Representation Theory

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  • Prove Engel's theorem. (Serganova)
  • Prove Lie's theorem. What would happen if the hypothesis was not that g is solvable but that g \ne [g,g]? (Serganova)
  • Why is g = [g,g] for g semisimple? (Weinstein)
  • What is the exponential map, and what is it good for? (Serganova)
  • Classify the real connected abelian Lie groups. (Serganova)
  • Prove that a Lie group homomorphism \phi\colon H \to G for H connected is determined by the derivative at the identity. (Serganova)
  • Give an example of a Lie group G where the exponential map is not surjective. (Weinstein)
  • Given the standard representation of sl_n({\mathbb C}) identify the simple roots and explain the correlation between the height of the root and the corresponding "location" in the matrix. (Frenkel)
  • Decompose \operatorname{Sym}_n(V)\otimes \operatorname{Sym}_m(V) where V is the 2-dimensional irreducible representation of sl_2({\mathbb C}). (Frenkel)
  • Do the calculation above using a character formula. (Reshetikhin)
  • State and explain the Harish-Chandra isomorphism. (Wodzicki)
  • Explain how to write down the Weyl group of SLn using generators and relations. (Frenkel)
  • What is a Verma module? (Reshetikhin)
  • When is a Verma module finite-dimensional? (Wodzicki)
  • What is the exponential map for sl_2({\mathbb C})? What is it a map from and to? Is it a homomorphism, is it surjective?
  • What proofs of the Weyl character formula do you know? (Reshetikhin)
  • What is Weyl's Integration formula? How do you use it to prove WCF? (Reshetikhin)
  • What is the dimension of E8? (Borcherds)
  • Decompose E8 as a representation of E7. (Borcherds)
  • Given a point in a semisimple Lie-algebra, how can we tell whether it lies in a Cartan subalgebra? (Knutsen)
  • What is the relation between the Lie groups SU(2) and SO(3,\R)? Prove that the center of SU(2) is \Z/2\Z using Schur's lemma. (Weinstein)
  • Under what condition on G is every discrete normal subgroup of a Lie group G contained in the center of G? Prove this. (Reshetikhin)
  • List all the irreducible complex representations of the Lie group SO(3,\R). (Knutsen)
  • Decompose the square of the adjoint representation of sl(3) into irreducibles. (Hint: Weyl Character Formula.) (Haiman)
  • State a theorem that explains the basic relationship between Lie algebras and Lie groups. Say some words about the proof. (Haiman)

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