Matrix representation of Lie Algebra $B_2$

Question

I'm writing some practical examples where to calculate the Killing form, the Cartan Matrix, Dynkin diagrams etc. Does anybody have on or two nice matrix representations of the $B_2$ Algebra? It would be really apreciated.

Answer 1

Two nice matrix representation are the natural and the adjoint representation. In fact, $B_2$ is the simple Lie algebra $\mathfrak{so}(5)$ of dimension $10$, so the natural representation is given by skew symmetric matrices of order $5$, i.e., $E_{ij}-E_{ji}$ for $i\neq j$ and $1\le i,j\le 5$. For example $$ E_{12}-E_{21}=\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \cr -1 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \end{pmatrix} $$ The adjoint representation is given by the linear operators $ad(x)$ defined by $ad(x)(y)=[x,y]$, which we can compute from the basis $(e_1,\ldots ,e_{10})$ of $\mathfrak{so}(5)$. There are many references for more concrete matrices, e.g., see here.