On rational symplectic parametrization of the coadjoint orbit of $\mathrm{GL}(N)$. Diagonalizable case

Abstract

Abstract. A method for constructing birational Darboux coordinates on a coadjoint orbit of the general linear group is presented. This method is based on the Gauss decomposition of a matrix in the product of an upper-triangular and a lower-triangular matrix. The method works uniformly for the orbits formed by the diagonalizable matrices of any size and for arbitrary dimensions of the eigenspaces. §0. Introduction Our aim in this paper is to present a method for canonical parametrization of an important algebraic symplectic manifold, namely, a coadjoint orbit of the complex general linear group; see [4,5,10].The problem of description of any manifold consists of two steps. First, we should construct charts. They are sets keeping the information about the local structure of the manifold. The charts should have a global structure as simple as possible. The second step is creation of the atlas. We should glue the charts in a proper way. The law of gluing is stated by the transition functions, which identify the overlapping parts of the charts. The transition functions set the global structure of the manifold.Largely, this article is devoted to the first step. We construct one chart, a Zariski open subset of the orbit. Such a domain covers the entire orbit except for several submanifolds of dimension smaller than the dimension of the orbit. The parametrization of the charts is given analytically in Theorems 2 and 3.To describe the covering, we point out what subspaces must be in general position with the coordinate subspaces. Different charts are parametrized by renumberings of the coordinates. The transition functions can be obtained by reparametrization of the domain already parametrized in the renumerated basis. We do not present these formulas: they are bulky and useless.It should be noted that the relative arrangement of the coordinate domains of the orbit should be well understood for the following reason. There are problems where we need to glue different orbits (i.e., orbits that differ by the spectral structure of matrices they involve) to one algebraic symplectic manifold. The organization of the maps in the atlases of these manifolds is similar to the organization of the maps in one orbit. As important examples, we mention the phase spaces of the systems of equations of the isomonodromic deformations [6]- [8].We identify gl(N, C) with its dual gl * (N, C) by using the nondegenerate form A, B = tr AB. Then the coadjoint orbits are identified with the adjoint orbits. Let O J be the 2010 Mathematics Subject Classification. Primary 53D05.

Isomonodromic deformations: Confluence, Reduction $\&$ Quantisation

“…In this case Takiff algebra coincide with the ordinary Lie algebra. Parametrisation in such situation was obtained in works [5,6] 2.5.2. Second order pole.…”

“…Harnad generalised these coordinates to allow rectangular m 1 × m 2 matrices and used them to introduce an interesting duality between two systems of linear ODEs: one of dimension m 1 and the other of dimension m 2 [24] and [53]. Similar coordinates were also introduced and partly used in the context of non-autonomous Hamiltonian description of Garnier-Painlevé differential systems by M. Babich and Derkachov [5,6]. However in these latter works, the authors restricted to the case of rational parametrisation of co-adjoint orbits of Gl n (C) and other semi-simple Lie groups and did not consider loop algebras.…”

Rational version of Archimedes symplectomorphysm and birational Darboux coordinates on coadjoint orbit of $GL(N,C)$

“…The following observation (see [9,10]) forms a basement of the construction: the canonical symplectic structure on an orbit and the hierarchic structure (12) which I present below are coordinated.…”

“…The idea belongs to S. E. Derkachov and A. N. Manashov, they use it for the needs of quantum field theory [9]. Recently the method was applied for the parameterization of the orbits swept by the diagonalizable matrices [10]. In the present article we are giving the evaluation of the method of [9,10] to the general Jordan case.…”

“…Recently the method was applied for the parameterization of the orbits swept by the diagonalizable matrices [10]. In the present article we are giving the evaluation of the method of [9,10] to the general Jordan case.…”