18.747: Problem set 6; due Thursday, March 23

1. Show that the Virasoro algebra Vir is perfect (coincides with its commutator subalgebra). Derive that Vir is not isomorphic to W ⊕ C as a Lie algebra.

2. Prove Proposition 1.2 in Kac-Raina (criterion for V′_{α,β} to be unitary).

3. Let c, h be complex numbers, V (c, h) the irreducible Vir module with highest weight (c, h), and v the highest weight vector of V (c, h). Show that L_{−1}v = 0 if and only if h = 0.

4. Consider the Heisenberg algebra A generated by a 2-dimensional symplectic vector space V with basis x, y, Ω(x, y) = 1. Let V = ℂ[z] be the representation of A given by x ↦ ∂_{z}, y ↦ z, ℏ ↦ id. Consider the corresponding action of sl(2) on V.

a) Show that V, as a representation of sl(2), is a sum of two distinct irreducible representations. Show that, for an appropriate choice of a Borel subalgebra in sl(2), each of this representation is a Verma module. Find the highest weights.

b) (Optional) Let the real Heisenberg algebra A_{ℝ} be the subalgebra in A generated over ℝ by i∂_{z}, z. Let sl(2, ℝ) ⊂ sl(2, ℂ) be the normalizer of A_{ℝ}. Show that the action of sl(2, ℝ) on V can not be integrated to a projective action of SL(2, ℝ) on V; however, the similarly defined action on the space V̂ of ℂ-valued Schwartz functions on the real line z ∈ ℝ can. Show that one of the elements in the resulting central extension of SL(2, ℝ) acting in V̂ acts by Fourier transform, and another one acts by multiplication by the function exp(iz^{2}).