On the structure of an \(\alpha\)-stratified generalized Verma module over Lie algebra \(sl(n, \mathbb{C})\)
type: Item
instance of: scholarly article
title: On the structure of an \(\alpha\)-stratified generalized Verma module over Lie algebra \(sl(n, \mathbb{C})\)
zbMATH Open document ID: 0855.17003
DOI: 10.1007/BF02567805
author: Volodymyr Mazorchuk
published in: Manuscripta Mathematica
publication date: 1995-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/156110
review text: Let \({\mathfrak g}\) be a finite-dimensional simple complex Lie algebra. A \({\mathfrak g}\)-module \(V\) is said to be \(\alpha\)-stratified for some root \(\alpha\) if \(X_\alpha\) and \(X_{-\alpha}\) act injectively on \(V\), where \(X_{\pm \alpha}\) are root vectors for the one-dimensional root spaces \({\mathfrak g}_{\pm \alpha}\). Fixing \({\mathfrak g}\) to be \(sl(n, \mathbb{C})\), the author continues his investigation into the structure of \(\alpha\)-stratified generalized Verma modules \(M(\lambda, p)\) [Ukr. Math. J. 45, No. 9, 1360-1371 (1993); translation from Ukr. Mat. Zh. 45, No. 9, 1215-1224 (1993; Zbl 0827.17009)]. The Verma module \(M(\lambda, p)\) is generated by a weight vector \(v_\lambda\) of weight \(\lambda\), which is killed by all positive root vectors except \(X_\alpha\), and such that the `small Casimir operator' of the \(sl(2, \mathbb{C})\) subalgebra associated to the root \(\alpha\) acts on \(v_\lambda\) as multiplication by \(p^2\). The main result of the paper is to give a description of the submodules of an \(\alpha\)-stratified Verma module similar to that in the classic Bernstein-Gelfand-Gelfand paper [Funct. Anal. Appl. 5, 1-8 (1971); translation from Funkts. Anal. Prilozh. 5, No. 1, 1-9 (1971; Zbl 0246.17008)]. Once the \(\alpha\)-stratified technical machinery is constructed in a series of lemmas, the outline of the proof of the main theorem follows that of BGG.
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 866150
zbMATH Keywords: simple complex Lie algebra
zbMATH Keywords: \(\alpha\)-stratified generalized Verma modules
reviewed by: Duncan J. Melville
MaRDI profile type: Publication
cites work: Structure of representations generated by vectors of highest weight
cites work: A generalization of Verma modules, and irreducible representations of the Lie algebra sl(3)
cites work: Weight sl(3)-modules generated by semiprimitive elements
cites work: Generalized Verma modules, the Cartan-Helgason theorem, and the Harish- Chandra homomorphism
cites work: \(\alpha\)-stratified modules over the Lie algebra \(sl(n,\mathbb{C})\)
cites work: Structure of certain induced representations of complex semisimple Lie algebras
OpenAlex ID: W2079161690
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description: scientific article; zbMATH DE number 866150