CS-Baer correspondence

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Definition

The CS-Baer correspondence (short for central series Baer correspondence) is a generalization of the Baer correspondence that operates as follows:

CS-Baer Lie group G along with central subgroup H in which every element of [G,G] has a unique square root \leftrightarrow CS-Baer Lie ring L along with central subring M in which every element of [L,L] has a unique half

Here a CS-Baer Lie group G is simply a group of nilpotency class two that admits a central subgroup H as described above, and CS-Baer Lie ring L is simply a Lie ring of nilpotency class two that admits a central subring M as described above.

A special subcorrespondence is where H = Z(G) and where M = Z(L). These groups and Lie rings correspond with each other.

From group to Lie ring

Suppose G is a LCS-Baer Lie group and H is a central subgroup of G containing [G,G] such that every element of [G,G] has a unique square root in H. Then, G has

Lie ring operation that we need to define Definition in terms of the group operations Further comments
Addition, i.e., define x + y for x,y \in G x + y := \frac{xy}{\sqrt{[x,y]}} Since [x,y] \in [G,G], it has a unique square root in H. We denote this unique square root as \sqrt{[x,y]}. Note that H is in the center, hence the element \sqrt{[x,y]} is central. Thus, to divide by it, we do not need to specify whether we are dividing on the left or on the right.
Identity element for addition, denoted 0. Same as identity element for group multiplication, denoted e or 1. This automatically follows from the way addition is defined.
Additive inverse, i.e., define -x for x \in G. Same as x^{-1}, i.e., the multiplicative inverse in the group. This automatically follows from the way addition is defined.
Lie bracket, i.e., the [ , ] map in the Lie ring. Same as the group commutator [x,y] = xyx^{-1}y^{-1}.