Panferov Lie group

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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

References

Journal references

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