Good filtrations of representations of Lie algebras of Cartan type
type: Item
instance of: scholarly article
title: Good filtrations of representations of Lie algebras of Cartan type
zbMATH Open document ID: 0937.17007
DOI: 10.1016/S0022-4049(97)00032-7
author: Daniel K. Nakano
author: Zongzhu Lin
published in: Journal of Pure and Applied Algebra
publication date: 1999-08-16T00:00:00Z
review text: The authors continue the investigation of \(D(G)\mathfrak u(\mathfrak g)\)-modules, where \(\mathfrak g\) is a Lie \(p\)-algebra of Cartan type, \(G=\Aut\mathfrak g\), \(D(G)\) is the distribution algebra of \(G\), \(\mathfrak u(\mathfrak g)\) is the restricted enveloping algebra of \(\mathfrak g\), \(D(G)\mathfrak u(\mathfrak g)\) is a natural Hopf algebra containing \(\mathfrak u(\mathfrak g)\) as a normal Hopf subalgebra [see \textit{Z. Lin} and \textit{D. K. Nakano}, J. Algebra 179, 852-888 (1996; Zbl 0862.17015)]. Denote \(D(G)\mathfrak u(\mathfrak g)\) by \(A\). Let \(\mathfrak g=\bigoplus _{i=-s}^r \mathfrak g_i\) be the standard grading of \(\mathfrak g, \mathfrak n^-=\bigoplus _{i<0}\mathfrak g_i, \mathfrak n^+=\bigoplus _{i>0}\mathfrak g_i, \mathfrak b^{\pm }=\mathfrak g_0\oplus \mathfrak n^{\pm }.\) It is well known that \(G\cong G_0\ltimes U^+\) where \(G_0\) is reductive (\(G_0\) is one of the groups \(GL_m, GSP_{2r}\)), \(U^+\) is unipotent. Let \(T\) be a maximal torus in \(G_0\), \(\lambda \in X(T)_+\), \[ Z^-(\lambda)=H^0(\lambda)\cong \text{Ind}_{D(G_0)\mathfrak u(\mathfrak b^-)}^A H^0(\lambda), \] where \(H^0(\lambda)\) is the classical induced \(G_0\)-module. Note that \(Z^-(\lambda)\) is infinite-dimensional and \(Z^-(\lambda)\cong H^0(\lambda)\otimes k[U^+]\) as \(D(G_0)\)-module. A \(G_0\)-module \(M\) may be considered as the trivial \(U^+\)-module and one can build the coinduced module \(\text{Coind}_{D(G)}^AM=A\bigotimes _{D(G_0)}M.\) Let \(w_0\) be the longest element in the Weyl group of \(G_0, V(\lambda)=H^0(-w_0 (\lambda))^*\) the Weyl module of highest weight \(\lambda \) for \(G_0\) and \(Z^+(\lambda)=\text{Coind}_{D(G)}^AV(\lambda).\) The latter is finite-dimensional and isomorphic to \(\mathfrak u(\mathfrak n^-)\bigotimes_kV(\lambda)\) as \(D(G_0)\)-module. \(Z^+(\lambda)\) is an analog of the Weyl module for \(A\)-modules. Let \(\mathcal P\) be a collection of \(A\)-modules. A module \(M\) is said to have a good \(\mathcal P\)-filt…
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B55
zbMATH DE Number: 1308165
zbMATH Keywords: Lie algebra of Cartan type
zbMATH Keywords: representation
zbMATH Keywords: extension
zbMATH Keywords: good filtration
zbMATH Keywords: Hopf algebra
Wikidata QID: Q115340103
reviewed by: Michael Kuznetsov
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00032-7
OpenAlex ID: W2095064727
description: scientific article; zbMATH DE number 1308165