Formal series over an (un)twisted affine Lie algebra

Let $𝔤$ be quadratic Lie algebra, $𝜗 \in A u t 𝔤$ be an involutive isometry, and $˜ 𝔤$ and $˜ 𝔤 [𝜗]$ be the associated extended untwisted affine Lie algebra and extended twisted affine Lie algebra respectively. Then to each $𝑥 \in 𝔤$ , we associate the following formal series in $˜ 𝔤 [[𝑧, 𝑧 - 1]]$ and $˜ 𝔤 [𝜗] [[𝑧 1 / 2, 𝑧 - 1 / 2]]$ respectively: #m/def/lie

𝑥 ℤ (𝑧) = \sum 𝑛 \in ℤ (𝑥 \otimes 𝑡 𝑛) 𝑧 - 𝑛 \in ˜ 𝔤 [[𝑧, 𝑧 - 1]] 𝑥 ℤ + 1 / 2 (𝑧) = \sum 𝑛 \in ℤ (𝑥 (𝑛) \otimes 𝑡 𝑛 / 2) 𝑧 - 𝑛 / 2 \in ˜ 𝔤 [𝜗] [[𝑧 1 / 2, 𝑧 - 1 / 2]]

and we will drop the subscripts when it is clear. In the latter

𝑥 \mapsto 𝑥 (𝑖) = 12 (𝑥 + (- 1) 𝑖 𝜗 𝑥)

denotes the appropriate projection into the $𝜗$ -eigenspace decomposition $𝔤 = 𝔤 (0) \oplus 𝔤 (1)$ .¹

Commutation relations

For $𝑥, 𝑦 \in 𝔤$ , we may recast the commutation relations of $˜ 𝔤$ using the Formal delta and $𝐷$ operator as

[𝑥 (𝑧 1), 𝑥 (𝑧 2)] = [𝑥, 𝑦] (𝑧 2) 𝛿 (𝑧 1 / 𝑧 2) - ⟨ 𝑥, 𝑦 ⟩ (𝐷 𝛿) (𝑧 1 / 𝑧 2) 𝑐 [𝑐, 𝑥 (𝑧)] = 0 [𝑑, 𝑥 (𝑧)] = - 𝐷 𝑥 (𝑧) [𝑐, 𝑑] = 0

and in $˜ 𝔤 [𝜗]$ as

[𝑥 (𝑧 1), 𝑥 (𝑧 2)] = 12 \sum 𝑖 \in ℤ 2 [𝜗 𝑖 𝑥, 𝑦] (𝑧 2) 𝛿 ((- 1) 𝑖 𝑧 1 / 2 1 / 𝑧 1 / 2 2) - 12 \sum 𝑖 \in ℤ 2 ⟨ 𝜗 𝑖 𝑥, 𝑦 ⟩ 𝐷 1 𝛿 ((- 1) 𝑖) 𝑧 1 / 2 1 / 𝑧 1 / 2 2) 𝑐 [𝑐, 𝑥 (𝑧)] = 0 [𝑑, 𝑥 (𝑧)] = - 𝐷 𝑥 (𝑧) [𝑐, 𝑑] = 0

#state/develop | #lang/en | #SemBr

1988. Vertex operator algebras and the Monster, §2.3, pp. 58–60 ↩

Formal series over an (un)twisted affine Lie algebra

Commutation relations

Footnotes