MS-E1200 - Lie groups and Lie algebras, 29.10.2019-18.12.2019

Lectures

• Lectures
  

  Lectures

  A set of lecture notes is being written, which correspond quite accurately to the topics covered in the course.
  

  
  

  • lecture notes

  
  

  These notes are not in a polished form, and they are somewhat incomplete, and you should expect them to be updated with corrections, additions, and modifications regularly. In the meantime, you can also check my hand-written notes from an eariler course:

  
  

  • Hand-written notes from an old course.
    

  
  

  
  

  
  

  Lecture I: Tuesday, October 29, at 10-12 in M2
  

  Introduction and practical arrangements.
  

  Groups, group homomorphisms, representations, invariant subspaces and subrepresentations.

  
  

  Lecture II: Wednesday, October 30, at 12-14
  

  Operations on representations of finite groups (direct sums, tensor products, duals), irreducible representations, complete reducibility for complex representations of finite groups, Schur's lemma.

  
  

  Lecture III: Tuesday, November 5, at 10-12
  

  Character theory for representations of finite groups: class functions, how characters are affected by operations (direct sum, dual, tensor product), orthogonality relation for irreducible characters.

  
  

  Lecture IV: Wednesday, November 6, at 12-14
  

  Character theory for representations of finite groups: formula for multiplicities of irreps. Regular representation. Restricted and induced representations.

  
  

  Lecture V: Tuesday, November 12, at 10-12
  

  Lie groups. Examples of matrix Lie groups. Topological  considerations: compactness, connectedness, simply-connectedness.
  

  
  

  Lecture VI: Wednesday, November 13, at 12-14
  

  Lie algebras of Lie groups: Lie's theorems and Von Neumann's theorem. Examples of Lie algebras of matrix Lie groups. Lie algebra homomorphisms.

  
  

  Lecture VII: Tuesday, November 19, at 10-12
  

  Relationship between homomorphisms of Lie groups and homomorphisms between their Lie algebras. Adjoint representations of Lie groups and of Lie algebras.

  
  

  Lecture VIII: Wednesday, November 20, at 12-14
  

  The Lie groups \( \mathrm{SO}_3 \) and \( \mathrm{SU}_2 \) and their Lie algebras \( \mathfrak{so}_3 \) and \( \mathfrak{su}_2 \). Representation theory of \(\mathfrak{sl}_2(\mathbb{C}) \): irreducible finite dimensional representations, highest weight representations \( L(\lambda) \).
  

  
  

  Lecture IX: Tuesday, November 26, at 10-12
  

  Complexification \( \mathfrak{g}_{\mathbb{C}} \) of a real Lie algebra \( \mathfrak{g} \). Complex representations of the Lie algebras \( \mathfrak{so}_3 \) and \( \mathfrak{su}_2 \) and the Lie groups \( \mathrm{SU}_2 \) and \( \mathrm{SO}_3 \). Complete reducibility of finite-dimensional representations of \(\mathfrak{sl}_2(\mathbb{C}) \).

  
  

  Lecture IX: Wednesday, November 27, at 12-14
  

  Representation theory of \(\mathfrak{sl}_3(\mathbb{C}) \): roots and root spaces, weight spaces in representations, highest weight vectors.

  
  

  Lecture XI: Tuesday, December 3, at 10-12
  

  More about tepresentation theory of \(\mathfrak{sl}_3(\mathbb{C}) \): irreducible finite-dimensional representations are characterized by the highest weight. Lie algebras of compact type.
  

  
  

  Lecture XII: Wednesday, December 4, at 12-14
  

  Lie algebras of compact semisimple type: structure and classification.