Twisted affine Lie algebra

Let $𝔤$ be a quadratic Lie algebra over $𝕂$ with a symmetric $𝔤$ -invariant bilinear form $⟨ \cdot, \cdot ⟩$ , and $𝜗 \in A u t 𝔤$ be an involutive isometry of $⟨ \cdot, \cdot ⟩$ . The corresponding twisted affine Lie algebra $ˆ 𝔤 [𝜗]$ and extended twisted affine Lie algebra $˜ 𝔤 [𝜗]$ are generalizations of the corresponding untwisted counterparts. #to/motivate

Construction

Let $𝔤$ be a Lie algebra over $𝕂$ with an involution $𝜗 \in A u t 𝔤$ , and let $⟨ \cdot, \cdot ⟩$ be a $𝔤$ -invariant bilinear form which is also invariant under $𝜗$ in the sense that

⟨ 𝜗 𝑥, 𝜗 𝑦 ⟩ = ⟨ 𝑥, 𝑦 ⟩

for all $𝑥, 𝑦 \in 𝔤$ .¹ Then $𝔤 = 𝔤 (0) \oplus 𝔤 (1)$ is $ℤ 2$ -graded into orthogonal² even and odd subspaces

𝔤 (𝑖) = {𝑥 \in 𝔤 : 𝜗 𝑥 = (- 1) 𝑖 𝑥}

Let $𝕂 [𝑡 1 / 2, 𝑡 - 1 / 2]$ be the $12 ℤ$ -graded algebra of Laurent polynomials in indeterminate $𝑡 1 / 2$ and $𝑑$ be its degree derivation. Constructing

𝔩 = 𝔤 \otimes 𝕂 𝕂 [𝑡 1 / 2, 𝑡 - 1 / 2] \oplus 𝕂 𝑐

with the same bilinear product defined for the (untwisted) affine Lie algebra gives a Lie algebra. Defining the involution $𝑣 : 𝑡 1 / 2 \mapsto - 𝑡 1 / 2$ on $𝕂 [𝑡 1 / 2, - 𝑡 1 / 2]$ we extend $𝜗$ to the following involution on $𝔩$

𝜗 : 𝑐 \mapsto 𝑐 𝜗 : 𝑥 \otimes 𝑓 \mapsto 𝜗 𝑥 \otimes 𝑣 𝑓

The twisted affine Lie algebra $ˆ 𝔤 [𝜗]$ associated with $𝔤$ , $⟨ \cdot, \cdot ⟩$ , and $𝜗$ is the even subalgebra of $𝔩$ under $𝜗$ #m/def/lie

ˆ 𝔤 [𝜗] = {𝑥 \in 𝔩 : 𝜗 𝑥 = 𝑥} = 𝔤 (0) \otimes 𝕂 [𝑡, 𝑡 - 1] \oplus 𝔤 (1) \otimes 𝑡 1 / 2 𝕂 [𝑡, 𝑡 - 1] \oplus 𝕂 𝑐

As in the untwisted case, $𝑑$ extends to a derivation of $ˆ 𝔤 [𝜗]$

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

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

˜ 𝔤 [𝜗] = ˆ 𝔤 [𝜗] ⋊ 𝕂 𝑑

Further generalizations

This may be generalized to automorphisms of any finite order.

Properties

In case $𝜗 = 1$ , these constructions yield their untwisted counterparts.

Functoriality

Let $𝖨 𝗇 𝗏 𝖰 𝖫 𝗂 𝖾 𝕂$ denote the category where an object is a Quadratic Lie algebra with an involutive isometric, and a morphism $𝑓 : (𝔤, 𝜗) \to (𝔤, 𝜑)$ is an isometric homomorphism of Lie algebras such that $𝑓 𝜗 = 𝜑 𝑓$ . Then this constructions forms a functor $𝖨 𝗇 𝗏 𝖰 𝖫 𝗂 𝖾 𝕂 \to 𝖦 𝗋 12 ℤ 𝖫 𝗂 𝖾 𝕂$ .

Particular examples

Affine Lie algebras of $𝔰 𝔩 2 𝕂$

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

1988. Vertex operator algebras and the Monster, §1.6, p. 19–20 ↩
In the sense $⟨ 𝔤 (0), 𝔤 (1) ⟩ = 0$ . ↩