The second Gelfand-Dickey bracket as a bracket on a Poisson-Lie Grassmannian (Q1315003)

The author presents a new geometric interpretation of the second Gel'fand-Dikij bracket [\textit{I. M. Gel'fand} and \textit{L. A. Dikij}: On the second Hamiltonian structure for equations of Korteweg-de Vries type, Preprint, 1978, Institute of Applied Mathematics, Moskva]. Another interpretation has been given by \textit{A. O. Radul} [Phys. Lett. B 265, 86-91 (1991)]. The author considers the group GL\((C^ \infty(\mathbb{R}))\), the usual GL-type Poisson-Lie structure on it and the Grassmannian \(\text{Gr}_ n(C^ \infty(\mathbb{R}))\) of \(n\)-dimensional subspaces in \(C^ \infty(\mathbb{R})\). Then a Poisson structure is induced on \(\text{Gr}_ n(C^ \infty(\mathbb{R}))\) from that GL\((C^ \infty(\mathbb{R}))\), when \(\text{Gr}_ n(C^ \infty(\mathbb{R}))\) is thought of as a quotient by a parabolic subgroup. A differential operator of order \(N\) with leading term \(\partial^ n\) is identified with the (\(n\)-dimensional) subspace its solutions in \(C^ \infty(\mathbb{R})\). Then the second Gel'fand-Dikij structure does coincide with the induced Poisson structure on \(\text{Gr}_ n(C^ \infty(\mathbb{R}))\).