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

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.