Let $W(n,m)$ be generalized JacobsonWitt algebra over a field of characteristic $p>3$. According to the grading of $W(n,m)$, we know that it inherits the grading from $A(n,m)$ as follows: $$W(n,m)_i=\sum_{j=1}^{n} A(n,m)_{i+1}D_j\;.$$ Consequently, $$W(n,m)=\bigoplus_{i=1}^{s} W(n,m)_{i}$$ with $ s=\sum_ {i=1}^{n} (p^{m_i}1)1$. I want to know the reason of writing the index of $A(n,m)$ in $W(n,m)_i=\sum_{j=1}^{n} A(n,m)_{i+1}D_j$ ? in fact why we must put $i+1$ ? In the case of $p=2$ how is the grading for $W(n,m)$?

1$\begingroup$ because the $D_j$ lower degree by one $\endgroup$ – მამუკა ჯიბლაძე Jan 31 '15 at 5:33

$\begingroup$ it means that $D_j$ acts on $A(n,m)_{i+1}$ from left? sorry I am confused of that? $\endgroup$ – user118746 Jan 31 '15 at 13:12