Recognize some 4-dimensional Lie algebra

Question

I am looking for the matrix representations of the Lie Algebra given by the commutators:

$[G_1,G_2]=0,\\ [G_1,G_3]=-[G_2,G_3]=G_4,\\ [G_1,G_4]=G_3, [G_2,G_4]=-G_3,\\ [G_3,G_4]=-2G_1+2G_2$

But I don't know if they form some well-known Lie Algebra.

Answer 1

First of all, you have a typo. We need $[G_2,G_4]=-G_3$ and not $+G_3$. Otherwise the Jacobi identity is not satisfied: $$ [G_1,[G_2,G_3]]+[G_2,[G_3,G_1]]+[G_3,[G_1,G_2]]=-G_3-G_3+0\neq 0. $$

For all real and complex Lie Algebras of dimension $4$, faithful matrix representations of minimal degree have been determined. See for example the paper by Ghanam and Thompson "Minimal Matrix Representations of Four-Dimensional Lie Algebras".

For a general method to compute this, at least for nilpotent Lie algebras, see my paper here.

It is easy to find out which algebra from the list is yours in dimension $4$ by comparing isomorphism invariants, like $\dim [L,L]$, or $\dim Z(L)$ or $\dim [[L,L],[L,L]]$ or $\dim \operatorname{Der}(L)$, etc.

So your Lie algebra is $\mathfrak{sl}_2(\Bbb C)\oplus \Bbb C=\mathfrak{gl}_2(\Bbb C)$, where the center is generated by $2(G_2-G_1)$.

Answer 2

The list of all 4-dim real Lie algebras can be found here:

https://en.m.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras