We present a series of results at the interface of cluster algebras and integrable systems, discussing various connections to the broader world of representation theory, geometry, and mathematical physics.

In chapter 3 we develop a rigorous theory of Poisson-Lie structures on ind-algebraic groups and treat the case of symmetrizable Kac-Moody groups within this framework. We use this as a setting for the construction of integrable systems on Hamiltonian reductions of symplectic leaves of affine Lie groups, providing generalizations of the periodic relativistic Toda chain to all affine types.

In chapter 4 we formulate and prove a precise relationship between the Chamber Ansatz of Fomin and Zelevinsky and the general phenomenon of duality between cluster varieties. We also extend the construction of cluster structures on double Bruhat cells of algebraic groups to the setting of symmetrizable Kac-Moody groups, in particular encompassing the examples considered in chapter 3.

In chapter 5 we realize the cluster structures associated with Q-systems as amalgamations of those on double Bruhat cells of simple algebraic groups. We use this to identify Q-system dynamics with those of a factorization mapping, thus deducing their integrability in a uniform way for various Dynkin types, and relate them to the Fomin-Zelevinsky twist automorphism. In the process we also provide cluster realizations of twisted Q-systems.

In chapter 6 we identify the Hamiltonians of the open relativistic Toda system (equivalently the conserved quantities of the Q-systems studied in chapter 5) as cluster characters, certain generating functions of Euler characteristics of quiver Grassmannians. Heuristically this means the Hamiltonians should be interpreted as generalized canonical basis elements, and we explain how such an expression is predicted by the appearance of the relevant cluster structures in supersymmetric gauge theory.