603B -- Lie groups (Fall 2024)

Class schedule and homework

Week 16: Topology of solvable groups, wrap-up

Week 15: Radical of a Lie group, Lie's theorem, Levi decomposition of Lie groups

Week 14: Verma modules, highest weight irreps, Levi decomposition

Week 13: Simplicity of Lie algebras via irreducibility of root systems, root systems overview

Homework 3 (sign into BU account for access)

Week 12: Cartan subalgebras, roots, Weyl groups, root systems

Week 11: Weyl group of sl(3;C), reductive and semisimple Lie algebras

Week 10: Lie's Third Theorem, construction of universal enveloping algebra, representations of sl(3;C)

Week 9: Consequences of BCH, extension of Lie algebra morphisms to Lie groups, analytic subgroups

Weeks 7&8: Statement of Baker-Campbell-Hausdorff, tangent spaces of smooth manifolds

Week 6: Unitary representations, Schur's lemma, representations of sl(2;C), nonrepresentations

Week 5: Representations, examples, constructions, complete reducibility

Week 4: Functoriality of Lie algebras, local homeomorphism of Lie algebras and matrix Lie groups, consequences thereof

Homework 2 (sign into BU account for access)

Week 3: Lie product formula, Lie algebras definition and basic properties

Week 2: Topology of matrix Lie groups, manifolds, matrix exponential

Homework 1 (due sooner than later) (sign in to BU account for access)

Week 1: Overview, introduction to matrix groups, examples