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18.755

Lie Groups and Lie Algebras II

3-0-9 units · Graduate

A more in-depth treatment of Lie groups and Lie algebras. Topics may include homogeneous spaces and groups of automorphisms; representations of compact groups and their geometric realizations, Peter-Weyl theorem; invariant differential forms and cohomology of Lie groups and homogeneous spaces; complex reductive Lie groups, classification of real reductive groups.

Spring 2024

This course is the second half of the year-long introductory graduate sequence {{% resource_link "93cfd9d8-3baa-4130-abc9-01f73ebe2527" "*Lie Groups and Lie Algebras I*" %}} *& Lie Groups and Lie Algebras II*. Topics include classical groups, Haar measure on locally compact groups, the representation-theoretic understanding of the hydrogen atom, representations of compact (in particular, finite) groups, the Peter-Weyl theorem with proof, maximal tori, Cartan and Iwasawa decompositions, classification of real reductive Lie groups, topology of Lie groups, proof of the third fundamental theorem of Lie theory, Levi decomposition, Ado's theorem, Borel subgroups, and flag manifolds.

Fall 2004 · Introduction to Lie Groups

Topics

  • Historical Background and Informal Introduction to Lie Theory
  • Differentiable Manifolds, Differentiable Functions, Vector Fields, Tangent Spaces
  • Tangent Spaces; Mappings and Coordinate Representation
  • Affine Connections
  • Normal Coordinates
  • Definition of Lie Groups
  • Left-invariant Affine Connections
  • Further Analysis of the Universal Enveloping Algebra
  • Lie Subgroups and Lie Subalgebras
  • Lie Algebras of some Classical Groups
  • Lie Transformation Groups
  • Homogeneous Spaces as Manifolds
  • Examples
  • The Killing Form
  • Compact Semisimple Lie Groups
  • The Universal Covering Group
  • Semi-direct Products
  • Solvable Lie Algebras
  • Differential 1-forms
  • Exterior Differentiation and Effect of Mappings
  • Integration of Forms
  • Maurer-Cartan Forms
  • Invariant Forms and Harmonic Forms
  • Real Forms
  • The Classical Groups and the Classification of Simple Lie Algebras, Real and Complex

Readings

Prerequisites & course information

Prereq: 18.745 or permission of instructor G (Spring) 3-0-9 units

Sources