Exercises Week 5

Complex Lie Algebras in Dimensions 1, 2, 3:

  1. Let L_2 be the unique non-abelian Lie algebra of dimension 2. What is the realisation of L_2 as a Lie subalgebra of a matrix algebra?
  2. Suppose we have a Lie algebra of dimension 3 and rank 1. Show that we can choose a basis \{X, Y, Z\} such that [X, Y] = [X, Z] = 0. We know that the resulting Lie algebra has [Y, Z] = X (WLOG). Show this is isomorphic to the Lie algebra of strictly upper-triangular 3 \times 3 matrices. Write an explicit isomorphism.
  3. Let \mathfrak{g}_\alpha be the dimension 3 Lie algebra with [X, Y] = Y, [X, Z] = \alpha Z, [Y, Z] = 0. Show that \mathfrak{g}_\alpha \simeq \mathfrak{g}_{\alpha'} iff \alpha = \alpha' or \alpha = \frac{1}{\alpha'}. What is the realisation of this as a Lie subalgebra of a matrix algebra?
  4. What is the realisation of the dimension 3 Lie algebra with [X, Y] = Y, [X, Z] = Y + Z, [Y, Z] = 0? Show that this is not isomorphic to the above.

Rep Theory of \mathfrak{sl}_2(\mathbb{C}):

  1. Construct one infinite-dimensional irreducible representation of \mathfrak{sl}_2(\mathbb{C}).
  2. Show that the standard representation, with H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, X= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, is isomorphic to V(1).
  3.  Show that \textrm{Sym}^n S \simeq V(n).
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