Linear quivers and the geometric setting of quantum \(\mathrm{GL}_n\).
type: Item
instance of: scholarly article
title: Linear quivers and the geometric setting of quantum \(\mathrm{GL}_n\).
zbMATH Open document ID: 1060.16044
DOI: 10.1016/S0019-3577(02)80026-X
published in: Indagationes Mathematicae. New Series
publication date: 2003-08-07T00:00:00Z
review text: Let \({\mathcal Z}=\mathbb Z[\nu,\nu^{-1}]\) and \(\Delta=(\Delta_0,\Delta_1)\) a quiver such that the underlying graph is a simply-laced connected Dynkin graph. Let \(\Phi^+\) be the set of positive roots and \(\Theta=\{f\colon\Phi^+\to\mathbb N\}\). The twisted generic Hall algebra \(H(\Delta)\) over \(\mathcal Z\) has a \(\mathcal Z\)-free basis \(\{u_f\}_{f\in\Theta}\) and the product is defined in terms of Hall polynomials and the Euler form. \textit{C. M. Ringel} [Aportaciones Mat., Comun. 15, 85--114 (1995)] proved that \(H(\Delta)\) is isomorphic to the Lusztig form (a \(\mathcal Z\)-subalgebra) of the positive part of the quantized enveloping algebra over \(\mathbb Q(\nu)\) associated to the semisimple Lie algebra corresponding to \(\Delta\). When \(\Delta\) is a linear quiver the authors provide better relations for the product in \(H(\Delta)\). There is an algebra isomorphism between \(\mathbb H(\Delta)=H(\Delta)\otimes_{\mathcal Z}\mathbb Q(\nu)\) and the possitive part \(\mathbb V^+\) of the BLM algebra defined by \textit{A. A. Beilinson, G. Lusztig} and \textit{R. MacPherson} [Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)]. The authors extend also this isomorphism to \(H(\Delta)\) and a suitable \(\mathcal Z\)-subalgebra \(V^+\). This allows the computation of a PBW basis for \(V^+\). At this point the authors present two posets defined in terms of some algebraic varieties; the first one is the opposite of the degeneration order on \(\Phi^+\) defined by \textit{K. Bongartz} [Adv. Math. 121, No. 2, 245-.287 (1996; Zbl 0862.16007)], and the second one is the Bruhat order on some matrices with non-negative entries. Both orders are compared. The last family of algebras covered by the authors are the \(q\)-Schur algebras \(\mathbb U_r\). The authors remember a basis of \(\mathbb U_r\) coming from BLM algebras and they extend it to a \(\mathcal Z\)-subalgebra denoted by \(U_r\). In the next section the kernel of the canonical epimorphism from the …
reviewed by: F. J. Lobillo
Mathematics Subject Classification ID: 16T20
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 20C08
zbMATH DE Number: 1961120
zbMATH Keywords: canonical bases
zbMATH Keywords: Hall algebras
zbMATH Keywords: BLM algebras
zbMATH Keywords: quantized enveloping algebras
zbMATH Keywords: \(q\)-Schur algebras
author: Jie Du
author: Brian J. Parshall
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1961120