Symplectic volumes of certain symplectic quotients associated with the special unitary group of degree three
type: Item
instance of: scholarly article
title: Symplectic volumes of certain symplectic quotients associated with the special unitary group of degree three
zbMATH Open document ID: 1157.53045
author: Taro Suzuki
author: Tatsuru Takakura
published in: Tokyo Journal of Mathematics
publication date: 2008-09-29T00:00:00Z
review text: Let \(G\) be a compact Lie group with Lie algebra \(\mathfrak g\). Let \(\mathcal O_1\), \(\mathcal O_2\),\(\cdots\),\(\mathcal O_n\subset \mathfrak g^*\) be coadjoint orbits, which are symplectic manifolds under the Kirillov-Kostant-Souriau symplectic forms. The action of \(G\) on the product symplectic manifold \(\mathcal O_1\times \mathcal O_2\times \cdots\times \mathcal O_n\) is Hamiltonian, with moment map \(\Phi:\mathcal O_1\times \mathcal O_2\times \cdots\times \mathcal O_n\rightarrow \mathfrak g^*\), \((x_1,x_2,\cdots,x_n)\mapsto x_1+x_2+\cdots+x_n\). Assume that \(0\) is a regular value of \(\Phi\), and the quotient \(\mathcal M:=G\backslash\Phi^{-1}(0)\) is a smooth manifold. Then \(\mathcal M\) is canonically a symplectic manifold (the symplectic quotient of a Hamiltonian action). It is an interesting problem to calculate explicitly the volume of the compact symplectic manifold \(\mathcal M\). When \(G=SU(2)\), the calculation is done by the second author in [\textit{T. Takakura}, Adv. Studies in Pure Math. 34, 255--259 (2002; Zbl 1029.53091)]. In the paper under review, the authors carry out the calculation for \(G=SU(3)\), under certain restrictions on the coadjoint orbits.
Mathematics Subject Classification ID: 53D20
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 5347982
zbMATH Keywords: coadjoint orbit
zbMATH Keywords: symplectic quotient
zbMATH Keywords: symplectic volume
zbMATH Keywords: SU(3)
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1219844821
OpenAlex ID: W2030557518
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DOI: 10.3836/TJM/1219844821
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description: scientific article; zbMATH DE number 5347982