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

2022/07/13 by Jean Thierry‐Mieg, Peter Jarvis, Thierry-Mieg, Jean +3
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2207.06545

openalex publication_date 2022/07/13 · openalex created_date 2022/07/16 · openalex updated_date 2026/07/28

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 C₂ and C₃, 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 C₂ and we define a new factorizable chiral-Casimir T^-=C₂(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.

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