Central group extension

Central extension of an abelian group

Let (𝐴, +) be an abelian group and consider a central extension

1→𝐶exp↪ˆ𝐴𝜋↠𝐴→1

Then ˆ𝐴 is nilpotent with its commutator subgroup central group since

[ˆ𝐴,ˆ𝐴]⊴exp⁡(𝐶)⊴𝑍(ˆ𝐴)

and given a \Set-section 𝑠(−) :𝐴 ↪ˆ𝐴 of 𝜋 we have the associated commutator map

𝑐0:𝐴×𝐴→𝐶(𝑎,𝑏)↦ln⁡[𝑠𝑎,𝑠𝑏]

which is alternating ℤ-bilinear and independent of 𝑠(−).¹

Properties

In what follows, if 𝐵 ≤𝐴 is a subgroup let ˆ𝐵 =𝜋−1𝐵.

1. ˆ𝐵 is abelian iff 𝑐0(𝐵,𝐵) =0.
2. Consider the radical of 𝑐0

𝑅={𝑎∈𝐴:𝑐0(𝑎,𝐴)=0}

Then ˆ𝑅 =𝑍(ˆ𝐴). ^P2 3. The associated 2-cycle 𝜀0 :𝐴 ×𝐴 →𝐶 and associated commutator 𝑐0 :𝐴 ×𝐴 →𝐶 are related by

𝑐0(𝑎,𝑏)=𝜀0(𝑎,𝑏)−𝜀0(𝑏,𝑎)

that is, 𝑐0 is the antisymmetrization of 𝜀0.

Proof

Note ˆ𝐵 =𝑠𝐵e𝐶, and [𝑠𝐵e𝐶,s𝐵e𝐶] =[𝑠𝐵,𝑠𝐵], from which one easily verifies ^P1.

Assume 𝑠𝑎e𝑝 ∈ˆ𝑅, so 𝑎 ∈𝑅. Then 𝑐0(𝑎,𝐴) =ln⁡[𝑠𝑎,𝑠𝐴] =0. Given any 𝑠𝑏e𝑞 ∈ˆ𝐴,

𝑠𝑎e𝑝𝑠𝑏e𝑞=e𝑞𝑠𝑎𝑠𝑏e𝑝=e𝑞𝑠𝑏𝑠𝑎e𝑝=𝑠𝑏e𝑞𝑠𝑎e𝑝

so 𝑠𝑎e𝑝 ∈𝑍(ˆ𝐴). Similarly, if 𝑠𝑎 ∈𝑍(ˆ𝐴) then 𝑐0(𝑎,𝐴) =ln⁡[𝑠𝑎,𝐴] =0. Therefore ˆ𝑅 =𝑍(ˆ𝐴), proving ^P2.

Finally, noting that exp is a group monomorphism,

e𝑐0(𝑎,𝑏)=[𝑠𝑎,𝑠𝑏]=𝑠𝑎𝑠𝑏𝑠−1𝑎𝑠−1𝑏=𝑠𝑎+𝑏e𝜀0(𝑎,𝑏)𝑠−1𝑏+𝑎e−𝜀0(𝑏,𝑎)=e𝜀0(𝑎,𝑏)−𝜀0(𝑎,𝑏)

proving ^P3.

Special cases

• Cyclic central extension of a free abelian group
  • 2 central extension of a free abelian group
• 2 central extension of an elementary abelian 2-group (includes extraspecial 2-groups)

---

tidy | en | SemBr

Footnotes

1. 1988. Vertex operator algebras and the Monster, §5.2, pp. 104ff. ↩