Cyclic central extension of a free abelian group

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

𝑐 0 : 𝐴 \times 𝐴 \to ℤ + 𝑝

and equivalence classes of central extensions

1 \to ℤ + 𝑝 e x p ↪ ˆ 𝐴 𝜋 ↠ 𝐴 \to 1

given by taking $𝑐 0$ as the associated commutator map, and using the Correspondence between 2-cocycles and central extensions.¹ #m/thm/group

Proof

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

Now let

𝑐 0 : 𝐴 \times 𝐴 \to ℤ + 𝑝

be an alternating $ℤ$ -bilinear map and let $𝑆 = {𝑎 𝑖} 𝑛 𝑖 = 1$ . Let

𝜀 0 : 𝐴 \times 𝐴 \to ℤ + 𝑝 (𝛼 𝑖, 𝛼 𝑗) \mapsto {𝑐 0 (𝛼 𝑖, 𝛼 𝑗) 𝑖 > 𝑗 0 𝑖 \leq 𝑗

Then $𝜀 0$ is $ℤ$ -bilinear and thus a 2-cocycle, amd $𝜀 0 (𝑎, 𝑏) - 𝜀 0 (𝑏, 𝑎) = 𝑐 0 (𝑎, 𝑏)$ . By the Correspondence between 2-cocycles and central extensions, there is a central extension of the form above with 2-cocycle $𝜀 0$ and thus commutator map $𝑐 0$ .

Finally let

1 \to ℤ + 𝑝 ↪ 𝐵 𝜋' ↠ 𝐴 \to 1

be a central extension with the same commutator map $𝑐 0$ . Define a $𝖲 𝖾 𝗍$ -section $𝑠 (-)$ of $𝜋'$ so that

𝑠 (-) : 𝑛 \sum 𝑘 = 1 𝑚 𝑘 𝑎 𝑘 = 𝑛 \prod 𝑘 = 1 𝑠 𝑚 𝑘 𝑎 𝑘

Then

𝑠 𝑎 𝑠 𝑏 = 𝑠 𝑎 + 𝑏 e 𝜀 0 (𝑎, 𝑏)

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

Automorphisms

Letting $𝜋 𝑥 = ―― 𝑥$ and

A u t (ˆ 𝐴; e) = {𝜑 \in A u t ˆ 𝐴 : 𝜑 e x p = e x p} A u t (𝐴; 𝑐 0) = {𝜓 \in A u t 𝐴 : 𝑐 0 (𝜓, 𝜓) = 𝑐 0}

we have the group extension²

1 \to 𝖠 𝖻 (𝐴, ℤ + 𝑝) * ↪ A u t (ˆ 𝐴; e) 𝜋 ↠ A u t (𝐴; 𝑐 0) \to 1

where for $𝜆 \in 𝖠 𝖻 (𝐴, ℤ + 𝑝) ≅ (ℤ + 𝑝) 𝑛$ , #m/thm/group

𝜆 * : ˆ 𝐴 \to ˆ 𝐴 𝑥 \mapsto 𝑥 e 𝜆 ¯ 𝑥

Proof

Note $𝜆 *$ is a group homomorphism for any $𝜆 \in 𝖠 𝖻 (𝐴, ℤ + 𝑝)$ since

𝜆 * (𝑥 𝑦) = 𝑥 e 𝜆 ―― 𝑥 𝑦 e 𝜆 ―― 𝑦 = 𝑥 𝑦 e 𝜆 ―― 𝑥 + 𝜆 ―― 𝑦 = 𝑥 𝑦 e 𝜆 ――― 𝑥 𝑦 = (𝜆 * 𝑥) (𝜆 * 𝑦)

and that $*$ is itself a group homomorphism since for $𝜆, 𝜇 \in 𝖠 𝖻 (𝐴, ℤ + 𝑝)$

(𝜆 + 𝜇) * 𝑥 = 𝑥 e 𝜆 ―― 𝑥 + 𝜇 ―― 𝑥 = 𝜆 * (𝑥 e 𝜇 ―― 𝑥) = 𝜆 * 𝜇 * 𝑥

Furthermore, the induced automorphism $――― 𝜆 * = i d 𝐴$ for any $𝜆 \in 𝖠 𝖻 (𝐴, ℤ + 𝑝)$ .

Now let $𝜑 \in A u t (ˆ 𝐴; e)$ be an automorphism such that $―― 𝜑 = i d 𝐴$ . It follows that $𝜑 𝑥 = 𝑥 e 𝑓 (𝑥)$ for some function $𝑓 : ˆ 𝐴 \to ℤ + 𝑝$ . Noting that

𝑥 e 𝑓 (𝑥) = 𝜑 𝑥 = 𝜑 (𝑥 e 𝑠) = 𝑥 e 𝑓 (𝑥 e 𝑠)

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

𝑥 𝑦 e 𝜆 ――― 𝑥 𝑦 = 𝜑 (𝑥 𝑦) = (𝜑 𝑥) (𝜑 𝑦) = 𝑥 e 𝜆 ―― 𝑥 𝑦 e 𝜆 ―― 𝑦 = 𝑥 𝑦 e 𝜆 ―― 𝑥 + 𝜆 ―― 𝑦

it follows $𝜆 \in 𝖠 𝖻 (𝐴, ℤ + 𝑝)$ .

Now consider a general $𝜑 \in A u t (ˆ 𝐴; e)$ . Then

𝑐 0 (¯ 𝜑 ¯ 𝑥, ¯ 𝜑 ¯ 𝑦) = l n [𝜑 𝑥, 𝜑 𝑦] = l n 𝜑 [𝑥, 𝑦] = l n [𝑥, 𝑦] = 𝑐 0 (―― 𝑥, ―― 𝑦)

for all $𝑥, 𝑦 \in ˆ 𝐴$ , so $―― 𝜑 \in A u t (𝐴; 𝑐 0)$ . Conversely, given $𝜓 \in A u t (𝐴; 𝑐 0)$ we consider the central extension

1 \to ℤ + 𝑝 ↪ ˆ 𝐴 𝜓 𝜋 ↠ 𝐴 \to 1

which has the commutator map $𝑐 0$ , and thus from the above correspondence, this extension is equivalent to the original one, giving an automorphism in $A u t (ˆ 𝐴; e)$ .

Furthermore, any automorphism $𝜗 \in A u t (ˆ 𝐴; e)$ such that $―― 𝜗 = - 1$ is itself an involution.

Special cases

2-central extension of a free abelian group

#state/tidy | #lang/en | #SemBr

1988. Vertex operator algebras and the Monster, ¶5.2.3, pp. 106–107 ↩
1988. Vertex operator algebras and the Monster, ¶5.4.1, p. 112 ↩

Cyclic central extension of a free abelian group

Automorphisms

Special cases

Footnotes