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 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 ^degreeDerivation 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 $⟨ \cdot, \cdot ⟩$ , lie iff $𝔤$ is a Lie algebra and $⟨ \cdot, \cdot ⟩$ is a symmetric $𝔤$ -invariant bilinear form, and we have the ^central

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 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 $𝑑$ 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 Category of quadratic Lie algebras to [[Category of graded Lie algebras| $𝖦 𝗋 ℤ 𝖫 𝗂 𝖾 𝕂$ ]], 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 $𝜄 : 1 \to ˆ \cdot : 𝖫 𝗂 𝖾 𝕂 \to 𝖫 𝗂 𝖾 𝕂$ .

tidy | 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 $𝔤 \otimes 𝑡 0$ . ↩

Table of Contents

Affine Lie algebra

Construction

Properties

Functoriality

Graph View

Backlinks

Table of Contents

Affine Lie algebra

Construction

Properties

Functoriality

Particular affine Lie algebras and related constructions

Footnotes

Graph View

Backlinks