On the quantum KPRV determinants for semisimple and affine Lie algebras
type: Item
instance of: scholarly article
title: On the quantum KPRV determinants for semisimple and affine Lie algebras
zbMATH Open document ID: 1043.17008
DOI: 10.1023/A:1014410819083
author: Anthony Joseph
author: Dragana Todoric
published in: Algebras and Representation Theory
publication date: 2002-06-24T00:00:00Z
review text: \textit{K. R. Parthasarathy, R. Ranga Rao}, and \textit{V. S. Varadarajan} [Ann. Math. (2) 85, 383--429 (1967; Zbl 0177.18004)] defined the now so-called PRV determinants relative to a standard Borel subalgebra of a finite-dimensional complex simple Lie algebra. Later, \textit{B. Kostant} [Lie groups and their representations (ed. I. Gelfand), Summer School Conference, Budapest, 1971, Halsted/Wiley, New York, 231--331 (1975; Zbl 0327.22010)] introduced similar determinants relative to certain parabolic subalgebras while having in mind applications to principal series representations. The goal of the paper under review is to define and study analogs of KPRV determinants for the quantized enveloping algebra \(U_q(\mathfrak g)\) of a split finite-dimensional semisimple or affine Lie algebra \(\mathfrak g\) over a field of characteristic zero. (Note that it does not seem to be not clear how to define KPRV determinants for the universal enveloping algebra \(U(\mathfrak g)\) of an affine Lie algebra \(\mathfrak g\).) The first section contains an overview of the paper and some of the motivation for introducing quantum KPRV determinants. The second section is a review of the necessary background and notation for quantized enveloping algebras and their representation theory. In the third section the authors consider simple integrable admissible \(U_q(\mathfrak g)\)-modules \(V\) with non-vanishing zero weight space \(V_0\). They define a filtration of \(V_0\) with a corresponding basis which they call Baumann filtration and Baumann basis, respectively. The latter is then used to define the quantum KPRV determinant \(\mathfrak p_{\pi^\prime,V}\) associated to \(V\) and a set \(\pi^\prime\) of simple roots corresponding to a parabolic subalgebra of \(\mathfrak g\). The authors show that \(\mathfrak p_{\pi^\prime,V}\) is non-zero and up to a constant multiple independent of the choice of the Baumann basis. The rest of the paper almost exclusively is concerned with the case of …
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 1758597
zbMATH Keywords: semisimple Lie algebra
zbMATH Keywords: affine Lie algebra
zbMATH Keywords: quantized enveloping algebra
zbMATH Keywords: parabolic subalgebra
zbMATH Keywords: admissible module
zbMATH Keywords: integrable module
zbMATH Keywords: simple module
zbMATH Keywords: Baumann filtration
zbMATH Keywords: Baumann basis
zbMATH Keywords: quantum KPRV determinant
zbMATH Keywords: generalized Verma module
zbMATH Keywords: annihilator
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1758597