Linear Hamiltonian circle actions that generate minimal Hilbert bases
type: Item
instance of: scholarly article
title: Linear Hamiltonian circle actions that generate minimal Hilbert bases
zbMATH Open document ID: 0973.53067
author: Ágúst Sverrir Egilsson
publication date: 2000-03-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75417
review text: The following theorem is the main result of the paper. Let \(S^1\times {\mathbb C}^k\to {\mathbb C}^k\) be a linear Hamiltonian circle action with relatively prime weights \(n_1,\dots,n_k\). Assume that the algebra \(C^\infty({\mathbb R}^{2k})^{S^1}\) is generated by \(k^2\) polynomials \(f_1,\dots,f_{k^2}\) which is the minimal possible number of such polynomials for a fixed number \(k\). Denote the greatest common divisor of all the weights except \(n_i\) by \(d_i\). Assume furthermore that at least three of the numbers \(d_1,\dots, d_k\) are not equal to one. Then the Poisson structure on the singular orbit space \({\mathbb C}^k/S^1\), embedded into \({\mathbb R}^{k^2}\) by \(F=(f_1,\dots,f_{k^2})\), cannot be extended to \({\mathbb R}^{k^2}\). The paper is a continuation of [Electron. Res. Announc. Am. Math. Soc. 1, No. 2, 48-56 (1995; Zbl 0849.58029)] by the author and answers a question of Cushman and Weinstein in the negative.
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 53D20
Mathematics Subject Classification ID: 37K05
zbMATH DE Number: 1409072
zbMATH Keywords: singular Poisson structure
zbMATH Keywords: reduction
zbMATH Keywords: Hamiltonian action
reviewed by: Gudlaugur Thorbergsson
MaRDI profile type: Publication
OpenAlex ID: W2326680866
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DOI: 10.5802/AIF.1755
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1409072