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Lie-Poisson theory for direct limit Lie algebras

In this paper, we develop the fundamentals of Lie-Poisson theory for direct limits $G=\dirlim G_{n}$ of complex algebraic groups $G_{n}$ and their Lie algebras $\fg=\dirlim \fg_{n}$. We show that $\fg^{*}=\invlim\fg_{n}^{*}$ has the structure of a Poisson provariety and that each coadjoint orbit of $G$ on $\fg^{*}$ has the structure of an ind-variety. We construct a weak symplectic form on every coadjoint orbit and prove that the coadjoint orbits form a weak symplectic foliation of the Poisson provariety $\fg^{*}$. We apply our results to the specific setting of $G=GL(\infty)=\dirlim GL(n,\C)$ and $\fg^{*}= M(\infty)=\invlim \fgl(n,\C)$, the space of infinite complex matrices with arbitrary entries. We construct a Gelfand-Zeitlin integrable system on $M(\infty)$, which generalizes the one constructed by Kostant and Wallach on $\fgl(n,\C)$. The system integrates to an action of a direct limit group $A(\infty)$ on $M(\infty)$, whose generic orbits are Lagrangian ind-subvarieties of the corresponding coadjoint orbit of $GL(\infty)$ on $M(\infty)$.