# ISU MATH 620X Spring 2018

## Lie algebras and their representations

Course Material

Main text: Kirillov, Jr., An introduction to Lie Groups and Lie Algebras

### Lecture summary

Lecture 1: Definition of manifolds and (real and complex) Lie groups. Implicit function theorem. Examples. Further reading: Spivak, Calculus on Manifolds. In particular Chapter 5.

Lecture 2: Connected manifolds and Lie groups. Discrete Lie groups. Open submanifolds. The connected component at the identity element. G/G^0 is discrete.

Lecture 3: Fundamental group of a manifold. Simply connected manifolds and Lie groups. Universal cover of a manifold and Lie group. Further reading: Hatcher, Algebraic Topology. Chapter 1.1 and 1.3. In particular Theorem 1.38 and the paragraph that follows.

Lecture 4: Tangent spaces and vector fields on manifolds, derivative (differential) of a morphism, left-invariant vector fields, Lie algebra (as a vector space) of a Lie group.

Lecture 5-6: Classical groups

Lecture 7: Open, immersed, embedded submanifolds. Closed Lie subgroups.

Lecture 8: Quotient groups and homogenous spaces

Lecture 9: One-parameter subgroups, the exponential map.

Lecture 10: Classes of manifolds. The commutator (bracket).

Lecture 11-12: Computing differentials using curves. Differential of Ad. Jacobi identity. Definition of Lie algebra. Derivations. Action of SL(2,K) and sl(2,K) on polynomials.

Lecture 13: Abelian Lie algebras. Lie subalgebras and Lie ideals. Lie algebras of subgroups and quotients. Vect(M).

Lecture 14: Baker-Campbell-Hausdorff formula. Fundamental theorems 3.40,3.41,3.42.

Lecture 15: Equivalence of categories (connected simply-connected Lie groups)/(finite-dimensional Lie algebras). Complexification and real forms.

Lecture 16: Subrepresentations, direct sum, tensor product, and the dual of representations. Invariants.

Lecture 17: Irreducible and completely reducible representations. Intertwining operators.

Lecture 18: Schur's Lemma. Unitary representations. The Haar measure on a compact real Lie group.

Lecture 19: Unitarizability of representations of compact real Lie groups. Characters. Matrix coefficients. Hilbert space of square-integrable functions on G.

Lecture 20: Orthogonality of matrix coefficients. Peter-Weyl Theorem.

Lecture 21: Representations of sl(2,C)

Lecture 22: The universal enveloping algebra

Lecture 23: The Poincare-Birkhoff-Witt theorem

Lecture 24: The classification problem. Solvable and nilpotent Lie algebras

Lecture 25: Lie's theorem and Engel's theorem.

Lecture 26: The radical. Semisimple and reductive Lie algebras.

Lecture 27: The Killing form and Cartan's Criteria

Lecture 28: Jordan decomposition

Lecture 29: (discussion: Clebsch-Gordan formula and affine Lie algebras)

Lecture 30: Semi-simple Lie algebras

Lecture 31: Weyl's Theorem on complete reducibility

Lecture 32: Cartan subalgebras

Lecture 33: Root space decomposition

Lecture 34: Structure of semisimple Lie algebras

Lecture 35: Root systems

Lecture 36-38: Positive roots - simple roots - sp(4)

Lecture 39: Cartan matrices and Dynkin diagrams

Lecture 40: Serre's Theorem

Lecture 41-43: Highest weight theory