# CS-Baer correspondence

## 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 $[x,y] = xyx^{-1}y^{-1}$.