Lie Groups and Lie Algebras

Syllabus

• Lie algebras, generators, subalgebras and ideals. Solvable, nilpotent, semisimple and abelian Lie algebras. Representations. Schur's lemma. The Cartan-Weyl basis. Cartan subalgebras; roots; the Killing form. Structure constants; positive roots; simple roots and the Cartan matrix. The Dynkin basis. Dynkin diagrams. Root strings, root systems and the Weyl group. Weight systems. Highest weight representations. Characters and the Weyl formula. Universal enveloping algebra and the Poincaré-Birkhoff-Witt theorem. Casimir operators and Racah's theorem. Tensor products of representations. The Racah-Speiser algorithm. Manifolds, tangent spaces, vector fields. Homotopy. Fundamental group and covering spaces. Lie groups. The exponential map. The universal covering group. Group actions and representations on function spaces. Group integration: the Haar measure. The Peter-Weyl theorem.

References

• Symmetries, Lie Algebras and Representations, J. Fuchs and C. Schweigert, Cambridge University Press
• Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics, D.H. Sattinger and O.L. Weaver, Springer-Verlag
• Lie Groups Beyond an Introduction, A.W. Knapp, Birkhauser

Homeworks

• Homework 1
• Homework 2
• Homework 3