The structure of \({\mathfrak s\mathfrak l}(2,1)\)-supersymmetry: Irreducible representations and primitive ideals
The purpose of this paper is to give a detailed study of the enveloping algebra \(U\) of the Lie superalgebra \({\mathfrak g} = {\mathfrak s}{\mathfrak l} (2,1)\), including in particular a classification of simple Harish-Chandra modules, a detailed computation of the center \(Z\) of \(U\) (which turns out \textit{not} to be finitely generated), and a classification of the primitive spectrum of \(U\). Irreducible representations of \(U\) turn out to correspond bijectively to irreducible representations of the enveloping algebra \(U'\) of the even part \({\mathfrak g}_0 = {\mathfrak g}{\mathfrak l}(2)\) of \(\mathfrak g\), and indeed the so-called ``regular'' representations of \(U\) are obtained by induction from a subalgebra of \(\mathfrak g\) containing \({\mathfrak g}_0\). The remaining irreducible representations of \(U\) are called ``degenerate'' and remain irreducible on restriction to \({\mathfrak h} = {\mathfrak osp} (2,1)\). It turns out that primitive ideals also fall into two disjoint classes, also called regular and degenerate. The regular primitive ideals are parametrized by a certain piece of the primitive spectrum of \(U'\) while the degenerate ones are parametrized by the primitive spectrum of \(U({\mathfrak h})\), which is known by [\textit{G. Pinczon}, J. Algebra 132, 219-242 (1990; Zbl 0702.17005)]. This classification of primitive ideals of \(\mathfrak g\), unlike one recently obtained by E. Letzter in terms of the primitive spectrum of \(U'\) alone, respects the inclusion relations among the primitive ideals.
scientific article; zbMATH DE number 695062