Casimir of a three dimensional solvable lie algebra

Question

Good morning everyone. I've encountered recently during my computations the following lie algebra

$$\mathfrak g=\text{span}(f_0,f_1,f_2),$$ with $$\begin{eqnarray}[f_2,f_1]&=&f_0+a f_2,\\ [f_1,f_0]&=&bf_2,\\ [f_2,f_0]&=&0,\qquad a,b\in \mathbb R\setminus\{0\}\end{eqnarray}$$

Of course $$\dim[\mathfrak g,\mathfrak g]=2,$$ and the algebra is solvable, moreover I know we can interpret it as the operator $$\text{ad}f_1$$ acting on $$\text{span}(f_0,f_2).$$ I was wondering to know if there are known casimirs for this algebra or any suggestions or references on how to find them.

Thanks anyone for helping me.

Best wishes

Answer 1

Casimirs for low-dimensional Lie algebras are given explicitly in Invariants of real low-dimensional Lie algebras by J Patera, RT Sharp, P Winternitz and H Zassenhaus.

As you point out, the Lie algebra in question is the semidirect product of the two-dimensional abelian Lie algebra spanned by $(f_0,f_2)$ by the one-dimensional Lie algebra spanned by $f_1$, where the action of $f_1$ is given by the matrix $$ A = \begin{pmatrix} 0 & 1 \cr b & a \end{pmatrix} $$

Since $a\neq 0$, the matrix has nonzero trace and hence the Lie algebra is not unimodular. The characteristic polynomial of the above matrix is $\chi_A(t) = t^2 - a t - b$. We have to to consider three different cases:

1. $b > -1/4 a^2$, in which case the Lie algebra is isomorphic to $A_{3,5}$ in that paper,

2. $b=-1/4 a^2$, in which case it is isomorphic to $A_{3,2}$, and

3. $b < -1/4 a^2$, in which case it is isomorphic to $A_{3,7}$.

I leave you the exercise of relating your basis to the ones in the paper (hint: bring the matrix to normal form) so that you can read off the invariants from their Table I.