Universal PBW-bases of Hall-Ringel algebras and Hall polynomials
type: Item
instance of: scholarly article
title: Universal PBW-bases of Hall-Ringel algebras and Hall polynomials
zbMATH Open document ID: 0893.16005
author: Liangang Peng
author: Jin Yun Guo
published in: Journal of Algebra
publication date: 1998-06-23T00:00:00Z
review text: By results of C. M. Ringel and of J. A. Green, the positive part \(U^+\) of a quantized enveloping algebra of a symmetrizable Kac-Moody algebra is isomorphic to the composition algebra \(\mathcal C\) of an associated quiver \(\Gamma\). This composition algebra is contained in and in the case of a Dynkin diagram isomorphic to the Hall algebra \(\mathcal H\) (defined by Ringel and hence also named `Hall-Ringel algebra') of the same quiver. There is a more general definition of such a Hall algebra which works for finitary rings. The authors generalize several results which were known before in more special situations. The Hall algebra by definition has a basis given by isomorphism classes of representations of \(\Gamma\). The authors show that over the field of rational numbers there is a PBW-like basis consisting of ordered products of indecomposable representations. Moreover, it is shown that \(\mathcal C\) coincides with \(\mathcal H\) if the category of representations of \(\Gamma\) has no short cycle (in which case indecomposable modules are determined by their composition factors). Furthermore, the existence of Hall polynomials is proved in some cases, in particular in the case of module categories without short cycles.
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 1106613
zbMATH Keywords: quantum groups
zbMATH Keywords: quantized enveloping algebras
zbMATH Keywords: symmetrizable Kac-Moody algebras
zbMATH Keywords: composition algebras
zbMATH Keywords: Dynkin diagrams
zbMATH Keywords: Hall algebras
zbMATH Keywords: quivers
zbMATH Keywords: PBW-like bases
zbMATH Keywords: indecomposable representations
zbMATH Keywords: categories of representations
zbMATH Keywords: indecomposable modules
zbMATH Keywords: Hall polynomials
zbMATH Keywords: module categories
reviewed by: Steffen Koenig
MaRDI profile type: Publication
full work available at URL: jabr.1997.7065
OpenAlex ID: W2082954310
cites work: Hall algebras, hereditary algebras and quantum groups
cites work: The hall polynomials of a cyclic serial algebra
cites work: Short Chains and Short Cycles of Modules
cites work: Tame algebras and integral quadratic forms
cites work: Q3972806
cites work: Hall algebras and quantum groups
cites work: Q4028610
cites work: The Composition Algebra of a Cyclic Quiver
DOI: 10.1006/JABR.1997.7065
description: scientific article; zbMATH DE number 1106613