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 sln(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).

Course Coordinators

Suggested References

  1. J. E. Humphreys, Introduction to Lie Algebras and Representation theory, Springer Graduate Texts in Mathematics, 1972
  2. W Fulton & J Harris, Representation Theory, Grad. Texts in Maths, Springer 1991
  3. B C Hall Lie Groups, Lie Algebras and Representations, Grad. Texts in Maths. Springer 2003
Credit Units (ECTS): 
Α2, 0634