Findings

Classification of irreducible lowest weight modules achieved

Explicit formulas for all singular vectors provided

Framework established for representation theory of $Z_2^2$-graded superalgebras

Abstract

We investigate representations of the $\mathbb{Z}_2^2$-graded extension of $osp(1|2)$ which is the spectrum generating algebra of the recently introduced $\mathbb{Z}_2^2$-graded version of superconformal mechanics. The main result is a classification of irreducible lowest weight modules of the $\mathbb{Z}_2^2$-graded extension of $osp(1|2)$. This is done via introduction of Verma modules and its maximal invariant submodule generated by singular vectors. Explicit formula of all singular vectors are also presented.