Week 1: General linear Lie algebras (gl_n): definition, root space decomposition, and the reductive Lie algebra structure of gl_n.
Week 2: Representations of gl_n: definitions, basic constructions: subrepresentation, direct sum, irreducible, indecomposable representations, examples. Homomorphisms, quotients, and isomorphism theorems
Week 3: Representation theory of sl_2: classification of finite dimensional irreducible representations of sl_2, complete reducibility
Week 4: Representation theory of sl_3: examples. Casimir elements, complete reducibility of finite dimensional representations of gl_n
Week 5: The universal enveloping algebra: definition, existence, PBW theorem, integral forms.
Week 6: Correspondence between representations of general linear Lie algebra and special linear Lie algebra
Week 7: Highest weight representations: Verma modules,classification of finite dimensional irreducible representations of gl_n
Week 8: Generators and relations of irreducible representations
Week 9: Character theory: definition, basic properties, characters of Verma modules, examples
Week 10: Weyl denominator identity, Weyl character formula
Week 11: Fundamental modules of gl_n and sl_n
Week 12: Tensor product of irreducible representations: Steinberg’s formula, Littlewood–Richardson coefficients
DOWNLOAD APP
FOLLOW US