# 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 along with central subgroup in which every element of has a unique square root CS-Baer Lie ring along with central subring in which every element of has a unique half

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

A special subcorrespondence is where and where . These groups and Lie rings correspond with each other.

### From group to Lie ring

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

Lie ring operation that we need to define | Definition in terms of the group operations | Further comments |
---|---|---|

Addition, i.e., define for | Since , it has a unique square root in . We denote this unique square root as . Note that is in the center, hence the element 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 . | Same as identity element for group multiplication, denoted or . | This automatically follows from the way addition is defined. |

Additive inverse, i.e., define for . | Same as , 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 . |