Simple Witt algebra

Definition
Suppose $$p$$ is a prime number greater than 3. Suppose $$F$$ is a field of characteristic $$p$$. The $$p$$-dimensional simple Witt algebra over $$F$$, denoted $$W(1;\underline{1})$$, is a Lie algebra over $$F$$defined as follows.


 * The additive group has basis $$e_{-1},e_0,\dots,e_{p-2}$$. Explicitly, it is $$\bigoplus_{i=-1}^{p-2} Fe_i$$.
 * The Lie bracket is defined as follows on the basis:

$$[e_i,e_j] := \left \lbrace\begin{array}{rl} (j - i)e_{i+j}, & -1 \le i + j \le p - 2 \\0, & \mbox{otherwise}\\\end{array}\right.$$

Note in particular that we can take $$F$$ to be the prime field $$\mathbb{F}_p$$, or we can take $$F$$ to be the finite field $$\mathbb{F}_q$$ where $$q$$ is a power of the prime $$p$$, or we can take $$F$$ as an infinite algebraically closed field of characteristic $$p$$.

However, although we can define a bracket of this sort in characteristics other than $$p$$, we do not get a Lie algebra in those characteristics, because the Jacobi identity fails.

Verification of alternation
This is obvious from the definition:


 * $$[e_i,e_i] = 0$$ both in the case $$2i \notin \{ -1,\dots,p-2 \}$$ and in the case $$2i \in \{-1,\dots,p-2\}$$.
 * $$[e_i,e_j] = -[e_j,e_i]$$ for $$i \ne j$$ both in the case $$i + j \notin \{ -1,\dots,p-2 \}$$ and in the case $$i + j \in \{-1,\dots,p-2\}$$.

Verification of Jacobi identity
Note that one of the cases in the verification crucially uses that the characteristic is $$p$$.

It suffices to verify the Jacobi identity on triples of the form $$(e_i,e_j,e_k)$$. Explicitly, it suffices to check that for all $$i,j,k \in \{-1,0,\dots,p-2\}$$, we have:

$$[[e_i,e_j],e_k] + [[e_j,e_k],e_i] + [[e_k,e_i],e_j] = 0$$

We make cases:

Related notions

 * Zassenhaus algebra is a subalgebra of the simple Witt algebra corresponding to an additive subgroup $$G$$ of $$F$$.
 * Panferov Lie algebra is defined similarly, except that the indices used to label the algebras are somewhat shifted.