Totally positive sequences and \(R\)-matrix quadratic algebras
type: Item
instance of: scholarly article
title: Totally positive sequences and \(R\)-matrix quadratic algebras
zbMATH Open document ID: 1026.16013
DOI: 10.1007/BF02677498
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-09-14T00:00:00Z
review text: The author studies the Hilbert series of the quadratic algebra associated with a unitary \(R\)-matrix. The main result is that the Hilbert series is rational with real positive poles and real negative zeros. This result has also been obtained for a more general \(R\)-matrix-Hecke operator by the reviewer [Acta Math. Vietnam. 24, No. 2, 235-246 (1999; Zbl 0949.16025)] and for a closed Hecke operator, it is shown [\textit{Nguyên Phuong Dung} and the reviewer, Int. Math. Res. Not. 2003, No. 40, 2193-2203 (2003; Zbl 1041.16035)] that the numerator and the denominator of the Poincaré series are reciprocal polynomials. Using the main result, the author gives an example of a Koszul algebra that is not associated to any unitary \(R\)-matrix. He also proposes an application of the main theorem in classifying cotriangular Hopf algebras using a superversion of a theorem of Deligne. Note that this superversion of Deligne's theorem has recently been proved by \textit{P. Deligne} himself [Mosc. Math. J. 2, No. 2, 227-248 (2002; Zbl 1005.18009)]. Note also that the author's idea has also been carried out in more detail by \textit{P. Etingof} and \textit{S. Gelaki} [Am. J. Math. 123, No. 4, 699-713 (2001; Zbl 0990.16030)] using the above mentioned result of Deligne.
Mathematics Subject Classification ID: 16S37
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1506082
zbMATH Keywords: \(R\)-matrices
zbMATH Keywords: Koszul algebras
zbMATH Keywords: Hilbert series
zbMATH Keywords: quadratic algebras
zbMATH Keywords: Poincaré series
zbMATH Keywords: cotriangular Hopf algebras
author: Alexei Davydov
MaRDI profile type: Publication
cites work: Q3708148
cites work: Q3932925
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cites work: Koszul Resolutions
cites work: Power series with integral coefficients
full work available at URL: https://doi.org/10.1007/bf02677498
OpenAlex ID: W2009819744
description: scientific article; zbMATH DE number 1506082