Matrix representations of $\mathfrak{g}^*$.

Question

Let $g = sl_2$. Then there is a matrix representation of $g$ as follows. The Lie algebra $g$ is a three dimensional vector space with a basis $E, F, H$ such that $[E,F]=H$,$[H,E]=2E$,$[H,F]=-2F$. The elements $E, F, H$ have matrix representations: $E = \left( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right)$, $F = \left( \begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix} \right)$, $H = \left( \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right)$.

Let $g^*$ be the dual vector space of $g$. The vector space $g^*$ is has a dual basis $E^*, F^*, H^*$. Under the bracket $[E^*, F^*]=0$, $[H^*, E^*]=\frac{1}{4}E^*$, $[H^*, F^*]=\frac{1}{4}F^*$, $g^*$ becomes a Lie algebra.

My question is: are there some matrix representations of $E^*, F^*, H^*$? Thank you very much.

Answer 1

The Lie algebra you described doesn't really have much to do with $\mathfrak{sl}_{2}$ because you just described an arbitrary bracket on a 3 dimensional vector space. This Lie algebra has a fairly simple faithful matrix representation as

$$H \mapsto \left(\begin{array}{cc} \frac{1}{4} & 0 & 0\\ 0 & \frac{1}{4} & 0\\ 0 & 0 & 0 \end{array}\right)$$

$$E \mapsto \left(\begin{array}{cc} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right)$$

$$F \mapsto \left(\begin{array}{cc} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{array}\right)$$

Alternatively, you can also view the Lie algebra as the Lie algebra of the group of transformations of the plane that are built out of scaling and translation.

Answer 2

Every finite-dimensional Lie algebra has a faithful matrix representation by a theorem of Ado and Iwasawa. The proof is not so easy. However, if the Lie algebra has trivial center, we can just take the adjoint representation given by $x\mapsto ad(x)$, where $ad(x)(y)=[x,y]$.
The solvable Lie algebra of dimension $3$, which you have given, has trivial center. Hence the adjoint representation is a faithful matrix representation of dimension $3$.