An equivalence of two categories of \(\text{sl}(n,\mathbb{C})\)-modules
Let \(\mathfrak g\) be a complex finite-dimensional semi-simple Lie algebra with a fixed triangular decomposition \({\mathfrak g}={\mathfrak n}_-\oplus{\mathfrak h}\oplus{\mathfrak n}_+\) and let \({\mathcal O}={\mathcal O}({\mathfrak g})\) denote the category of finitely generated, \(\mathfrak h\)-diagonalizable, and \({\mathfrak n}_+\)-locally finite \(\mathfrak g\)-modules. A module in \(\mathcal O\) is said to have a quasi-Verma flag if it has a descending separating filtration such that each factor is a (non-zero) submodule in a Verma module. Let \(\mathcal K\) denote the full subcategory of \(\mathcal O\) which consists of all modules with integral support having a quasi-Verma flag and being complete with respect to Enright's completion functor. As was shown by \textit{V. Futorny} and the authors of the paper under review [in Manuscr. Math. 102, No. 4, 487-503 (2000; Zbl 1018.17005)], Enright's completion functor can be used to make \(\mathcal K\) into an Abelian category (which is not the Abelian structure inherited from that on \(\mathcal O\)) such that tensoring with any finite-dimensional module is an exact endofunctor. Every full subcategory of the category of all finitely generated \(\mathfrak g\)-modules having this property is called admissible. In the following let \({\mathfrak g}=\text{sl}(n,\mathbb{C})\) and consider the full subcategory \({\mathcal F}={\mathcal F}(GZ)\) in the category of all \(\mathfrak g\)-modules consisting of all subfactors of modules \(F\otimes GZ\) where \(F\) is an arbitrary finite-dimensional \(\mathfrak g\)-module and \(GZ\) is a fixed simple generic Gelfand-Zetlin module. Then \(\mathcal F\) (with its natural Abelian structure) is admissible. The main result of the paper under review is an explicit construction of an equivalence between the categories \(\mathcal K\) and \(\mathcal F\) which commutes with translation functors. Moreover, the authors apply this equivalence to certain generalizations of the category \(\mathcal O\) introduced by them jointly with \textit{V. Futorny} [Algebr. Represent. Theory 5, No. 3, 259-276 (2002; Zbl 1031.17008)] and to obtain some new information on Gelfand-Zetlin modules.
complex finite-dimensional semi-simple Lie algebras
scientific article; zbMATH DE number 1802384