Lie algebras MOC

Twisted affine Lie algebra

Let 𝔤 be a quadratic Lie algebra over 𝕂 with a symmetric 𝔤-invariant bilinear form ⟨ ⋅, ⋅⟩, and 𝜗 ∈Aut⁡𝔤 be an involutive isometry of ⟨ ⋅, ⋅⟩. The corresponding twisted affine Lie algebra ˆ𝔤[𝜗] and extended twisted affine Lie algebra ˜𝔤[𝜗] are generalizations of the corresponding untwisted counterparts. motivate

Construction

Let 𝔤 be a Lie algebra over 𝕂 with an involution 𝜗 ∈Aut⁡𝔤, and let ⟨ ⋅, ⋅⟩ be a 𝔤-invariant bilinear form which is also invariant under 𝜗 in the sense that

⟨𝜗𝑥,𝜗𝑦⟩=⟨𝑥,𝑦⟩

for all 𝑥,𝑦 ∈𝔤.¹ Then 𝔤 =𝔤(0) ⊕𝔤(1) is ℤ2-graded into orthogonal² even and odd subspaces

𝔤(𝑖)={𝑥∈𝔤:𝜗𝑥=(−1)𝑖𝑥}

Let 𝕂[𝑡1/2,𝑡−1/2] be the 12ℤ-graded algebra of Laurent polynomials in indeterminate 𝑡1/2 and 𝑑 be its degree derivation. Constructing

𝔩=𝔤⊗𝕂𝕂[𝑡1/2,𝑡−1/2]⊕𝕂𝑐

with the same bilinear product defined for the (untwisted) affine Lie algebra gives a Lie algebra. Defining the involution 𝑣 :𝑡1/2 ↦ −𝑡1/2 on 𝕂[𝑡1/2, −𝑡1/2] we extend 𝜗 to the following involution on 𝔩

𝜗:𝑐↦𝑐𝜗:𝑥⊗𝑓↦𝜗𝑥⊗𝑣𝑓

The twisted affine Lie algebra ˆ𝔤[𝜗] associated with 𝔤, ⟨ ⋅, ⋅⟩, and 𝜗 is the even subalgebra of 𝔩 under 𝜗 lie

ˆ𝔤[𝜗]={𝑥∈𝔩:𝜗𝑥=𝑥}=𝔤(0)⊗𝕂[𝑡,𝑡−1]⊕𝔤(1)⊗𝑡1/2𝕂[𝑡,𝑡−1]⊕𝕂𝑐

As in the untwisted case, 𝑑 extends to a derivation of ˆ𝔤[𝜗]

𝑑(𝑐)=0𝑑(𝑥⊗𝑓)=𝑥⊗𝑑𝑓

so that homogenous subspaces are the eigenspaces of 𝑑. One obtains the extended twisted affine Lie algebra associated with 𝔤, ⟨ ⋅, ⋅⟩, and 𝜗 by adjoining the derivation 𝑑 lie

˜𝔤[𝜗]=ˆ𝔤[𝜗]⋊𝕂𝑑

Further generalizations

This may be generalized to automorphisms of any finite order.

Properties

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

Functoriality

Let 𝖨𝗇𝗏𝖰𝖫𝗂𝖾𝕂 denote the category where an object is a Quadratic Lie algebra with an involutive isometric, and a morphism 𝑓 :(𝔤,𝜗) →(𝔤,𝜑) is an isometric homomorphism of Lie algebras such that 𝑓𝜗 =𝜑𝑓. Then this constructions forms a functor 𝖨𝗇𝗏𝖰𝖫𝗂𝖾𝕂 →𝖦𝗋12ℤ𝖫𝗂𝖾𝕂.

Particular examples

• Affine Lie algebras of sl_2

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §1.6, p. 19–20 ↩

2. In the sense ⟨𝔤(0),𝔤(1)⟩ =0. ↩