Hamiltonian evolutions of twisted gons in $RP^n$

In this paper we describe a well-chosen discrete moving frame and their associated invariants along projective polygons in $\RP^n$, and we use them to write explicit general expressions for invariant evolutions of projective $N$-gons. We then use a reduction process inspired by a discrete Drinfeld-Sokolov reduction to obtain a natural Hamiltonian structure on the space of projective invariants, and we establish a close relationship between the projective $N$-gon evolutions and the Hamiltonian evolutions on the invariants of the flow. We prove that {any} Hamiltonian evolution is induced on invariants by an evolution of $N$-gons - what we call a projective realization - and we give the direct connection. Finally, in the planar case we provide completely integrable evolutions (the Boussinesq lattice related to the lattice $W_3$-algebra), their projective realizations and their Hamiltonian pencil. We generalize both structures to $n$-dimensions and we prove that they are Poisson. We define explicitly the $n$-dimensional generalization of the planar evolution (the discretization of the $W_n$-algebra) and prove that it is completely integrable, providing also its projective realization.

💡 Research Summary

The paper develops a comprehensive discrete differential‑geometric framework for twisted N‑gons in real projective space RPⁿ and shows how their invariant evolutions give rise to Hamiltonian flows on the space of projective invariants. The authors begin by constructing a discrete moving frame: to each vertex of a twisted polygon they associate a (n + 1) × (n + 1) matrix in GL(n + 1) and fix the frame by a set of normalization conditions. This procedure yields a complete set of projective invariants (generalized cross‑ratios) that encode the geometry of the polygon up to projective equivalence.

With the invariants in hand, the authors write the most general invariant evolution of the polygon in explicit matrix form. By differentiating the moving frame and projecting onto the invariant coordinates they obtain a system of difference equations for the invariants. Crucially, they introduce the notion of a “projective realization”: any prescribed evolution of the invariants can be lifted to an actual motion of the polygon’s vertices, so that the invariant flow is not merely formal but has a concrete geometric incarnation.

The second major contribution is the construction of a natural Hamiltonian structure on the invariant space. Inspired by the continuous Drinfeld‑Sokolov reduction, the authors perform an analogous discrete reduction. Starting from the standard Lie‑Poisson bracket on the loop algebra of GL(n + 1) and imposing the moving‑frame constraints, they derive two compatible Poisson brackets on the invariants. These brackets are expressed in terms of difference operators P₁ and P₂, which satisfy the Magri compatibility condition, thus providing a Hamiltonian pencil λ P₁ + P₂. Consequently, any Hamiltonian functional generates an invariant evolution that can be realized by a polygon flow.

In the planar case (n = 2) the theory reproduces the well‑known Boussinesq lattice, which is the discrete analogue of the lattice W₃‑algebra. The authors exhibit its two Poisson structures, construct a Lax pair (L, M), and verify the zero‑curvature condition dL/dt =