Affine Lie algebras of $𝔰 𝔩 2 𝕂$

Let $𝔞 =$ $𝔰 𝔩 2 𝕂$ with its Chevalley basis and let $𝜎 𝑖$ be the suggestively named¹ involutive isometries of $𝔞$ defined by #m/def/lie

𝜎 1 : 𝛼 1 \mapsto - 𝛼 1 𝜎 2 : 𝛼 1 \mapsto - 𝛼 1 𝜎 3 : 𝛼 1 \mapsto 𝛼 1 𝑥 \pm 𝛼 1 \mapsto 𝑥 \mp 𝛼 1 𝑥 \pm 𝛼 1 \mapsto - 𝑥 \mp 𝛼 1 𝑥 \pm 𝛼 1 \mapsto - 𝑥 \pm 𝛼

and let $𝜎 0 = 1$ . Furthermore we let

𝑥 + 𝛼 1 = 𝑥 𝛼 1 + 𝑥 - 𝛼 1 = 𝑥 𝛼 1 + 𝑥 𝜎 1 𝛼 1 = 𝑥 𝛼 1 - 𝑥 𝜎 2 𝛼 1 𝑥 - 𝛼 1 = 𝑥 𝛼 1 - 𝑥 - 𝛼 1 = 𝑥 𝛼 1 - 𝑥 𝜎 1 𝛼 1 = 𝑥 𝛼 1 + 𝑥 𝜎 2 𝛼 1

We consider the untwisted or twisted affine Lie algebra $ˆ 𝔞 𝑖 = ˆ 𝔞 [𝜎 𝑖]$ ² which have bases

ˆ 𝔞 0 = ⟨ 𝑐, 𝛼 1 \otimes 𝑡 𝑚, 𝑥 \pm 𝛼 1 \otimes 𝑡 𝑚 : 𝑚 \in ℤ ⟩ ˆ 𝔞 1 = ⟨ 𝑐, 𝛼 1 \otimes 𝑡 𝑚 + 1 / 2, 𝑥 + 𝛼 1 \otimes 𝑡 𝑚, 𝑥 - 𝛼 1 \otimes 𝑡 𝑚 + 1 / 2 : 𝑚 \in ℤ ⟩ ˆ 𝔞 2 = ⟨ 𝑐, 𝛼 1 \otimes 𝑡 𝑚 + 1 / 2, 𝑥 + 𝛼 1 \otimes 𝑡 𝑚 + 1 / 2, 𝑥 - 𝛼 1 \otimes 𝑡 𝑚 : 𝑚 \in ℤ ⟩ ˆ 𝔞 3 = ⟨ 𝑐, 𝛼 1 \otimes 𝑡 𝑚, 𝑥 \pm 𝛼 1 \otimes 𝑡 𝑚 + 1 / 2 : 𝑚 \in ℤ ⟩

The 1-dimensional subalgebra $𝔥 = 𝕂 𝛼 \leq 𝔞$ generates the natural Heisenberg algebras

ˆ 𝔥 ℤ = 𝕂 𝑐 \oplus ⨁ 𝑛 \in ℤ ∖ {0} 𝛼 1 \otimes 𝑡 𝑛 \leq ˜ 𝔥 \leq ˆ 𝔞 0, ˆ 𝔞 3 ˆ 𝔥 ℤ + 12 = 𝕂 𝑐 \oplus ⨁ 𝑛 \in ℤ + 12 𝛼 1 \otimes 𝑡 𝑛 \leq ˜ 𝔥 [- 1] \leq ˆ 𝔞 1, ˆ 𝔞 2

[𝑐, ˆ 𝔞 𝑖] = 0 [𝛼 1 \otimes 𝑡 𝑚, 𝛼 1 \otimes 𝑡 𝑛] = 2 𝑚 𝛿 𝑚 + 𝑛 𝑐

Isomorphism of extended Lie algebras

$˜ 𝔞$ and $˜ 𝔞 [𝜎 3]$ are isomorphic (but not as graded Lie algebras) under

𝑐 \mapsto 𝑐 𝑑 \mapsto 𝑑 - 14 𝛼 1 𝛼 1 \otimes 𝑡 𝑛 \mapsto 𝛼 1 \otimes 𝑡 𝑛 + 12 𝛿 𝑛 𝑐 𝑥 \pm 𝛼 1 \mapsto 𝑥 \pm 𝛼 1 \otimes 𝑡 𝑛 \pm 1 / 2

There also exist grade-preserving isomorphisms between $˜ 𝔞 [𝜎 𝑖]$ for $𝑖 = 1, 2, 3$ .

Via formal series

Taking a formal series approach on $ˆ 𝔞 𝑖$ , the exact characterization varies with $𝑖$ .

