Proof of Baer construction of Lie group for Baer Lie ring
This is a part of the Baer correspondence.
|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 .|
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
- Proof of generalized Baer construction of Lie group for class two 2-Lie ring with a suitable cocycle
Multiplication is associative
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 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 .
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.
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.