# Proof of Baer construction of Lie group for Baer Lie ring

## Contents

## Statement

This is a part of the Baer correspondence.

Suppose is a Baer Lie ring (?), i.e., a uniquely 2-divisible class two Lie ring, with addition denoted and Lie bracket denoted . We give the structure of a 2-powered class two group as follows:

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

Group multiplication | Since center of uniquely 2-divisible Lie ring is uniquely 2-divisible, we obtain that the element is central. | |

Identity element for multiplication | Same as the zero element of the Lie ring. | |

Multiplicative inverse . | Same as the additive inverse . | |

Group commutator | Same as the Lie bracket . |

## Related facts

### Baer correspondence and its other parts

- Baer correspondence
- Proof of Baer construction of Lie ring for Baer Lie group
- Proof of mutual inverse nature of the Baer constructions between group and Lie ring

### Generalizations

- Proof of generalized Baer construction of Lie group for class two 2-Lie ring with a suitable cocycle

## Proof

### Multiplication is associative

**To prove**:

**Proof**: We have:

We use that the linearity of the Lie bracket, and also use that the Lie ring has nilpotency class two to simplify to zero. The expression simplifies to:

Similarly, we can show that:

Thus, associativity holds.

### Agreement of identity and inverses

Since , we have , and similarly, .

Further, since , we have , and similarly, .

### Commutator agrees with Lie bracket

**To prove**: The multiplicative commutator equals the Lie bracket

**Proof**: We have:

which becomes:

which simplifies to:

Simplifying, and using the fact that the Lie ring has nilpotency class two (so applying a Lie bracket twice gives zero), we are left with the Lie bracket .

### Class two

Since the commutator agrees with the Lie bracket, the class two condition on the Lie ring forces the class two condition on the Lie ring.

### 2-powered

The squaring map for the group operation agrees with the doubling map of the additive group of the Lie ring. Explicitly:

We know that the Lie ring is 2-powered, i.e., the doubling map of the additive group of the Lie ring is bijective. Hence, the squaring map of the group is also bijective, i.e., the group is 2-powered.