On the Hall algebra of coherent sheaves on \(\mathbb P^1\) over \(\mathbb F^1\)
type: Item
instance of: scholarly article
title: On the Hall algebra of coherent sheaves on \(\mathbb P^1\) over \(\mathbb F^1\)
zbMATH Open document ID: 1279.14022
published in: Journal of Pure and Applied Algebra
publication date: 2012-05-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.3570
review text: In the paper the category \(Coh_n(\mathbb{P}^1)\) of normal coherent sheaves on the monoid scheme \(\mathbb{P}^1\) is defined and studied. This category resembles in most ways a finitary abelian category (but not additive) and hence the Ringel-Hall algebra on it may be defined. The algebra is proved to be isomorphic as a Hopf algebra to the enveloping algebra of the product of a non-standard Borel in the loop algebra \(Lgl_2\) and an abelian Lie algebra on infinitely many generators. This may be viewed as a (\(q=1\)) version of Kapranov's result relating (a certain subalgebra of) the Ringel-Hall algebra of \(\mathbb{P}^1\) over \(\mathbb{F}_q\) to a non-standard quantum Borel inside the quantum loop algebra \(\mathbb{U}_v(\hat{sl}_2),\) where \(v^2=q.\)
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 14A20
zbMATH DE Number: 6039251
zbMATH Keywords: coherent sheave
zbMATH Keywords: monoid scheme
zbMATH Keywords: Hall algebra
author: Matt Szczesny
reviewed by: Huang Hua-Lin
MaRDI profile type: Publication
OpenAlex ID: W2964259105
arXiv ID: 1009.3570
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DOI: 10.1016/J.JPAA.2011.08.001
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description: scientific article; zbMATH DE number 6039251