Rational symplectic coordinates on the space of Fuchs equations $m \times m$-case

Details Rational symplectic coordinates on the space of Fuchs equations $m \times m$-case Rational symplectic coordinates on the space of Fuchs equations $m \times m$-case

2008-07-12

arxiv.org/abs/0807.1958v1

Physics

Mathematical Physics

11 pages

Scientific paper

10.1007/s11005-008-0274-3

A method of constructing of Darboux coordinates on a space that is a symplectic reduction with respect to a diagonal action of $GL(m})$ on a Cartesian product of $N$ orbits of coadjoint representation of $GL(m)$ is presented. The method gives an explicit symplectic birational isomorphism between the reduced space on the one hand and a Cartesian product of $N-3$ coadjoint orbits of dimension $m(m-1)$ on an orbit of dimension $(m-1)(m-2)$ on the other hand. In a generic case of the diagonalizable matrices it gives just the isomorphism that is birational and symplectic between some open, in a Zariski topology, domain of the reduced space and the Cartesian product of the orbits in question. The method is based on a Gauss decomposition of a matrix on a product of upper-triangular, lower-triangular and diagonal matrices. It works uniformly for the orbits formed by diagonalizable or not-diagonalizable matrices. It is elaborated for the orbits of maximal dimension that is $m(m-1)$.

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