Representation Theory of Lie Algebras

1. Introduction - Introduction to Lie groups - construction of Lie algebras from Lie groups - Basic definitions - derivations - ideals solvable and nilpotent Lie algebras - example of Lie algebra sl_n (C). 2. Simple and semisimple Lie Algebras - Cartan subalgebras - Killing forms - Weyl group - Dynkin diagrams - classification of semisimple Lie algebras. 3. Enveloping Algebras - Definition of enveloping algebras - Poincaré-Birkhoff-Witt theorem - Exponential embeding of Lie algebras to Lie groups - Casimirs - Hopf structure of enveloping algebra. 4. Representations and modules - Theorem Ado-Isawa - finite dimensional irreducible representations - adjoint representation - tensor representations - Inducible representations - Representations of solvable - nipotent and semisimple algebras - Verma modules. 5. Applications - Symmetries of integrable systems - Backlund-Lie symmetries -Lax operators in Hamiltonian systems - Lie-Poisson algebras - Symmetries of quantum systems and Lie agebras su(2), su(3).

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