Homework Set #11

1. Determine the weights of the natural n-dimensional representation of sl(n).
2. If L is semisimple with maximal toral subalgebra H and V is an irreducible L-representation, λ∈H^*, and v is a non-zero vector in the λ weight space of V, prove that V is the direct sum of its weight spaces.
3. Let L be a simple Lie algebra and V a non-trivial finite-dimensional representation of L. Prove that the abelian group generated by the weights of V contains the abelian group generated by the roots of L.
4. Prove that if V(n) denotes the n-dimensional irreducible representation of sl(2) then V(n)⊗V(2) = V(n+1)⊕V(n-1), for all n≥2.