On the coloured graph structure of Lusztig's canonical basis
type: Item
instance of: scholarly article
title: On the coloured graph structure of Lusztig's canonical basis
zbMATH Open document ID: 0881.17010
author: Markus Reineke
published in: Mathematische Annalen
publication date: 1998-02-17T00:00:00Z
review text: Let \({\mathfrak g}\) be a semisimple finite dimensional Lie algebra over the complex numbers, \(\Delta\) its Dynkin diagram, \(U\) the associated quantized enveloping algebra and \(U^+\) its positive part. Lusztig has shown the existence of what is called a canonical basis \({\mathfrak B}\) of \(U^+\). From Kashiwara's theory of crystal bases one gets a coloured graph structure on \({\mathcal B}\), which contains all the crystal graphs of finite dimensional simple representations. The paper under review gives a handy description of the coloured graph structure of \({\mathfrak B}\) under the additional assumption that \({\mathfrak g}\) does not contain a component of type \(E_8\). The method is based on \textit{C. M. Ringel}'s [Invent. Math. 101, 583-591 (1990; Zbl 0735.16009)] description of \(U^+\) as a Hall algebra of a quiver. There is a bijection between elements of the canonical basis \({\mathfrak B}\) and isomorphism classes of representations of certain quivers which are oriented graphs with underlying diagram \(\Delta\). Ringel defined a multiplicative structure on these representations. Thus the action of Kashiwara's operators can be given an interpretation within the category of representations of a quiver. Indecomposable representations and homomorphisms or extensions between them are conveniently encoded in the (finite) Auslander-Reiten quiver \(\Gamma\). Since the orientation of \(\Delta\) can be chosen arbitrarily, one has a choice between several Auslander-Reiten quivers to work with. For the cases \(A_n\), \(D_n\), \(E_6\) and \(E_7\), the author finds an orientation of \(\Delta\) such that the space of homomorphisms from any indecomposable module to a simple one has dimension zero or one. (The cases \(B_n\), \(C_n\), \(F_4\) and \(G_2\) are covered by a slight modification of the method.) This implies that the crucial set of representations to be considered in this context becomes a partially ordered set. The computation of the coloured graph struc…
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 1031515
zbMATH Keywords: semisimple Lie algebra
zbMATH Keywords: indecomposable representations
zbMATH Keywords: canonical basis
zbMATH Keywords: coloured graph structure
zbMATH Keywords: Hall algebra
zbMATH Keywords: quivers
zbMATH Keywords: Auslander Reiten quiver
zbMATH Keywords: antichains
zbMATH Keywords: crystal graphs
reviewed by: Steffen Koenig
MaRDI profile type: Publication
full work available at URL: s002080050058
OpenAlex ID: W2036900976
DOI: 10.1007/S002080050058
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description: scientific article; zbMATH DE number 1031515