Generalized Baer correspondence

The term generalized Baer correspondence can be used for any of a number of possible generalizations of the Baer correspondence, some of which are more general than others.

The Baer correspondence (which we will call the Baer correspondence proper to distinguish it from generalizations) is a correspondence:

Uniquely 2-divisible class (at most) two Lie rings (called Baer Lie rings) $$\leftrightarrow$$ Uniquely 2-divisible class (at most) two groups (called Baer Lie groups)

All our generalizations share the following features:


 * Each generalization has some class (at most) two Lie rings on one side and some class (at most) two groups on the other side.
 * In each generalization, the Lie ring and the group have the same order, and their structure is roughly similar. In most of the generalizations discussed below, the additive group of the Lie ring is 1-isomorphic to the group, but this 1-isomorphism is not given by a unique formula as is the case with the Baer correspondence proper. Rather, there is some degree of choice which ends up not mattering.

In order to understand each generalization, we recall that for the Baer correspondence proper, the formulas are as follows:


 * Group multiplication in terms of Lie ring addition: $$xy := x + y + \frac{1}{2}[x,y]$$ where $$[, ]$$ denotes the commutator.
 * Lie ring addition in terms of group multiplication: $$x + y : = \frac{xy}{\sqrt{[x,y]}}$$ where $$[, ]$$ denotes the Lie bracket.

The elements $$\frac{1}{2}[x,y]$$ and $$\sqrt{[x,y]}$$ lie in the respective centers because center of uniquely p-divisible Lie ring is uniquely p-divisible and center of uniquely p-divisible group is uniquely p-divisible. Hence, in the latter case, it is okay to divide by this element without specifying whether the division is on the left or on the right.

Each generalization has to tackle the fundamental problem:

Summary of generalizations
With the exception of the first two generalizations (that are incomparable with each other in terms of generality) these generalizations are in increasing order of generality: each one generalizes all its predecessors.

Derived subgroup/subring is uniquely 2-divisible
This is a slight generalization of the Baer correspondence:

LCS-Baer Lie rings $$\leftrightarrow$$ LCS-Baer Lie groups

Here:


 * A LCS-Baer Lie ring is a Lie ring of nilpotency class two whose derived subring is uniquely 2-divisible.
 * A LCS-Baer Lie group is a group of nilpotency class two whose derived subgroup is uniquely 2-divisible.

Elements in the derived subgroup/subring can be uniquely divided by 2 in a slightly bigger central subgroup/subring
This is a further slight generalization:

CS-Baer Lie rings $$\leftrightarrow$$ CS-Baer Lie groups

Here:


 * A CS-Baer Lie ring is a Lie ring of nilpotency class two $$L$$ such that there exists a Lie subring $$K$$ such that $$[L,L] \le K \le Z(L)$$, and every element of $$L$$ has a unique half in $$K$$.
 * A CS-Baer Lie group is a group of nilpotency class two $$G$$ such that there exists a subgroup $$H$$ such that $$[G,G] \le H \le Z(G)$$, and every element of $$[G,G]$$ has a unique square root within $$H$$.

Linear halving generalization
The key idea behind this generalization is to find a new candidate for the commutator (or Lie bracket) which still gives a class two structure, and whose square (or double) is the original commutator (or Lie bracket). The correspondence is thus:

Lie rings whose brackets arise as the double of a Lie bracket giving nilpotency class two $$\leftrightarrow$$ Groups whose commutator arises as the square of an alternating bihomomorphism of class two

From Lie ring to group
Suppose $$L$$ is a Lie ring of class at most two. Suppose we find a function:

$$*:L \times L \to L$$

satisfying the following conditions:


 * $$x * y \in Z(L)$$ for all $$x,y \in L$$.
 * $$x * y = 0$$ whenever $$x \in L$$ and $$y \in Z(L)$$.
 * $$*$$ is biadditive, i.e., it is $$\mathbb{Z}$$-bilinear. In other words, $$a * (b + c) = (a * b) + (a * c)$$ and $$(a + b) * c = (a * c) + (b * c)$$ for all $$a,b,c \in L$$.
 * $$*$$ is alternating, i.e., $$a * a = 0$$ for all $$a \in L$$. This also implies that $$*$$ is skew symmetric, i.e., $$a * b = -(b * a)$$ for all $$a,b \in L$$.
 * $$2(x * y) = [x,y]$$ for all $$x,y \in L$$.

Equivalently, we can state the above as: $$*$$ is a Lie bracket on $$L$$ that gives $$L$$ the structure of a Lie ring of class at most two, and such that $$2(*) = [, ]$$.

Then, we can define the Lie group corresponding to $$L$$ by the multiplication $$\cdot$$:

$$x \cdot y := x + y + (x * y)$$

From group to Lie ring
Similar to the above.

From Lie ring to group
Suppose $$L$$ is a Lie ring of class at most two. Suppose we find a function:

$$*:L \times L \to L$$

satisfying the following conditions:


 * $$x * y \in Z(L)$$ for all $$x,y \in L$$.
 * $$x * y = 0$$ whenever $$x \in L$$ and $$y \in Z(L)$$.
 * $$*$$ is a 2-cocycle for trivial group action of $$L$$ on itself. (Along with the previous two conditions, this would imply that it descends to a 2-cocycle for trivial group action of $$L/Z(L)$$ on $$Z(L)$$).
 * $$x * y = 0$$ whenever $$x, y$$ additively generate a cyclic subgroup of $$L$$.
 * $$x * y = -(y * x)$$ for all $$x,y \in L$$.
 * $$2(x * y) = [x,y]$$ for all $$x,y \in L$$.

Then, we define the group operation $$\cdot$$ by:

$$x \cdot y := x + y + (x * y)$$

From Lie ring to group
Suppose $$L$$ is a Lie ring of class at most two. Suppose we find a function:

$$*:L \times L \to L$$

satisfying the following conditions:


 * $$x * y \in Z(L)$$ for all $$x,y \in L$$.
 * $$x * y = 0$$ whenever $$x \in L$$ and $$y \in Z(L)$$.
 * $$*$$ is a 2-cocycle for trivial group action of $$L$$ on itself. (Along with the previous two conditions, this would imply that it descends to a 2-cocycle for trivial group action of $$L/Z(L)$$ on $$Z(L)$$).
 * $$x * y = 0$$ whenever $$x, y$$ additively generate a cyclic subgroup of $$L$$.
 * $$(x * y) -(y * x) = [x,y]$$ for all $$x,y \in L$$.

Then, we define the group operation $$\cdot$$ by:

$$x \cdot y := x + y + (x * y)$$