Some Poisson Structures and Lax Equations Associated with the Toeplitz Lattice and the Schur Lattice

The Toeplitz lattice is a Hamiltonian system whose Poisson structure is known. In this paper, we unveil the origins of this Poisson structure and derive from it the associated Lax equations for this lattice. We first construct a Poisson subvariety $\mathcal{H}_n$ of $\mathbf{GL}_n(\mathbb{C})$, which we view as a real or complex Poisson–Lie group whose Poisson structure comes from a quadratic $R$-bracket on $\mathfrak{gl}_n(\mathbb{C})$ for a fixed $R$-matrix. The existence of Hamiltonians, associated to the Toeplitz lattice for the Poisson structure on $\mathcal{H}_n$, combined with the properties of the quadratic $R$-bracket allow us to give explicit formulas for the Lax equation. Then we derive from it the integrability in the sense of Liouville of the Toeplitz lattice. When we view the lattice as being defined over $\mathbb{R}$, we can construct a Poisson subvariety $\mathcal{H}_n^\tau$ of $U_n$ which is itself a Poisson–Dirac subvariety of $\mathbf{GL}^\mathbb{R}_n(\mathbb{C})$. We then construct a Hamiltonian for the Poisson structure induced on $\mathcal{H}^\tau_n$, corresponding to another system which derives from the Toeplitz lattice the modified Schur lattice. Thanks to the properties of Poisson–Dirac subvarieties, we give an explicit Lax equation for the new system and derive from it a Lax equation for the Schur lattice. We also deduce the integrability in the sense of Liouville of the modified Schur lattice.