Complex Lie Algebras in Dimensions 1, 2, 3:
- Let be the unique non-abelian Lie algebra of dimension 2. What is the realisation of as a Lie subalgebra of a matrix algebra?
- Suppose we have a Lie algebra of dimension 3 and rank 1. Show that we can choose a basis such that . We know that the resulting Lie algebra has (WLOG). Show this is isomorphic to the Lie algebra of strictly upper-triangular matrices. Write an explicit isomorphism.
- Let be the dimension 3 Lie algebra with , , . Show that iff or . What is the realisation of this as a Lie subalgebra of a matrix algebra?
- What is the realisation of the dimension 3 Lie algebra with , , ? Show that this is not isomorphic to the above.
Rep Theory of :
- Construct one infinite-dimensional irreducible representation of .
- Show that the standard representation, with , , , is isomorphic to .
- Show that .