Affine Lie algebra

Let $𝔤$ be a quadratic Lie algebra with a symmetric $𝔤$ -invariant bilinear form $⟨ \cdot, \cdot ⟩$ . The corresponding affine Lie algebra $ˆ 𝔤$ is a certain graded central extension of the Loop algebra $𝔤 [𝑡, 𝑡 - 1]$ .¹ Thence one can construct the corresponding extended affine Lie algebra $˜ 𝔤$ by adjoining the degree derivation. A generalization is the Twisted affine Lie algebra.

Construction

Let $𝔤$ be an algebra over $𝕂$ with some bilinear form $⟨ \cdot, \cdot ⟩ : 𝔤 \times 𝔤 \to 𝕂$ .² Further let $𝑑 = 𝑡 𝑑 𝑑 𝑡$ be the degree derivation on $𝕂 [𝑡, 𝑡 - 1]$ , and construct the vector space

ˆ 𝔤 = 𝔤 [𝑡, 𝑡 - 1] \oplus 𝕂 𝑐

with the bilinear product $[\cdot, \cdot] : ˆ 𝔤 \times ˆ 𝔤 \to ˆ 𝔤$ defined by the conditions

[𝑐, ˆ 𝔤] = [ˆ 𝔤, 𝑐] = 0 [𝑥 \otimes 𝑓, 𝑦 \otimes 𝑔] = [𝑥, 𝑦] \otimes 𝑓 𝑔 + ⟨ 𝑥, 𝑦 ⟩ (𝑑 𝑓 \cdot 𝑔) 0 𝑐

the latter being equivalent to

[𝑥 \otimes 𝑡 𝑛, 𝑦 \otimes 𝑡 𝑚] = [𝑥, 𝑦] \otimes 𝑡 𝑛 + 𝑚 + ⟨ 𝑥, 𝑦 ⟩ 𝑛 𝛿 𝑛 + 𝑚 𝑐

Then $ˆ 𝔤$ is a Lie algebra, called the affine Lie algebra associated with $𝔤$ and $⟨ \cdot, \cdot ⟩$ , #m/def/lie iff $𝔤$ is a Lie algebra and $⟨ \cdot, \cdot ⟩$ is a symmetric $𝔤$ -invariant bilinear form, and we have the central extension

0 \to 𝕂 𝑐 ↪ ˆ 𝔤 ↠ 𝔤 [𝑡, 𝑡 - 1] \to 0

Proof

First note the bracket on $ˆ 𝔤$ is alternating iff that on $𝔤$ is. Let $𝑁 = 𝑛 + 𝑚 + 𝑘$ . Then the Jacobi identity on $ˆ 𝔤$ is equivalent to

0 = [𝑥 \otimes 𝑡 𝑛, [𝑦 \otimes 𝑡 𝑚, 𝑧 \otimes 𝑡 𝑘]] + [𝑦 \otimes 𝑡 𝑚, [𝑧 \otimes 𝑡 𝑘, 𝑥 \otimes 𝑡 𝑛]] + [𝑧 \otimes 𝑡 𝑘, [𝑥 \otimes 𝑡 𝑛, 𝑦 \otimes 𝑡 𝑚]] = [𝑥 \otimes 𝑡 𝑛, [𝑦, 𝑧] \otimes 𝑡 𝑚 + 𝑘 + 𝐶 1 𝑐] + [𝑦 \otimes 𝑡 𝑚, [𝑧, 𝑥] \otimes 𝑡 𝑘 + 𝑛 + 𝐶 2 𝑐] + [𝑧 \otimes 𝑡 𝑘, [𝑥, 𝑦] \otimes 𝑡 𝑛 + 𝑚 + 𝐶 3 𝑐] = ([𝑥, [𝑦, 𝑧]] + [𝑦, [𝑧, 𝑥]] + [𝑧, [𝑥, 𝑦]]) \otimes 𝑡 𝑁 + (⟨ 𝑥, [𝑦, 𝑧] ⟩ 𝑛 + ⟨ 𝑦, [𝑧, 𝑥] ⟩ 𝑚 + ⟨ 𝑧, [𝑥, 𝑦] ⟩ 𝑘) 𝛿 𝑁, 0 𝑐

