How to calculate ad(X)?

Question

I have looked everywhere, and maybe its really simple, and I am just being stupid, but I really don't know how to calculate ad(X). I understand that ad_x(y)=[x,y], but i just want to calculate ad(x)? I also know that Ad(g)(X) = g^(-1)Xg. "g inverse multiplied by X multiplied by g", but the determinant for my g is 0, so it can't have an inverse, hence why I can't do it this way. My g is \begin{bmatrix}0&x&y\\x&0&z\\y&-z&0\end{bmatrix}And I have to work out ad(x1), where x1 is one of the basis of the g. I already have the basis, it is \begin{bmatrix}0&1&0\\1&0&0\\0&0&0\end{bmatrix} Thank you.

Answer 1

The adjoint operators ${\rm ad}(x)$ for $x\in L$ are linear maps, so we can compute them by evaluating them on a basis of the Lie algebra $L$. For example, let $L$ be the Heienberg Lie algebra with basis $(e_1,e_2,e_3)$ and brackets $[e_1,e_2]=e_3$. Then $$ {\rm ad}(e_1)=\begin{pmatrix} 0&0&0 \\ 0&0&0 \\ 0&1&0 \end{pmatrix}, $$ since ${\rm ad}(e_1)(e_2)=[e_1,e_2]=e_3$. The images of the basis vectors are the colums of the matrix. Similarly for ${\rm ad}(e_2)$. Finally ${\rm ad}(e_3)=0$, since $e_3$ generates the center of $L$, and $Z(L)=\ker(ad(L))$.

For your Lie algebra, you first have to fix a basis $(e_1,e_2,e_3)$, then compute all Lie brackets, and from them write down the adjoint operators.

Answer 2

If you already know structure constants in the base you're using, then the coefficients of the adjoint action are easily computed as $$(ad(e_i))^k_j=c_{ij}^k,$$ since $$ad(e_i)(e_j)=[e_i, e_j]=\sum{c_{ij}^k}e_k.$$