Frieze patterns and a symplectic structure

Abstract

Frieze patterns, first introduced and studied by H. Coexter in 1971, are certain infinite arrays of elements in a commutative ring satisfying some simple algebraic relations among their entries. Coxeter’s celebrated theorem says that frieze patterns with positive integer entries and finite widths are in one-to-one correspondence with triangulations of polygons. Frieze patterns have drawn more attention recently, and the connections with other fields of mathematics, such as projective geometry, discrete integrable systems, and cluster algebras have been investigated in the literature.

The first part of the thesis contains a review of some basic facts on frieze patterns in any integral domain. The second part of the thesis concentrates on continuant frieze patterns in a field K with finite widths. For any integer n ≥ 1, the set Zn of all frieze patterns in K with width n was proved to be an algebraic sub-variety of Kn+1. When n is even, it was proved that Zn is smooth. Moreover, a Poisson structure on Cn was introduced for any integer n, and it was proved that when n is even, Zn is a symplectic submanifold of Cn+1. For even n, two integrable systems on Zn were identified using Euler continuants, and their Hamiltonian flows were explicitly computed.