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Explore a comprehensive mathematics lecture on integrable master systems and their reductions in infinite-dimensional symplectic geometry. Delve into the analysis of classical doubles of semisimple, connected and simply connected compact Lie groups, examining three key doubles: the cotangent bundle, Heisenberg double, and internally fused quasi-Poisson double. Learn how each double supports two natural integrable systems, with particular focus on the cotangent bundle case where one system is generated by class functions and the other by invariant functions of its Lie algebra. Understand the reduction process through quotient by cotangent lift of conjugation action, and discover how this extends to other doubles. Examine the moduli space of flat principal G-connections on the torus with a hole, and investigate how degenerate integrability transfers to smooth components of Poisson quotients. Master explicit formulas for reduced Poisson structures and equations of motion using dynamical r-matrices, while exploring applications in Ruijsenaars-Schneider type many-body systems and their compact counterparts for G=SU(n).
