Panferov Lie algebra

Definition
Suppose $$n$$ is a natural number greater than or equal to 2 and $$R$$ is a commutative unital ring. The Panferov Lie algebra of degree $$n$$ over the ring $$R$$ is a $$R$$-Lie algebra (and hence also is a Lie ring) defined as follows:


 * The additive group is a free module of rank $$n$$ with basis $$e_1,e_2,\dots,e_n$$. Explicitly, the Lie algebra is $$\bigoplus_{i=1}^n Re_i$$
 * The Lie bracket is defined as follows:

$$[e_i,e_j] = \left\lbrace \begin{array}{rl} (i - j)e_{i+j}, & i + j \le n \\ 0, & i + j > n \\\end{array}\right.$$

Verification of alternation
This is obvious from the definition:


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

Verification of Jacobi identity
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 $$i,j,k \in \{1,2,\dots,n-1,n\}$$, 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 algebras

 * Simple Witt algebra has a very similar presentation, but turns out to be simple instead of nilpotent. Note that this crucially relies on the characteristic of the field being equal to the $$n$$-value.
 * Zassenhaus algebra is a generalization of the simple Witt algebra.
 * Panferov Lie group is the corresponding Lazard Lie group via the Lazard correspondence. This does not always make sense. Rather, it makes sense when $$R$$ is powered over all primes strictly less than $$n$$.