This is an introductory course to the theory of Lie algebras. The prerequisites for this course are Linear Algebra and Abstract Algebra. The book for this course is "Introduction to Lie Algebras and Representation
Theory" by James Humphreys. The grades will be based on weekly homework assignments.
Course outline:
- Lie algebras, Lie subalgebras, Lie ideals, quotient Lie algebras,
homomorphisms
- Representations, representation theory of sl(2)
- Solvable and Nilpotent Lie Algebras, Engel's Theorem, Lie's Theorem
- Simplicity and semi-simplicity, Killing form, Cartan's criterion
- Complete reducibility of representations, Casimir element
- Root systems, Weyl group
- Cartan matrix, Dynkin diagrams
- Cartan subalgebras, Borel subalgebras
- Classification of complex semi-simple Lie algebras
- Universal enveloping algebra, Poincare-Birkoff-Witt Theorem
- Weights, Highest weight, Verma modules, Finite dimensional modules
- Characters, Weyl Character formula
The course will be given by Crystal Hoyt.