Some examples for Lie algebras

Question

I need some small examples for Lie algebras over finite fields ( GF(2) or GF(3)) including some simple Lie algebras and some others which are not simple. And I would be thankful if anyone could give the details for their basis and the matrices of adjoint representations?

Answer 1

There are examples of old and new simple Lie algebras over $GF(2)$ in the paper Some new simple Lie algebras in characteristic 2 of Bettina Eick, in particular in section $5.4$, where explicit generators are given. In low dimensions there is the paper Simple Lie algebras of low dimension of Michael Vaughan-Lee. As for non-simple ones, see the article Classification of solvable Lie algebras by Willem de Graaf.

Here is a $3$-dimensional example, the Jacobson-Witt Lie algebra $W(1,\underline{2})^{(1)}$ in characteristic $2$ with basis $x,y,h$ and Lie brackets $[x,y]=h,\;[h,x]=x,\;[h,y]=y$. Surely you know to to construct the adjoint operators from here: $$ {\rm ad}(x)=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \; {\rm ad}(y)=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \; {\rm ad}(h)=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \; $$