Lazard correspondence

Definition

The Lazard correspondence generalizes the Lie correspondence to the situation where we are not over fields of characteristic zero.

General correspondence

The Lazard correspondence is a correspondence between the collection of Lazard Lie rings and the collection of Lazard Lie groups, which associates, to each Lazard Lie ring, a Lazard Lie group on the same underlying set, and to each Lazard Lie group, a Lazard Lie ring on its underlying set.

Here, a Lazard Lie ring refers to a Lie ring for which there exists a natural number $c$ such that the subring generated by any subset of size at most three is nilpotent of class at most $c$ and every element is uniquely $p$-divisible for all primes $p \le c$.

A Lazard Lie group refers to a group for which there exists a natural number $c$ such that the subgroup generated by any subset of size at most three is nilpotent of class at most $c$ and every element is uniquely $p$-divisible for all primes $p \le c$.

The Lazard correspondence defines the group operations using formulas in terms of the Lie ring operations and division by numbers all of whose prime factors are less than or equal to $c$, using the Baker-Campbell-Hausdorff formula. Conversely, it defines the Lie ring operations in terms of the group operations and division by numbers all of whose prime factors are less than or equal to $c$, using the inverse Baker-Campbell-Hausdorff formula.

Correspondence for a particular prime number $p$

For a fixed prime number $p$, the Lazard correspondence restricts to a correspondence between the p-Lie rings that are Lazard p-Lie rings and the p-groups that are Lazard Lie p-groups. Specifically, these are the Lie rings (respectively groups) in which the subring (respectively subgroup) generated by any subset of size at most three has class at most $p - 1$.

Particular cases

The case $c = 1$

Further information: abelian Lie correspondence

Restricting to the case $c = 1$, we get a correspondence:

Abelian groups $\leftrightarrow$ Abelian Lie rings

The correspondence is as follows: each abelian group becomes an abelian Lie ring by taking the same additive sturcture and using a trivial Lie bracket. Conversely, each abelian Lie ring becomes an abelian group by forgetting the bracket structure and looking only at the underlying group.

The case $c = 2$

Further information: Baer correspondence

In the case $c = 2$, both the group and the Lie ring have class at most two. Note that the 3-local condition (the condition on subgroups/subrings generated by three elements) in this case puts a global bound on class since the condition for having class two is a 3-local condition, i.e., it involves only three variables. Thus, the correspondence is:

2-powered class two Lie rings $\leftrightarrow$ 2-powered class two groups

For any odd prime number $p$, the correspondence restricts to a correspondence:

Class two $p$-Lie rings $\leftrightarrow$ Class two $p$-groups

The correspondence is termed the Baer correspondence. It was described by Reinhold Baer prior to Lazard's general description of the Lazard correspondence. The explicit maps are:

Operation being built Operation in terms of which it is built Formula
Group multiplication Lie ring operations $xy := x + y + \frac{1}{2}[x,y]$
Group identity Lie ring operations Same as the 0 of the Lie ring.
Group inverse Lie ring operations $x^{-1} := -x$
Lie ring addition Group operations $x + y := \frac{xy}{\sqrt{[x,y]}}$
Lie ring identity ( $0$) Group operations Same as the group's multiplicative identity.
Lie ring additive inverses Group operations $-x := x^{-1}$
Lie bracket Group operations $[x,y] = xyx^{-1}y^{-1}$, i.e., same as the commutator in the group (because of class two, the commutator is the same in both left and right conventions).

The case $c = 3$

Further information: class three Lazard correspondence

This correspondence is between: $\{ 2,3 \}$-powered groups of nilpotency class at most three $\leftrightarrow$ $\{2, 3\}$-powered Lie rings of nilpotency class at most three

Note again that the constraint is on the global nilpotency class. On the Lie ring side, this is explained by the fact that nilpotency class three is 3-local for Lie rings. The corresponding statement from groups follows by tracing the correspondence.

For any prime $p > 3$, this restricts to a correspondence: $p$-groups where every subset of size at most three generates a subgroup of class at most three $\leftrightarrow$ $p$-Lie rings where every subset of size at most three generates a subring of class at most three

The explicit maps, arising from the Baker-Campbell-Hausdorff formula and inverse Baker-Campbell-Hausdorff formula, are:

Operation being built Operation in terms of which it is built Formula
Group multiplication Lie ring operations $xy := x + y + \frac{1}{2}[x,y] + \frac{1}{12}[x,[x,y]] - \frac{1}{12}[y,[x,y]]$
Group identity Lie ring operations Same as the 0 of the Lie ring.
Group inverse Lie ring operations $x^{-1} := -x$
Lie ring addition Group operations $x + y := xy\cdot \frac{1}{\sqrt{[x,y]}}\cdot \frac{1}{\sqrt{[x,[x,y]]}}\cdot \sqrt{[y,[x,y]]}$
Lie ring identity ( $0$) Group operations Same as the group's multiplicative identity.
Lie ring additive inverses Group operations $-x := x^{-1}$
Lie bracket Group operations $[x,y] := [x,y] \sqrt{[x,[x,y]]} \sqrt{[y,[x,y]]}$.

Note: The commutator on the left is in the Lie ring, the commutators on the right are in the group.

The case $c = 4$

This is the first case where we gain in generality in our examples by considering 3-local class instead of global class. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Examples

The finite $p$-groups are the primary examples for finite groups, since all other examples arise from these via direct products. For the prime $p = 2$, the only cases of the Lazard correspondence are those for $c = 1$, i.e., the abelian groups. Hence, we here restrict ourselves to the case of odd primes $p$.

Statistics at a glance for the prime 3

Order Logarithm of order to base 3 Total number of isomorphism classes of Lazard Lie groups Number for $c = 1$ (precisely the abelian groups) Number for $c = 2$ not covered under $c = 1$ (all are under the Baer correspondence)
3 1 1 1 0
9 2 2 2 0
27 3 5 3 2
81 4 11 5 6
243 5 35 7 28
729 6 144 11 133
2187 7 1772 15 1757

Statistics at a glance for the prime 5

Order Logarithm of order to base 3 Total number of isomorphism classes of Lazard Lie groups Number for $c = 1$ (precisely the abelian groups) Number for $c = 2$ not covered under $c = 1$ (all are under the Baer correspondence) Number for $c = 3$ not covered under $c \le 2$ Number for $c = 4$ not covered under $c \le 3$
5 1 1 1 0 0 0
25 2 2 2 0 0 0
125 3 5 3 2 0 0
625 4 15 5 6 4 0