Example of two-dimensional non-abelian Lie algebra?

Question

can some one give me an example of two-dimensional non-abelian Lie algebra?

Answer 1

Let $X=\begin{bmatrix} a&b\\0&0 \end{bmatrix} $ and $Y=\begin{bmatrix} x&y\\0&0 \end{bmatrix} $. Furthermore, let the Lie bracket be the matrix commutator: $[X,Y]:=XY-YX$. We get $XY= \begin{bmatrix} ax&ay\\0&0 \end{bmatrix}$, but $YX= \begin{bmatrix} ax&bx\\0&0 \end{bmatrix}$.

(That is the lie algebra of the affine group mentioned by Eric).

Answer 2

This is something easy to come up with: take a basis $\{X,Y\}$ of your space. Then to be non-abelian $[X,Y]$ has to be non-zero. So try $[X,Y] = X$. It's straightforward to verify this satisfies the axioms of a Lie algebra. With a little more work you can show this is the unique (up to isomorphism) two dimensional non-abelian Lie algebra.

This Lie algebra has a geometric interpretation as the Lie algebra of affine transformations of the real line, i.e. all maps $\mathbb R \to \mathbb R$ of the form $x \mapsto ax + b$, $a,b \in \mathbb R$.