Application of Koszul complex to Wronski relations for \(U(\mathfrak{gl}_n)\)
type: Item
instance of: scholarly article
title: Application of Koszul complex to Wronski relations for \(U(\mathfrak{gl}_n)\)
zbMATH Open document ID: 1058.17007
author: Tôru Umeda
published in: Commentarii Mathematici Helvetici
publication date: 2004-02-05T00:00:00Z
review text: It is well-known that the center \(Z({\mathfrak g}{\mathfrak l}_n)\) of the universal enveloping algebra \(U({\mathfrak g}{\mathfrak l}_n)\) of the general linear Lie algebra \({\mathfrak g}{\mathfrak l}_n\) is isomorphic to the polynomial ring in \(n\) variables. Furthermore, there are two classical sets of generators in \(Z({\mathfrak g}{\mathfrak l}_n)\), the first one consists of the so-called Capelli elements \(C_k\), which can be expressed via determinants, and the second one, discovered by \textit{M. L. Nazarov} in [Lett. Math. Phys. 21, No. 2, 123--131 (1991; Zbl 0722.17004)], consists of the elements \(D_k\), which can be expressed via permanents. There are classical formulae which express \(\{C_k\}\) via \(\{D_l\}\) and vice versa, usually called Wronski formulae, and there are also classical relations involving both \(\{C_k\}\) and \(\{D_l\}\), usually called Wronski relations. In the very interesting paper under review the author proves both Wronski formulae and Wronski relations using the exactness of the Koszul complex and the Euler-Poincaré principle.
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 2038402
zbMATH Keywords: center of the universal enveloping algebra
zbMATH Keywords: Capelli element
zbMATH Keywords: Koszul complex
zbMATH Keywords: Wronski relation
MaRDI profile type: Publication
full work available at URL: s00014-003-0784-7
OpenAlex ID: W2125124531
DOI: 10.1007/S00014-003-0784-7
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description: scientific article; zbMATH DE number 2038402