Panferov Lie group

Definition

For a ring and a natural number less than or equal to its powering threshold

Suppose $R$ is a commutative unital ring and $n$ is a natural number greater than or equal to 2 such that the additive group of $R$ is powered over the set of all natural numbers strictly less than $n$, or equivalently, $n$ is at most one more than the powering threshold of the additive group of $R$.

The Panferov Lie group for $R$ and $n$ is defined as the Lazard Lie group (via the Lazard correspondence) for the Panferov Lie algebra (defined below) with parameters $R$ and $n$.

For a prime number: the extreme case

Let $p$ be a prime number. The Panferov Lie group for $p$ is the Panferov Lie group corresponding to $R = \mathbb{F}_p$ (the prime field) and $n = p$.

Definition of Panferov Lie algebra being used

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

Particular cases

Powering thresholds for some ring choices

Choice of $R$ Maximum permissible value of $n$ = one more than the powering threshold of $R$ (this is the smallest prime number such that $R$ is not powered over that number) $R = \mathbb{F}_p$, the prime field for a prime number $p$ $p$ $R = \mathbb{Q}$, the field of rational numbers no maximum, i.e., $n$ could be any natural number. $R = \mathbb{F}_q$ where $q$ is a prime power that equals the size of the finite field $\mathbb{F}_q$ and $p$ is the underlying prime (the field characteristic). $p$ $R = \mathbb{Z}/p^k\mathbb{Z}$ where $p$ is a prime number and $k$ is a positive integer. $p$

Panferov Lie groups for some small choices of $n$ $n$ Description of the Panferov Lie group with parameter $n$ Nilpotency class (equals $n - 1$) Derived length (equals $\lceil \log_2(n + 2) - 1 \rceil$) Theoretical maximum value of derived length for the nilpotency class (the theoretical maximum is $\lfloor \log_2(n - 1) + 1 \rfloor$) Is the derived length at the theoretical maximum for the nilpotency class? (No except for powers of 2 and Mersenne numbers, but otherwise off by one
2 additive group of $R \times R$. It is an abelian group. 1 1 1 Yes
3 unitriangular matrix group of degree three $UT(3,R)$ over $R$. 2 2 2 Yes
4 Panferov Lie group of degree four 3 2 2 Yes
5 Panferov Lie group of degree five 4 2 3 No
6 Panferov Lie group of degree six 5 2 3 No
7 Panferov Lie group of degree seven 6 3 3 Yes

The extreme case

Prime number $p$ Panferov Lie group for $p$ Order (equals $p^p$) Nilpotency class (equals $p - 1$) Derived length (equals $\lceil \log_2(p + 2) - 1 \rceil$) Theoretical maximum value of derived length for the nilpotency class (the theoretical maximum is $\lfloor \log_2(p - 1) + 1 \rfloor$) Is the derived length at the theoretical maximum for the nilpotency class? (No except for $p = 2$ and Mersenne primes, in general it'll be off by one but still close enough)
2 Klein four-group 4 1 1 1 Yes
3 unitriangular matrix group:UT(3,3) 27 2 2 2 Yes
5 Panferov Lie group for 5 3125 4 2 3 No
7 Panferov Lie group for 7 823543 6 3 3 Yes

Arithmetic functions

Function Value in the general case Value in the case $R = \mathbb{F}_p, n = p$ Explanation
order $|R|^n$ $p^p$ same as the order of its Lazard Lie ring, which is $p$-dimensional vector space over $\mathbb{F}_p$.
exponent characteristic of the ring $R$ $p$ same as the exponent of its Lazard Lie ring, which is $p$ on account of it being a nonzero vector space over $\mathbb{F}_p$.
nilpotency class $n - 1$ $p - 1$ same as the nilpotency class of its Lazard Lie ring; see Panferov Lie algebra#Arithmetic functions
derived length $\lceil \log_2(n + 2) - 1 \rceil$ $\lceil \log_2(p + 2) - 1 \rceil$ same as the derived length of its Lazard Lie ring, see Panferov Lie algebra#Arithmetic functions