For

ˆ 𝔞 0

In $ˆ 𝔞 0 [[𝑧, 𝑧 - 1]]$ we define the formal sums

𝑥 \pm 𝛼 1 (𝑧) = \sum 𝑛 \in ℤ (𝑥 \pm 𝛼 1 \otimes 𝑡 𝑛) 𝑧 - 𝑛 𝛼 1 (𝑧) = \sum 𝑛 \in ℤ (𝛼 1 \otimes 𝑡 𝑛) 𝑧 - 𝑛

the commutation relations are more conveniently expressed as

[𝛼 1 \otimes 𝑡 𝑚, 𝑥 \pm 𝛼 1 (𝑧)] = \pm 2 𝑧 𝑚 𝑥 \pm 𝛼 1 (𝑧) = ⟨ 𝛼 1, \pm 𝛼 1 ⟩ 𝑧 𝑚 𝑥 \pm 𝛼 1 (𝑧) [𝑥 \pm 𝛼 1 (𝑧 1), 𝑥 \pm 𝛼 (𝑧 2)] = 0 [𝑥 𝛼 1 (𝑧 1), 𝑥 - 𝛼 1 (𝑧 2)] = (𝛼 1 (𝑧 2) - 𝑐 𝐷 1) 𝛿 (𝑧 1 / 𝑧 2) [𝑑, 𝑥 \pm 𝛼 1 (𝑧)] = - 𝐷 𝑥 \pm 𝛼 1 (𝑧) [𝑑, 𝛼 1 (𝑧)] = - 𝐷 𝛼 1 (𝑧)

where $𝑚 \in ℤ$ .

For

ˆ 𝔞 3

In $ˆ 𝔞 3 [[𝑧 1 / 2, 𝑧 - 1 / 2]]$ we define the formal sums

𝑥 \pm 𝛼 1 (𝑧) = \sum 𝑛 \in ℤ + 12 (𝑥 \pm 𝛼 1 \otimes 𝑡 𝑛) 𝑧 - 𝑛 𝛼 1 (𝑧) = \sum 𝑛 \in ℤ (𝛼 1 \otimes 𝑡 𝑛) 𝑧 - 𝑛

the commutation relations are more conveniently expressed as

[𝛼 1 \otimes 𝑡 𝑚, 𝑥 \pm 𝛼 1 (𝑧)] = \pm 2 𝑧 𝑚 𝑥 \pm 𝛼 1 (𝑧) = ⟨ 𝛼 1, \pm 𝛼 1 ⟩ 𝑧 𝑚 𝑥 \pm 𝛼 1 (𝑧) [𝑥 \pm 𝛼 1 (𝑧 1), 𝑥 \pm 𝛼 (𝑧 2)] = 0 [𝑥 𝛼 1 (𝑧 1), 𝑥 - 𝛼 1 (𝑧 2)] = (𝛼 1 (𝑧 2) - 𝑐 𝐷 1) [(𝑧 1 / 𝑧 2) 1 / 2 𝛿 (𝑧 1 / 𝑧 2)] [𝑑, 𝑥 \pm 𝛼 1 (𝑧)] = - 𝐷 𝑥 \pm 𝛼 1 (𝑧) [𝑑, 𝛼 1 (𝑧)] = - 𝐷 𝛼 1 (𝑧)

For

ˆ 𝔞 1

In $ˆ 𝔞 1 [[𝑧 1 / 2, 𝑧 - 1 / 2]]$ we define the formal sums

𝑥 \pm 𝛼 1 (𝑧) = 12 \sum 𝑛 \in ℤ (𝑥 + 𝛼 1 \otimes 𝑡 𝑛) 𝑧 - 𝑛 \pm 12 \sum 𝑛 \in ℤ + 12 (𝑥 - 𝛼 1 \otimes 𝑡 𝑛) 𝑧 - 𝑛 𝛼 1 (𝑧) = \sum 𝑛 \in ℤ + 12 (𝛼 1 \otimes 𝑡 𝑛) 𝑧 - 𝑛

the commutation relations are more conveniently expressed as

[𝛼 1 \otimes 𝑡 𝑚, 𝑥 \pm 𝛼 1 (𝑧)] = \pm 2 𝑧 𝑚 𝑥 \pm 𝛼 1 (𝑧) = ⟨ 𝛼 1, \pm 𝛼 1 ⟩ 𝑧 𝑚 𝑥 \pm 𝛼 1 (𝑧) [𝑥 𝛼 1 (𝑧), 𝑥 - 𝛼 1 (𝑧 2)] = 12 (𝛼 1 (𝑧 2) - 𝑐 𝐷 1) 𝛿 (𝑧 1 / 2 1 / 𝑧 1 / 2 2)

and we also have

𝑥 - 𝛼 1 (𝑧) = l i m 𝑧 1 / 2 \to - 𝑧 1 / 2 𝑥 𝛼 1 (𝑧)

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