On double Ringel-Hall algebras
type: Item
instance of: scholarly article
title: On double Ringel-Hall algebras
zbMATH Open document ID: 1010.16011
author: Bangming Deng
author: Jie Xiao
published in: Journal of Algebra
publication date: 2002-09-08T00:00:00Z
review text: Due to the work of \textit{J. A. Green} [Invent. Math. 120, No. 2, 361-377 (1995; Zbl 0836.16021)] the Ringel-Hall algebra of a finite-dimensional hereditary algebra \(\Lambda\) augmented by a torus algebra can be endowed with a Hopf algebra structure. Therefore one can construct its Drinfeld double \({\mathcal D}(\Lambda)\). Analogous to a result of \textit{J. A. Green} [op. cit.] and \textit{C. M. Ringel} [in: Representation theory of algebras, CMS Conf. Proc. 19, 185-245 (1996; Zbl 0860.16026)] which states that the Drinfeld double of the composition subalgebra of a Ringel-Hall algebra augmented by a torus algebra is isomorphic to Lusztig's form of the quantized enveloping algebra of the corresponding Kac-Moody algebra, the authors of the paper under review show that \({\mathcal D}(\Lambda)\) is isomorphic to the quantized enveloping algebra of a generalized Kac-Moody algebra in the sense of \textit{R. Borcherds} [J. Algebra 115, No. 2, 501-512 (1988; Zbl 0644.17010)]. In the quiver case it was proved before by \textit{B. Sevenhant} and \textit{M. Van den Bergh} [J. Pure Appl. Algebra 160, No. 2-3, 319-332 (2001; Zbl 1025.17006)] that the Ringel-Hall algebra is isomorphic to the positive part of the quantized enveloping algebra of a generalized Kac-Moody algebra. The authors of the paper under review also observe that \({\mathcal D}(\Lambda)\) is a restricted non-degenerate member of some datum in the sense of \textit{J. A. Green} [op. cit.]. As a consequence, they obtain that the Ringel-Hall algebra is independent of the orientation of \(\Lambda\). In the quiver case this was pointed out by \textit{G. Lusztig} [J. Am. Math. Soc. 3, No. 2, 447-498 (1990; Zbl 0703.17008)] and proved by \textit{B. Sevenhant} and \textit{M. Van den Bergh} [J. Algebra 221, No. 1, 135-160 (1999; Zbl 0955.16016)] by using a notion of Fourier transform. Moreover, the authors of the paper under review define in a natural way a highest weight category \(\mathcal O\) and integrable modules…
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1797466
zbMATH Keywords: Ringel-Hall algebras
zbMATH Keywords: finite-dimensional hereditary algebras
zbMATH Keywords: torus algebras
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Drinfeld doubles
zbMATH Keywords: composition subalgebras
zbMATH Keywords: quantized enveloping algebras
zbMATH Keywords: Kac-Moody algebras
zbMATH Keywords: highest weight categories
zbMATH Keywords: integrable modules
zbMATH Keywords: \(R\)-matrix
zbMATH Keywords: quantum Yang-Baxter equation
zbMATH Keywords: tensor products
zbMATH Keywords: irreducible integrable highest weight modules
zbMATH Keywords: Kac-Weyl character formula
MaRDI profile type: Publication
full work available at URL: jabr.2001.9151
OpenAlex ID: W2055435624
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DOI: 10.1006/JABR.2001.9151
description: scientific article; zbMATH DE number 1797466