Global Lazard correspondence

Definition

The global Lazard correspondence is a subcorrespondence of the Lazard correspondence that puts a global condition on nilpotency class (in place of 3-local conditions) and is easier to work with for many proofs. Many sources simply define the Lazard correspondence as the global Lazard correspondence, and ignore the more general version.

Setup of the correspondence

The global Lazard correspondence is a correspondence:

Global Lazard Lie groups $\leftrightarrow$ Global Lazard Lie rings

Here:

• A global Lazard Lie group is a nilpotent group that is powered (i.e., uniquely divisible) for all primes less than or equal to its nilpotency class.
• A global Lazard Lie ring is a nilpotent Lie ring whose additive group is powered (i.e., uniquely divisible) for all primes less than or equal to its nilpotency class.

The global 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 formulas.

Breakdown by nilpotency class

Suppose $c$ is a positive integer. The global class $c$ Lazard correspondence is a correspondence:

Global class $c$ Lazard Lie groups $\leftrightarrow$ Global class $c$ Lazard Lie rings

Here:

• A global class $c$ Lazard Lie group is a nilpotent group of nilpotency class at most $c$ that is powered (i.e., uniquely divisible) by all primes less than or equal to $c$.
• A global class $c$ Lazard Lie ring is a nilpotent Lie ring of nilpotency class at most $c$ whose additive group is powered (i.e., uniquely divisible) by all primes less than or equal to $c$.

The following are true about the global class $c$ Lazard correspondences for different values of $c$:

• Interpolation of class: A group that is a global class $c_{1}$ Lazard Lie group as well as a global class $c_{2}$ Lazard Lie group is also a global class $c$ Lazard Lie group for all $c_{1}\leq c\leq c_{2}$. The same observation applies on the Lie ring side.
• List of permissible class ranges: Alternatively, a group is a global Lazard Lie group if and only if its nilpotency class is finite and less than or equal to its powering threshold. The group is a global class $c$ Lazard Lie group for all $c$ that are at least equal to the nilpotency class and at most equal to the powering threshold. Note that a powering threshold of $\infty$ corresponds to the case of a rationally powered nilpotent group) and in this case, the group is a global class $c$ Lazard Lie group for all $c$ that are at least equal to the nilpotency class. Similar observations apply on the Lie ring side.
• Global Lazard correspondences of different classes agree where they both make sense: Suppose a group is both a global class $c_{1}$ Lazard Lie group and a global class $c_{2}$ Lazard Lie group. Then, its Lazard Lie rings for the global class $c_{1}$ Lazard correspondence and global class $c_{2}$ Lazard correspondence coincide. The global Lazard correspondence can therefore be viewed as a union of the global class $c$ Lazard correspondences for all positive integer values of $c$.

Distinction with the general version (the 3-local case) of the Lazard correspondence

The (3-local) Lazard correspondence is a correspondence:

Lazard Lie groups $\leftrightarrow$ Lazard Lie rings

The key distinction between a Lazard Lie group and a global Lazard Lie group is that for the former, we only require powering for the primes that are less than or equal to the 3-local nilpotency class. This may be less than the nilpotency class, making the condition weaker. A similar distinction applies on the Lie ring side. Thus, the Lazard correspondence is more general than the global Lazard correspondence.

The global Lazard correspondence is a subcorrespondence of the Lazard correspondence, i.e., in cases where the global Lazard correspondence, it coincides with the Lazard correspondence.

Correspondence for a particular prime number $p$

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

p-groups of nilpotency class less than p $\leftrightarrow$ p-Lie rings of nilpotency class less than p

The rationally powered case

The Malcev correspondence, a correspondence between rationally powered nilpotent groups and rationally powered nilpotent Lie rings, is a subcorrespondence of the global Lazard correspondence.