# Format of grading Witt Lie Algebra

Let $$W(n,m)$$ be generalized Jacobson-Witt 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 $$W(n,m)$$?

• because the $D_j$ lower degree by one – მამუკა ჯიბლაძე Jan 31 '15 at 5:33
• it means that $D_j$ acts on $A(n,m)_{i+1}$ from left? sorry I am confused of that? – user118746 Jan 31 '15 at 13:12