Syllabus
- Lie algebra, homomorphisms of Lie algebra.
- Left-invariant vector fields on Lie groups, Lie algebra of a Lie group, one-parametric subgroups of a Lie group, exponential map.
- Correspondence between homomorphisms of Lie groups and homomorphisms of Lie algebras.
- Basic facts on representations of Lie groups and algberas (restrictions of representations, factor-representation, contragredient representation, sum and tensor product of representations, intertwining maps, isomorphism of representations).
- Irreducible representations of simple Lie algebras (classification of representations of sl(2,C), Cartan subalgebras, roots, positive roots, simple roots, weights, weight lattice, Weyl chambers, dominant weights, fundamental weights).
- Classification of irreducible representations of four classical series, construction of fundamental representations, spinor representations.
- Dynkin diagrams, classification of complex simple Lie algebras.
Course Website: https://kulietheory.wordpress.com/
Annotation
A basic course of structure and representation theory of Lie groups and algebras, with emphasis on complex semisimple Lie algebras. A recommended course for specialization Mathematical Structures within General Mathematics.