Gelfand-Zeitlin theory from the perspective of classical mechanics. I

Details Gelfand-Zeitlin theory from the perspective of classical mechanics. I Gelfand-Zeitlin theory from the perspective of classical mechanics. I

2004-08-24

arxiv.org/abs/math/0408342v2

Mathematics

Symplectic Geometry

plain tex, 54 pages

Scientific paper

A commutative Poisson subalgebra of the Poisson algebra of polynomials on the Lie algebra of n x n matrices over ${\Bbb C}$ is introduced which is the Poisson analogue of the Gelfand-Zeitlin subalgebra of the universal enveloping algebra. As a commutative algebra it is a polynomial ring in $n(n+1)/2$ generators, $n$ of which can be taken to be basic generators of the polynomial invariants. Any choice of the next $n(n-1)/2$ generators yields a Lie algebra of vector fields that generates a global holomorphic action of the additive group ${\Bbb C}^{n(n -1)/2}$. This paper proves several remarkable properties of this group action and relates it to the theory of orthogonal polynomials.

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