Hamiltonian evolutions of twisted polygons in \(\mathbb {RP}^n\) (Q2852520)

This impressive paper is the second of the authors on discrete group-based moving frames, an adaptation of the well-known theory of continuous moving frames to discrete systems. The first paper was [Found. Comput. Math. 13, No. 4, 545--582 (2013; Zbl 1279.14045)], with \textit{E. Mansfield} as a coauthor. The present paper relies heavily on the first for basic definitions, examples and proofs of fundamental theorems. For most of the paper, the setting is a homogeneous space \(M=G/H\), where \(G\) is a subgroup of the general linear group. The discrete objects are twisted \(N\)-gons, maps \(\phi:{\mathbb Z}\rightarrow M\) such that for some fixed \(g\in G\) we have \(\phi(p+N)=g\cdot \phi(p)\). The \(N\)-gon is described by its vertices \(x=(x_s)\), where \(x_s=\phi(s)\).NEWLINENEWLINENEWLINEA discrete moving frame \(\rho\) assigns an element of \(G\) to each vertex of the polygon in an equivariant fashion. Thus, \(\rho\) is a vector \(\rho=(\rho_s)\), where \(\rho_s(x)\in G\) for all \(s\), \(x=(x_s)\). Any \(g\in G\) can act uniformly on each component \(\rho_s\), either to the left or to the right, and on each vertex \(x_s\). Thus, a discrete moving frame is a sequence of interrelated moving frames in the usual sense. A scalar discrete invariant of the group is a function \(I\) defined on \(N\)-gons such that \(I((g\cdot x_s))=I((x_s))\) for every \(g\in G\) and every \(N\)-gon \(x\). The discrete analog of the Maurer-Cartan matrix is the sequence of group elements \(K_s=\rho_{s+1}\rho_s^{-1}\) evaluated along an \(N\)-gon, which can be shown to generate all discrete invariants of these polygons. Finally, an evolution equation \((x_s)_t=F_s(x)\) is said to be invariant under the group action if for any solution \(x(t)\) of the evolution equation, \(gx(t)\) is also a solution. As in the case of continuous moving frames, the interest is in the invariant evolution of discrete invariants. One wants to show that this evolution equation of the invariants is Hamiltonian, or better, that it is integrable. This latter can be achieved by exhibiting a bi-Hamiltonian structure or by showing the equivalence to an evolution equation already known to be integrable.NEWLINENEWLINENEWLINENext, the authors turn to their main object of study, that is, projective \(N\)-gons in \({\mathbb R \mathbb P}^n\). Here, \(G=\mathrm{PSL}(n+1)=\mathrm{SL}(n+1)/\pm I\). They show how to compute the discrete moving frames, the Maurer-Cartan matrices and invariants and the equations of invariant evolution for the invariants, with \(n=2\) as an example. Then, they employ a twisted Poisson structure on \(G^N\), defined by Semenov-Tian-Shansky, and a reduction process to obtain a natural Poisson bracket on the space of Maurer-Cartan matrices. They are able to show that any Hamiltonian evolution on this space can be induced by an invariant evolution of projective polygons in \({\mathbb R \mathbb P}^n\). The details are quite complicated. In the last part of the paper, they construct a family of bi-Hamiltonian structures and use them to show that integrable discretizations of \(W_n\)-algebras can be induced by invariant evolutions of projective polygons in \({\mathbb R \mathbb P}^n\).