Representation Theory of General Linear Lie Algebra

ABOUT THE COURSE: Representation theory has many applications in both mathematics and theoretical physics. The course will introduce representation theory via studying the finite dimensional representations of general Linear Lie algebra (gl_n). We prove all finite dimensional representations of gl_n is completely reducible. We develop Verma’s highest weight theory using the root space decomposition of gl_n, and using this we classify all finite dimensional irreducible representations of gl_n. Finally, we end with Weyl Character formula and some of its applications.INTENDED AUDIENCE:Students from Mathematics Postgraduate programsPREREQUISITES: First course in Linear algebra and Abstract algebra

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