The Hall algebra of the category of coherent sheaves on the projective line
type: Item
instance of: scholarly article
title: The Hall algebra of the category of coherent sheaves on the projective line
zbMATH Open document ID: 0967.18005
author: Pierre Baumann
author: Christian Kassel
publication date: 2001-04-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9906037
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 1589961
zbMATH Keywords: Hall algebra
zbMATH Keywords: projective line
zbMATH Keywords: coherent sheaves
zbMATH Keywords: representations of quivers
zbMATH Keywords: enveloping algebras
zbMATH Keywords: Kac-Moody algebras
zbMATH Keywords: quantum affine algebras
zbMATH Keywords: automorphic forms
zbMATH Keywords: Kronecker quiver
MaRDI profile type: Publication
OpenAlex ID: W2055947177
arXiv ID: math/9906037
cites work: Q3338360
cites work: Q4321790
cites work: Q4328336
cites work: Q4027068
DOI: 10.1515/CRLL.2001.031
published in: Journal für die Reine und Angewandte Mathematik
review text: In important work of \textit{C. M. Ringel, J. A. Green} and others, Hall algebras of representations of quivers have been shown to be closely related with and very useful for the study of quantized enveloping algebras of Kac-Moody algebras.NEWLINENEWLINENEWLINEIn the last few years, several authors started to study Hall algebras for other types of categories. \textit{L. Peng} and \textit{J. Xiao} used derived categories of quiver representations [J. Algebra 198, No. 1, 19-56 (1997; Zbl 0893.16007) and Invent. Math. 140, No. 3, 563-603 (2000; Zbl 0966.16006)]. \textit{M. M. Kapranov} looked at derived categories of coherent sheaves of smooth projective curves over finite fields [J. Math. Sci., New York 84, No. 5, 1311-1360 (1997; Zbl 0929.11015), see also J. Algebra 202, No. 2, 712-744 (1998; Zbl 0910.18005)]. Kapranov's results gave very interesting analogies between certain relations in quantum affine algebras and the behaviour of certain automorphic forms. NEWLINENEWLINENEWLINEBy a classical result of \textit{A. A. Beilinson} [Funct. Anal. Appl. 12, 214-216 (1979; Zbl 0424.14003)], in the special case of the curve being the projective line, coherent sheaves are closely related to representations of the Kronecker quiver. The authors work out this example in great detail, thus illustrating both the situation for coherent sheaves and (implicitly) for Kronecker modules (where everything is known already from Ringel's work). They do not use any of the above references, except for definitions; instead they work out the structure of this Hall algebra from first principles.
reviewed by: Steffen Koenig
description: scientific article