More tight monomials in quantized enveloping algebras
type: Item
instance of: scholarly article
title: More tight monomials in quantized enveloping algebras
zbMATH Open document ID: 0912.17007
published in: Journal of Algebra
publication date: 1999-05-26T00:00:00Z
review text: Let \(U=U_q(g)\) be the quantum group associated to a semisimple Lie algebra \(g\) of rank \(n\) by Drinfeld and Jimbo. The negative part \(U^-\) of \(U\) has a canonical basis \(B\) with some nice properties. To calculate the elements of \(B\) explicitly is a hard problem; it was solved completely only in types \(A_1\) and \(A_2\). In [Isr. Math. Conf. Proc. 7, 117-132 (1993)], \textit{G. Lusztig} describes a method to calculate certain elements of \(B\). Fixing an orientation of the Dynkin diagram of \(g\), Lusztig associates a quadratic form and shows that a monomial is tight (resp. semi-tight) provided this quadratic form satisfies a certain positivity condition (resp. nonnegativity condition) and that the positivity condition for tightness always holds in type \(A_3\), and asks when one has (semi-)tightness in type \(A_n\). In this paper, the author shows that for a certain orientation of the Dynkin diagram, the positivity condition is always satisfied in type \(A_4\). It is also demonstrated that Lusztig's comment that things get more complicated in cases with higher rank in type \(A\), is true. In particular, the author exhibits a reduced expression for \(w_0\) in type \(A_r\) \((r\geq 6)\) with a quadratic form that does not even satisfy the conditions for semi-tightness, for any orientation of the Dynkin diagram. In Section 2, the author describes the situation he is working in, and in Section 3, he summarizes the results about tight monomials from Lusztig's paper which are needed. In Section 4, the author shows how they can be applied to the type \(A_4\) case, and in Section 5, he explains how some techniques of linear programming providing spanning sets in a certain sense for cones can be applied to understanding the quadratic form. These techniques are applied to a tractable reduced expression in type \(A_5\). In Section 6, the counterexample mentioned above is given. In Section 7, the author gives a general description in type \(A_n\) of the spanning ve…
reviewed by: Fang Li
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1218681
zbMATH Keywords: quantum group
zbMATH Keywords: quantized enveloping algebra
zbMATH Keywords: tight monomial
zbMATH Keywords: quadratic form
author: Bethany R. Marsh
MaRDI profile type: Publication
Wikidata QID: Q56285891
full work available at URL: jabr.1997.7370
OpenAlex ID: W2121345434
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DOI: 10.1006/JABR.1997.7370
description: scientific article; zbMATH DE number 1218681