Homomorphisms between Verma modules in characteristic p
type: Item
instance of: scholarly article
title: Homomorphisms between Verma modules in characteristic p
zbMATH Open document ID: 0644.17003
DOI: 10.1016/0021-8693(88)90132-9
author: James Franklin
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathfrak g}\) be a complex semisimple Lie algebra, U its universal enveloping algebra and K an algebraically closed field of characteristic p. Then U has a \({\mathbb{Z}}\)-form \(U_{{\mathbb{Z}}}\) (that is, a subring such that \(U_{{\mathbb{Z}}}\otimes {\mathbb{C}}=U)\). Let \(U_ K=U_{{\mathbb{Z}}}\otimes K\) and \(V_{\lambda}\) be the Verma-module of \({\mathfrak g}\) with highest weight \(\lambda\). If \(V_{\mu}\) is a simple composition factor of \(V_{\lambda}\) then there are \({\mathbb{Z}}\)-forms (i.e. \({\mathbb{Z}}\)-lattices invariant under \(U_{{\mathbb{Z}}})\) of \(V_{\mu}\) and \(V_{\lambda}\) which can be tensored to yield \(U_ K\)-modules \(V_{\mu,K}\) and \(V_{\lambda,K}\), respectively. The author studies homomorphisms between \(V_{\mu,K}\) and \(V_{\lambda,K}\). The main result is a construction of a nonzero homomorphism from \(V_{\mu,K}\) to \(V_{\lambda,K}\) for \(\mu =\lambda -dr\), where r is a root and d is a positive integer which is small enough and satisfies a certain integrality condition with respect to \(\lambda\).
reviewed by: Joachim Hilgert
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 4049834
zbMATH Keywords: complex semisimple Lie algebra
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: characteristic p
zbMATH Keywords: Verma-module
zbMATH Keywords: highest weight
zbMATH Keywords: homomorphisms
Wikidata QID: Q60578971
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90132-9
OpenAlex ID: W2136708999
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description: scientific article; zbMATH DE number 4049834