Symplectic nature of representation theory. Lecture 1

Starting from the very definition of Lie algebras, we exhibit how representation theory of Lie algebras are connected to symplectic structures and integrable systems, mainly via examples. Topics might not include all of them.

• 1) Rise of a symplectic structure from a Lie algebra — Kostant-Kirillov form and moment map;
• 2) Representations and symplectic leaves — Poisson structures and associated varieties;
• 3) The McKay correspondence and its generalizations — singularities arising from simple Lie algebras;
• 4) The McKay correspondence and representation theory (of simple Lie algebras);
• 5) Integrable systems and simple Lie algebras — Kostant's construction of Toda lattice.