Variations on themes of Kostant
type: Item
instance of: scholarly article
title: Variations on themes of Kostant
zbMATH Open document ID: 1169.17002
author: Victor Ginzburg
published in: Transformation Groups
publication date: 2009-06-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/0710.1443
review text: Let \(\mathfrak{g}\) be a complex semi-simple Lie algebra and \({}^LG\) be the Langlands dual group for \(\mathfrak{g}\). In this paper the author proves that the cohomology algebra (with coefficients in \(\mathbb{C}\)) of an arbitrary spherical Schubert variety in the loop Grassmannian for \({}^LG\) is isomorphic to the quotient of the universal enveloping algebra of \(\mathfrak{g}\) modulo an appropriate ideal. The proof uses geometric Satake equivalence. Furthermore, it is shown that the quotient of the universal enveloping algebra which is mentioned above is isomorphic to a certain quotient of the symmetric algebra of \(\mathfrak{g}\) modulo the centralizer of a principal nilpotent in \(\mathfrak{g}\). At the end, the author gives a topological proof of a result (due to Kostant) describing the structure of the algebra \(\mathbb{C}[\mathfrak{g}]\).
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 5565874
zbMATH Keywords: Lie algebra
zbMATH Keywords: cohomology
zbMATH Keywords: Schubert variety
zbMATH Keywords: Satake equivalence
MaRDI profile type: Publication
OpenAlex ID: W1987569861
arXiv ID: 0710.1443
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DOI: 10.1007/S00031-008-9026-9
description: scientific article