Lie-Poisson theory for direct limit Lie algebras

In the first half of this paper, we develop the fundamentals of Lie-Poisson theory for direct limits G=lim→Gn of complex algebraic groups and their Lie algebras g=lim→gn. We describe the Poisson pro- and ind-variety structures on g⁎=lim←gn⁎ and the coadjoint orbits of G, respectively. While the existence of symplectic foliations remains an open question for most infinite-dimensional Poisson manifolds, we show that for direct limit algebras, the coadjoint orbits give a weak symplectic foliation of the Poisson provariety g⁎.The second half of the paper applies our general results to the concrete setting of G=GL(∞) and g⁎=M(∞), the space of infinite-by-infinite complex matrices with arbitrary entries. We use the Poisson structure of g⁎ to construct an integrable system on M(∞) that generalizes the Gelfand-Zeitlin system on gl(n,C) to the infinite-dimensional setting. We further show that this integrable system integrates to a global action of a direct limit group on M(∞), whose generic orbits are Lagrangian ind-subvarieties of the coadjoint orbits of GL(∞) on M(∞).