Central extension of an abelian group

Cyclic central extension of a free abelian group

Let 𝐴 =ℤ𝑆 be a free abelian group of finite rank 𝑛. Then there is a bijection between the set of alternating ℤ-bilinear maps

𝑐0:𝐴×𝐴→ℤ+𝑝

and equivalence classes of central extensions

1→ℤ+𝑝exp↪ˆ𝐴𝜋↠𝐴→1

given by taking 𝑐0 as the associated commutator map, and using the Correspondence between 2-cocycles and central extensions.¹ group

Proof

That equivalent central extensions determine the same commutator map follows from ^P3 and Correspondence between 2-cocycles and central extensions.

Now let

𝑐0:𝐴×𝐴→ℤ+𝑝

be an alternating ℤ-bilinear map and let 𝑆 ={𝑎𝑖}𝑛𝑖=1. Let

𝜀0:𝐴×𝐴→ℤ+𝑝(𝛼𝑖,𝛼𝑗)↦{𝑐0(𝛼𝑖,𝛼𝑗)𝑖>𝑗0𝑖≤𝑗

Then 𝜀0 is ℤ-bilinear and thus a 2-cocycle, amd 𝜀0(𝑎,𝑏) −𝜀0(𝑏,𝑎) =𝑐0(𝑎,𝑏). By the Correspondence between 2-cocycles and central extensions, there is a central extension of the form above with 2-cocycle 𝜀0 and thus commutator map 𝑐0.

Finally let

1→ℤ+𝑝↪𝐵𝜋′↠𝐴→1

be a central extension with the same commutator map 𝑐0. Define a \Set-section 𝑠(−) of 𝜋′ so that

𝑠(−):𝑛∑𝑘=1𝑚𝑘𝑎𝑘=𝑛∏𝑘=1𝑠𝑚𝑘𝑎𝑘

Then

𝑠𝑎𝑠𝑏=𝑠𝑎+𝑏e𝜀0(𝑎,𝑏)

for any 𝑎,𝑏 ∈𝐴, so by the Correspondence between 2-cocycles and central extensions these extensions are equivalent.

Automorphisms

Letting 𝜋𝑥 =――𝑥 and

Aut⁡(ˆ𝐴;e)={𝜑∈Aut⁡ˆ𝐴:𝜑exp=exp}Aut⁡(𝐴;𝑐0)={𝜓∈Aut⁡𝐴:𝑐0(𝜓,𝜓)=𝑐0}

we have the group extension²

1→𝖠𝖻(𝐴,ℤ+𝑝)∗↪Aut⁡(ˆ𝐴;e)𝜋↠Aut⁡(𝐴;𝑐0)→1

where for 𝜆 ∈𝖠𝖻(𝐴,ℤ+𝑝) ≅(ℤ+𝑝)𝑛, group

𝜆∗:ˆ𝐴→ˆ𝐴𝑥↦𝑥e𝜆¯𝑥

Proof

Note 𝜆∗ is a group homomorphism for any 𝜆 ∈𝖠𝖻(𝐴,ℤ+𝑝) since

𝜆∗(𝑥𝑦)=𝑥e𝜆――𝑥𝑦e𝜆――𝑦=𝑥𝑦e𝜆――𝑥+𝜆――𝑦=𝑥𝑦e𝜆―――𝑥𝑦=(𝜆∗𝑥)(𝜆∗𝑦)

and that ∗ is itself a group homomorphism since for 𝜆,𝜇 ∈𝖠𝖻(𝐴,ℤ+𝑝)

(𝜆+𝜇)∗𝑥=𝑥e𝜆――𝑥+𝜇――𝑥=𝜆∗(𝑥e𝜇――𝑥)=𝜆∗𝜇∗𝑥

Furthermore, the induced automorphism ――𝜆∗ =id𝐴 for any 𝜆 ∈𝖠𝖻(𝐴,ℤ+𝑝).

Now let 𝜑 ∈Aut⁡(ˆ𝐴;e) be an automorphism such that ――𝜑 =id𝐴. It follows that 𝜑𝑥 =𝑥e𝑓(𝑥) for some function 𝑓 :ˆ𝐴 →ℤ+𝑝. Noting that

𝑥e𝑓(𝑥)=𝜑𝑥=𝜑(𝑥e𝑠)=𝑥e𝑓(𝑥e𝑠)

it follows that 𝑓(𝑥) =𝜆――𝑥 for some function 𝜆 :𝐴 →ℤ+𝑝, and since

𝑥𝑦e𝜆―――𝑥𝑦=𝜑(𝑥𝑦)=(𝜑𝑥)(𝜑𝑦)=𝑥e𝜆――𝑥𝑦e𝜆――𝑦=𝑥𝑦e𝜆――𝑥+𝜆――𝑦

it follows 𝜆 ∈𝖠𝖻(𝐴,ℤ+𝑝).

Now consider a general 𝜑 ∈Aut⁡(ˆ𝐴;e). Then

𝑐0(¯𝜑¯𝑥,¯𝜑¯𝑦)=ln⁡[𝜑𝑥,𝜑𝑦]=ln⁡𝜑[𝑥,𝑦]=ln⁡[𝑥,𝑦]=𝑐0(――𝑥,――𝑦)

for all 𝑥,𝑦 ∈ˆ𝐴, so ――𝜑 ∈Aut⁡(𝐴;𝑐0). Conversely, given 𝜓 ∈Aut⁡(𝐴;𝑐0) we consider the central extension

1→ℤ+𝑝↪ˆ𝐴𝜓𝜋↠𝐴→1

which has the commutator map 𝑐0, and thus from the above correspondence, this extension is equivalent to the original one, giving an automorphism in Aut⁡(ˆ𝐴;e).

Furthermore, any automorphism 𝜗 ∈Aut⁡(ˆ𝐴;e) such that ――𝜗 = −1 is itself an involution.

Special cases

• 2 central extension of a free abelian group

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Footnotes

1. 1988. Vertex operator algebras and the Monster, ¶5.2.3, pp. 106–107 ↩

2. 1988. Vertex operator algebras and the Monster, ¶5.4.1, p. 112 ↩