Categories of finite dimensional weight modules over type I classical Lie superalgebras
\textit{V. G. Kac} classified the finite-dimensional irreducible representations of basic simple Lie superalgebras, and provided character formulas for the typical representations [Representations of classical Lie superalgebras, Lect. Notes Math. 676, 579-626 (1977; Zbl 0388.17002)]. Since then, important contributions have been made to understand the atypical modules of the type I superalgebras, \({sl} (m,n)\) and \(C(n)\), but apart from \(C(n)\) the character problem was never completely solved. The present paper is motivated by the conjectures in [\textit{J. Van der Jeugt}, \textit{J. W. B. Hughes}, \textit{R. C. King} and \textit{J. Thierry-Mieg}, J. Math. Phys. 31, 2278-2304 (1990; Zbl 0725.17004)]\ and in [\textit{V. Serganova}, Adv. Sov. Math. 16, No. 2, 151-165 (1993; Zbl 0795.17033)]. The author develops some Bernstein-Gelfand-Gelfand theory with Verma modules replaced by Kac modules. Kazhdan-Lusztig polynomials can be defined using the cohomology groups \(\text{Ext}^n (K(\lambda), L(\mu))\), where \(K(\lambda)\) is the Kac module with highest weight \(\lambda\) and \(L(\mu)\) is the simple module with highest weight \(\mu\). These polynomials are shown to coincide with those of Serganova. In the case of \({sl} (1,n)\) and \(C(n)\), these polynomials can be computed explicitly; the resulting character formula coincides with that given by Van der Jeugt, Hughes, King and Thierry-Mieg.
scientific article; zbMATH DE number 869788