Stratified Gradient Hamiltonian Vector Fields and Collective Integrable Systems
| Autor: | Hoffman, Benjamin, Lane, Jeremy |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We construct completely integrable torus actions on the dual Lie algebra of any compact Lie group $K$ with respect to the standard Lie-Poisson structure. These systems generalize properties of Gelfand-Zeitlin systems for unitary and orthogonal Lie groups: 1) the pullback to any Hamiltonian $K$-manifold is an integrable torus action, 2) if the $K$-manifold is multiplicity free, then the torus action is \textit{completely} integrable, and 3) the collective moment map has convexity and fiber connectedness properties. They also generalize the relationship between Gelfand-Zeitlin systems and canonical bases via geometric quantization by a real polarization. To construct these integrable systems, we generalize Harada and Kaveh's construction of integrable systems by toric degeneration to singular quasi-projective varieties. Under certain conditions, we show that the stratified-gradient Hamiltonian vector field of such a degeneration, which is defined piece-wise, has a flow whose limit exists and defines continuous degeneration map. Comment: Added survey of Gelfand-Zeitlin integrable systems to introduction. Added example of geometric quantization of cotangent bundles |
| Databáze: | arXiv |
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