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So I have a Lie algebra with two generators defined as:

$$ X=\begin{bmatrix} a&b\\0&0 \end{bmatrix}, \quad Y=\begin{bmatrix} x&y\\0&0 \end{bmatrix}. $$

The matrix commutator is:

$$ [X,Y]:=XY-YX. $$

In which:

$$ XY= \begin{bmatrix} ax&ay\\0&0 \end{bmatrix}, \quad YX= \begin{bmatrix} ax&bx\\0&0 \end{bmatrix}. $$

This was an example from another user.

My question is if it is appropriate to say that this algebra has two generators (as opposed to three), and also if the algebra has any unitary representations.

Dietrich Burde
  • 130,978
Mike Chang
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1 Answers1

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It was already remarked at the linked question that there is only one non-abelian Lie algebra of dimension $2$ over $K$ up to isomorphism and that we may take a basis $(x,y)$ such that $[x,y]=x$. In particular, we have only $2$ generators.

As for unitary representations we need to be precise what this is for Lie algebras, see this MO-post.

Dietrich Burde
  • 130,978