# 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 $\textrm{Sym}^n S \simeq V(n)$.