Explicit construction of the finite dimensional indecomposable representations of the simple Lie-Kac $SU(2/1)$ superalgebra and their low level non diagonal super Casimir operators

Abstract

All finite dimensional irreducible representations of the simple Lie-Kac super algebra SU(2/1) are explicitly constructed in the Chevalley basis as complex matrices. For typical representations, the distinguished Dynkin label is not quantized. We then construct the generic atypical indecomposable quivers classified by Marcu, Su and Germoni and typical indecomposable N-generations block triangular extensions for any irreducible module and any integer N. In addition to the quadratic and cubic super-Casimir operators $C_2$ and $C_3$, the supercenter of the enveloping algebra contains a chiral ghost super-Casimir operator T of mixed order (2,4)in the odd generators, proportional to the superidentity grading operator $χ$, and satisfying $T = χ\;C_2$ and we define a new factorizable chiral-Casimir $T^-=C_2(1-χ)/2=(UV+WX)(VU+XW)$ where (U,V,W,X) are the odd generators. In most indecomposable cases, the super-Casimirs are non diagonal. We compute their pseudo-eigenvalues.