On algebraic integrability of Gelfand-Zeitlin fields |
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Authors: | Mark Colarusso Sam Evens |
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Institution: | 1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA
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Abstract: | We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields
to the full set of strongly regular elements in
\mathfrakg\mathfrakl \mathfrak{g}\mathfrak{l} (n, ℂ). We use decomposition classes to stratify the strongly regular set by subvarieties XD {X_\mathcal{D}} . We construct an étale cover
^(\mathfrakg)]D {\hat{\mathfrak{g}}}_\mathcal{D} of XD {X_\mathcal{D}} and show that XD {X_\mathcal{D}} and
^(\mathfrakg)]D {\hat{\mathfrak{g}}}_\mathcal{D} are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on XD {X_\mathcal{D}} to Hamiltonian vector fields on
^(\mathfrakg)]D {\hat{\mathfrak{g}}}_\mathcal{D} and integrate these vector fields to an action of a connected, commutative algebraic group. |
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