Gelfand-Zeitlin theory from the perspective of classical mechanics II

Abstract

In this paper, Part II, of a two part paper we apply the results of [KW], Part I, to establish, with an explicit dual coordinate system, a commutative analogue of the Gelfand-Kirillov theorem for M(n), the algebra of $n\times n$ complex matrices. The function field F(n) of M(n) has a natural Poisson structure and an exact analogue would be to show that F(n) is isomorphic to the function field of a $n(n-1)$-dimensional phase space over a Poisson central rational function field in $n$ variables. Instead we show that this the case for a Galois extension, $F(n, {\frak e})$, of F(n). The techniques use a maximal Poisson commutative algebra of functions arising from Gelfand-Zeitlin theory, the algebraic action of a $n(n-1)/2$--dimensional torus on $F(n, {\frak e})$, and the structure of a Zariski open subset of M(n) as a $n(n-1)/2$--dimensional torus bundle over a $n(n+1)/2$--dimensional base space of Hessenberg matrices.