The $osp(1|2)$ \Z2 graded algebra and its irreducible representations

TL;DR

This paper constructs finite-dimensional irreducible representations of two versions of the -graded osp(1|2) algebra, analyzing their Casimir operators, and provides explicit matrix and differential operator realizations.

Contribution

It introduces new finite-dimensional irreducible representations for two variants of the -graded osp(1|2) algebra, detailing their Casimir operators and explicit realizations.

Findings

Two different second-order Casimir operators identified for the ten-generator algebra.

Eight-generator algebra has only one Casimir invariant.

Explicit matrix and differential operator realizations provided.

Abstract

In this paper, we construct finite dimensional irreducible representations for two different versions of \Z2 graded $osp(1|2)$ algebra based on eight- and ten-generators. We find that there are two different second-order Casimir operators for the ten generators version of the algebra one corresponding to the $(0,0)$ sector and another in the $(1,1)$ sector. Consequently, it is shown that the eight-generator version of the algebra has only one Casimir invariant in the $(0,0)$ sector. We present the differential operator realizations of these algebras. Starting with the highest weight, we construct the states of the irreducible finite-dimensional representations for both versions of the algebras. The matrix realizations of the generators on the representation space corresponding to these states are written down explicitly.