Triangulated categories and Kac-Moody algebras
type: Item
instance of: scholarly article
title: Triangulated categories and Kac-Moody algebras
zbMATH Open document ID: 0966.16006
author: Liangang Peng
author: Jie Xiao
published in: Inventiones Mathematicae
publication date: 2000-12-11T00:00:00Z
review text: The positive part of a finite dimensional semisimple complex Lie algebra can be constructed from the category of representations of an associated Dynkin quiver, by Ringel's Hall algebra method. Earlier work of \textit{L. Peng} and \textit{J. Xiao} [see J. Algebra 198, No. 1, 19-56 (1997; Zbl 0893.16007) and CMS Conf. Proc. 18, 539-554 (1996; Zbl 0870.16009)] used the derived category of quiver representations modulo the square of the shift functor to construct the full Lie algebra. The present paper follows a much more general approach; it starts with a triangulated category (over a finite field) such that \(T^2=1\) and constructs a Lie algebra from it. The most complicated part is checking the Jacobi identity, which involves a lot of counting of morphisms and triangles. Specializing to representation categories of quivers, one then passes to a generic form in order to replace the finite ground field by the complex numbers. In this way, the authors manage to construct all symmetrizable Kac-Moody Lie algebras.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 1462437
zbMATH Keywords: semisimple complex Lie algebras
zbMATH Keywords: quivers
zbMATH Keywords: triangulated categories
zbMATH Keywords: Hall algebras
zbMATH Keywords: categories of quiver representations
zbMATH Keywords: derived categories
zbMATH Keywords: symmetrizable Kac-Moody Lie algebras
reviewed by: Steffen Koenig
MaRDI profile type: Publication
full work available at URL: s002220000062
OpenAlex ID: W2020187653
DOI: 10.1007/S002220000062
description: scientific article; zbMATH DE number 1462437