Central extension of an abelian group

Cyclic central extension of a free abelian group

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

and equivalence classes of central extensions

given by taking 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

be an alternating -bilinear map and let . Let

Then is -bilinear and thus a 2-cocycle, amd . By the Correspondence between 2-cocycles and central extensions, there is a central extension of the form above with 2-cocycle and thus commutator map .

Finally let

be a central extension with the same commutator map . Define a -section of so that

Then

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

Automorphisms

Letting and

we have the group extension²

where for , group

Proof

Note is a group homomorphism for any since

and that is itself a group homomorphism since for

Furthermore, the induced automorphism for any .

Now let be an automorphism such that . It follows that for some function . Noting that

it follows that for some function , and since

it follows .

Now consider a general . Then

for all , so . Conversely, given we consider the central extension

which has the commutator map , and thus from the above correspondence, this extension is equivalent to the original one, giving an automorphism in .

Furthermore, any automorphism such that 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 ↩