The Witt algebra $W(n,m)$ is defined as the set of element $\{\sum f_j D_j$ such that $ f_j ∈ A(n,m)\}$ with usual Lie bracket. I am a bit confused about basis for $W(n,m)$? What is the meaning of "$f_j D_j$" ? I also know that $D_j$ is the partial derivation in $X_j$ on $A(n,m)$ and $A(n,m)$ is polynomial algebra. The set of $\{X^{(a)} D_i\}$ for i between $1$ and $n$, and a between $0$ and $(p^{m1-1},...,p^{mn-1})$. Here $p$ (prime number) is the characteristic of field.
Does This statement "$X^{(a)} D_i $" mean the $i$-th partial derivation of $X^{(a)}$? If so, why do not we write it $D(X^{(a)})$? May you give me more details on the format of basis of Witt algebras? Need to say, I have already studied the book from Helmut Strade on Modular Simple lie algebras.