which holds iff the Jacobi identity holds for $𝔤$ along with the identity

⟨ 𝑥, [𝑦, 𝑧] ⟩ 𝑛 + ⟨ 𝑦, [𝑧, 𝑥] ⟩ 𝑚 + ⟨ 𝑧, [𝑥, 𝑦] ⟩ 𝑘 = 0

for all $𝑛, 𝑚, 𝑘$ such that $𝑛 + 𝑚 + 𝑘 = 0$ . The latter is equivalent to the bilinear map being symmetric and $𝔤$ -invariant, as can be shown by varying $𝑛, 𝑚, 𝑘$ .

We extend $𝑑$ to a degree derivation of $ˆ 𝔤$ by

𝑑 (𝑐) = 0 𝑑 (𝑥 \otimes 𝑓) = 𝑥 \otimes 𝑑 𝑓

so that homogenous subspaces are the eigenspaces of $𝑑$ . One obtains the extended affine Lie algebra associated with $𝔤$ and $⟨ \cdot, \cdot ⟩$ by adjoining the degree derivation $𝑑$ #m/def/lie

˜ 𝔤 = ˆ 𝔤 ⋊ 𝕂 𝑑

giving the $ℤ$ -gradation³

˜ 𝔤 𝑛 = {𝔤 \otimes 𝑡 𝑛 𝑛 \neq 0 𝔤 \oplus 𝕂 𝑐 \oplus 𝕂 𝑑 𝑛 = 0

Properties

If $𝔤$ is an Abelian Lie algebra, the (extended) affine Lie algebra has a triangular decomposition $ˆ 𝔤 = ˆ 𝔤 - \oplus ˆ 𝔤 0 \oplus ˆ 𝔤 + ˜ 𝔤 = ˜ 𝔤 - \oplus ˜ 𝔤 0 \oplus ˜ 𝔤 +$
See Formal series over an (un)twisted affine Lie algebra

Functoriality

These constructions may be extended to functors from $𝖰 𝖫 𝗂 𝖾 𝕂$ to $𝖦 𝗋 ℤ 𝖫 𝗂 𝖾 𝕂$ , so that if $𝑓 \in 𝖰 𝖫 𝗂 𝖾 𝕂 (𝔤, 𝔥)$ is an isometric Lie algebra homomorphism, one defines $ˆ 𝑓, ˜ 𝑓 \in 𝖦 𝗋 ℤ 𝖫 𝗂 𝖾 𝕂 (ˆ 𝔤, ˆ 𝔥)$ by

ˆ 𝑓 : 𝑥 \otimes 𝑡 𝑛 \mapsto 𝑓 (𝑥) \otimes 𝑡 𝑛 𝑐 \mapsto 𝑐

and similarly

˜ 𝑓 : 𝑥 \otimes 𝑡 𝑛 \mapsto 𝑓 (𝑥) \otimes 𝑡 𝑛 𝑐 \mapsto 𝑐 𝑑 \mapsto 𝑑

We also have a natural inclusion $𝜄 : 1 \to ˆ \cdot : 𝖫 𝗂 𝖾 𝕂 \to 𝖫 𝗂 𝖾 𝕂$ .

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

This definition, following FLM, is much more general than the traditional one, which restricts $𝔤$ to be a semisimple Lie algebra. ↩
1988. Vertex operator algebras and the Monster, §1.6, p. 17ff. ↩
We identify $𝔤$ with $𝔤 \otimes 𝑡 0$ . ↩

Affine Lie algebra

Construction

Properties

Functoriality

Particular affine Lie algebras and related constructions

Footnotes