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:

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

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 $i\neq j$ 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:

No.	Case description	Details
1	$1\leq i+j+k\leq n$, forcing that $1\leq i+j,j+k,i+k\leq n$.	The expression becomes $((j-i)(k-(i+j))+(k-j)(i-(j+k))+(i-k)(j-(k+i)))e_{i+j+k}$, so showing it is zero reduces to showing that $(j-i)(k-(i+j))+(k-j)(i-(j+k))+(i-k)(j-(k+i))=0$. This can be broken down into two verifications: $(j-i)k+(k-j)i+(i-k)j=0$ $(j-i)(i+j)+(k-j)(j+k)+(i-k)(k+i)=(j^{2}-i^{2})+(k^{2}-j^{2})+(i^{2}-k^{2})=0$
2	$i+j+k>n$.	In this case, all the summands $[[e_{i},e_{j}],e_{k}],[[e_{j},e_{k}],e_{i}],[[e_{k},e_{i}],e_{j}]$ are zero. Note that the stage of the computation where each becomes zero depends on whether $i+j,j+k,i+k$ are in $\{1,2,\dots ,n\}$, but they all eventually become zero.

Arithmetic functions

Function	Value	Explanation
nilpotency class (as a Lie ring or as a Lie algebra)	$n-1$	The derived subring is $\sum _{i=3}^{p}Re_{i}$. Beyond that, $\gamma _{j}$ is $\sum _{i=j+1}^{n}Re_{i}$ for $j\leq n-1$. $\gamma _{p}=0$.
derived length (as a Lie ring or as a Lie algebra)	$\lceil \log _{2}(n+2)-1\rceil$	The $i^{th}$ member of the derived series is given as $\sum _{k=j}^{n}Re_{k}$ where $j=2^{i+1}-1$. We want to reach a situation where $j>n$, i.e., $2^{i+1}-1>np$, simplifying to $i\geq \log _{2}(n+2)-1$ and we get the expression.

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 $n$.

References

Journal references

• Nilpotent groups with lower central factors of minimal ranks by B. A. Panferov, Algebra and Logic, Volume 19, Page 455 - 459(Year 1980): ^{Gated copy (PDF)More info}

Textbook references

• p-automorphisms of finite p-groups by Evgenii I. Khukhro, 13-digit ISBN 978-0-521-59717-3, 10-digit ISBN 0-521-59717-X, Page 126, Example 10.28, ^{More info}