Let \begin{align} \mathfrak{g} = Span_{\mathbb{C}}\{ e_1, e_2, e_3, e_4, e_5: \text{ nonzero brackets are } [e_1, e_i]=e_{i+1}, i=2,3,4, [e_2, e_3]=e_5 \} \end{align} be a $5$dimensional Lie algebra. I want to write $e_1, \ldots, e_5$ as matrices. That is, I need to find an injective homomorphism of Lie algebras: $\mathfrak{g} \to \mathfrak{gl}(V)$, $V$ is some vector space. Are there some general method to do this? I know that Ado's theorem says that this homomorphism always exists. But I don't know how to construct these matrices explicitly. Can some software like GAP compute the matrices of $e_1, \ldots, e_5$? Thank you very much.

$\begingroup$ Take the invertible derivation mapping $e_i$ to $ie_i$, thus defining a 6dimensional semidirect product of your Lie algebra with a 1dimensional Lie algebra. The resulting Lie algebra has zero center and hence its adjoint representation is faithful (and easily computable). $\endgroup$– YCorAug 15, 2015 at 19:55
1 Answer
The Lie algebra is filiform nilpotent and is generated by $e_1$ and $e_2$. It is known that any faithful Lie algebra representation $\rho:\mathfrak{f}_n\rightarrow \mathbb{gl}(V)$ of a $n$dimensional filiform Lie algebra $\mathfrak{f}_n$ is of degree at least $n$. In the above example, we do not need to invoke Ado's theorem. It is enough to construct a faithful $5$dimensional Lie algebra representation as follows. By Lie's theorem, we may assume that the operators $\rho(e_i)$ are simultaneously uppertriangular. We may even assume that $\rho(e_1)$ satisfies further properties. Then it is easy to compute explicit matrices $\rho(e_1)$ and $\rho(e_2)$, which determine a faithful representation of degree $5$. For details on this construction see, e.g., here, and for computational aspects and other methods in general see here, with references to GAP programs.
Edit: A short computation shows that $$ \rho(e_1)=\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix},\; \rho(e_2)=\begin{pmatrix} 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} $$ defines a representation of the above filiform Lie algebra, where the center is represented nontrivially, i.e., $\rho(e_5)\neq 0$, so that the representation is faithful.

$\begingroup$ What are the furter properties that you may assume for $\rho(e_1)$? Jordan canonical form? Or something else? $\endgroup$ Aug 15, 2015 at 16:19

$\begingroup$ @HenrikWinther Unfortunately one cannot assume in general that $\rho(e_1)$ has Jordan canonical form and all operators remain simultaneously uppertriangular. But one can assume that in each row and each column of $\rho(e_1)$ is at most one nonzero entry, equal to $1$ (see section 2., "$\Delta$modules" in the first link above). $\endgroup$ Aug 15, 2015 at 17:36

1$\begingroup$ @DietrichBurde, thank you very much. In your reference and the manual of GAP, I didn't find the commands to compute $\rho(e_i)$. Where could we find these commands? $\endgroup$ Aug 16, 2015 at 1:56