Integrable Hamiltonian Systems from Poisson Reductions of Doubles - Part 1

Explore a detailed mathematical lecture that delves into the reduction of integrable master systems on classical doubles of semisimple, connected and simply connected compact Lie groups. Learn about the three types of doubles - cotangent bundle, Heisenberg double, and internally fused quasi-Poisson double - and their natural integrable systems. Examine how reduction is achieved through quotient by cotangent lift of conjugation action, with special focus on the cotangent bundle case in this first part of a three-lecture series. Understand how the quotient space of the internally fused double represents the moduli space of flat principal G-connections on the torus with a hole, and discover how degenerate integrability of master systems transfers to smooth components of Poisson quotients. Master explicit formulas for reduced Poisson structures and equations of motion using dynamical r-matrices within the context of infinite-dimensional geometry and its applications.