Formal sums over a Lie algebra

Formal series over an (un)twisted affine Lie algebra

Let 𝔤 be quadratic Lie algebra, 𝜗 ∈Aut⁡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 ∈𝔤, we associate the following formal series in ˜𝔤[[𝑧,𝑧−1]] and ˜𝔤[𝜗][[𝑧1/2,𝑧−1/2]] respectively: lie

𝑥ℤ(𝑧)=∑𝑛∈ℤ(𝑥⊗𝑡𝑛)𝑧−𝑛∈˜𝔤[[𝑧,𝑧−1]]𝑥ℤ+1/2(𝑧)=∑𝑛∈ℤ(𝑥(𝑛)⊗𝑡𝑛/2)𝑧−𝑛/2∈˜𝔤[𝜗][[𝑧1/2,𝑧−1/2]]

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

𝑥↦𝑥(𝑖)=12(𝑥+(−1)𝑖𝜗𝑥)

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

Commutation relations

For 𝑥,𝑦 ∈𝔤, we may recast the commutation relations of ˜𝔤 using the Formal delta and [[Degree operator on formal sums over a vector space|𝐷 operator]] as

[𝑥(𝑧1),𝑥(𝑧2)]=[𝑥,𝑦](𝑧2)𝛿(𝑧1/𝑧2)−⟨𝑥,𝑦⟩(𝐷𝛿)(𝑧1/𝑧2)𝑐[𝑐,𝑥(𝑧)]=0[𝑑,𝑥(𝑧)]=−𝐷𝑥(𝑧)[𝑐,𝑑]=0

and in ˜𝔤[𝜗] as

[𝑥(𝑧1),𝑥(𝑧2)]=12∑𝑖∈ℤ2[𝜗𝑖𝑥,𝑦](𝑧2)𝛿((−1)𝑖𝑧1/21/𝑧1/22)−12∑𝑖∈ℤ2⟨𝜗𝑖𝑥,𝑦⟩𝐷1𝛿((−1)𝑖)𝑧1/21/𝑧1/22)𝑐[𝑐,𝑥(𝑧)]=0[𝑑,𝑥(𝑧)]=−𝐷𝑥(𝑧)[𝑐,𝑑]=0

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §2.3, pp. 58–60 ↩