An equivalence of two categories of \(\text{sl}(n,\mathbb{C})\)-modules
type: Item
instance of: scholarly article
title: An equivalence of two categories of \(\text{sl}(n,\mathbb{C})\)-modules
zbMATH Open document ID: 1010.16006
DOI: 10.1023/A:1016531419024
author: Volodymyr Mazorchuk
author: Steffen Koenig
published in: Algebras and Representation Theory
publication date: 2002-09-18T00:00:00Z
review text: 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 \(\mathca…
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 1802384
zbMATH Keywords: category \(\mathcal O\)
zbMATH Keywords: Enright completion functors
zbMATH Keywords: equivalences of categories
zbMATH Keywords: complex finite-dimensional semi-simple Lie algebras
zbMATH Keywords: Verma modules
zbMATH Keywords: Abelian categories
zbMATH Keywords: finite-dimensional modules
zbMATH Keywords: exact endofunctors
zbMATH Keywords: Gelfand-Zetlin modules
zbMATH Keywords: translation functors
MaRDI profile type: Publication
full work available at URL: a:1016531419024
OpenAlex ID: W1492784833
description: scientific article; zbMATH DE number 1802384