## The affine Lie algebra as an extended loop Lie algebra

Last update: 27 July 2013

## The affine Lie algebra as an extended loop Lie algebra

A finite dimensional complex simple Lie algebra $\stackrel{\circ }{𝔤}$ is a Kac-Moody Lie algebra with Chevalley generators ${e}_{1},\dots ,{e}_{\ell },$ ${f}_{1},\dots ,{f}_{\ell },$ ${h}_{1},\dots ,{h}_{\ell },$ and Cartan matrix

$Å= (⟨αi∨,αj⟩) 1≤i,j≤ℓ ,with ⟨αi∨,αj⟩ =αj(hi).$

This means that $\stackrel{\circ }{𝔤}$ is presented by generators ${e}_{1},\dots ,{e}_{\ell },$ ${f}_{1},\dots ,{f}_{\ell },$ ${h}_{1},\dots ,{h}_{\ell },$ and “Serre relations”, where the relations are determined completely by the information in the Cartan matrix. Let

$𝔥∘ℂ=ℂ-span {h1,…,hℓ}.$

The restriction of the nondegenerate form $⟨,⟩:\stackrel{\circ }{𝔤}\otimes \stackrel{\circ }{𝔤}\to ℂ$ provides a nondegenerate form

$⟨,⟩:𝔥∘ℂ ⊗𝔥∘ℂ⟶ℂand ν: 𝔥∘ℂ ⟶∼ 𝔥∘ℂ* h ⟼ ⟨h,·⟩$

which provides a nondegenerate form $⟨,⟩:{\stackrel{\circ }{𝔥}}_{ℂ}^{*}\otimes {\stackrel{\circ }{𝔥}}_{ℂ}^{*}\to ℂ$ so that $⟨\nu \left({h}_{1}\right),\nu \left({h}_{2}\right)⟩=⟨{h}_{1},{h}_{2}⟩\text{.}$ for ${h}_{1},{h}_{2}\in {\stackrel{\circ }{𝔥}}_{ℂ}\text{.}$

The affine Lie algebra is

$𝔤=Lˆ(𝔤∘)= 𝔤∘[t,t-1] ⊕ℂK⊕ℂd,$

so that $𝔤$ is the $ℂ\text{-span}$ of $\left\{x{t}^{m},K,d | x\in \stackrel{\circ }{𝔤},m\in ℤ\right\}$ with bracket given by

$[xtm,ytn]= [x,y]tm+n+ δm,-n⟨x,y⟩ K,K∈Z(𝔤), [d,xtm]=mx tm,$

for $x,y\in \stackrel{\circ }{𝔤}$ and $m,n\in ℤ\text{.}$ The Cartan subalgebra of $𝔤$ and its dual are

$𝔥ℂ=ℂK⊕ 𝔥∘ℂ⊕ ℂdand write 𝔥ℂ*=ℂδ ⊕𝔥∘ℂ* ⊕ℂΛ0 (1.8)$

with

$δ(K)=0, δ(h)=0, δ(d)=1, λ‾(K)=0, λ‾(d)=0, Λ0(K)=1, Λ0(h)=0, Λ0(d)=0, (1.9)$

for $\stackrel{‾}{\lambda }\in {\stackrel{\circ }{𝔥}}_{ℂ}^{*}$ and $h\in {\stackrel{\circ }{𝔥}}_{ℂ}$ (see [Kac1104219, §6.2]).

If the root decomposition of $\stackrel{\circ }{𝔤}$ is

$𝔤∘=𝔥∘ℂ⊕ (⨁α∈R∘𝔤α)= (⨁α∈R∘-𝔤α) ⊕𝔥∘ℂ⊕ (⨁α∈R∘+𝔤α), (1.10)$

then the root decomposition of $𝔤$ is given by

$𝔤=𝔥ℂ⊕ (⨁k∈ℤ𝔤kδ)⊕ ( ⨁α∈R∘k∈ℤ 𝔤α+kδ ) ,where𝔤α+kδ =𝔤αtkand 𝔤kδ=𝔥∘ℂ tk,$

for $\alpha \in \stackrel{\circ }{R}$ and $k\in ℤ\text{.}$ The set of positive roots of $𝔤$ is

$R+= { kδ,α+kδ,-α+ (k-1)δ | α∈R∘+,k∈ ℤ>0 } . (1.11)$

Let $\theta$ be the highest root (highest weight of the adjoint representation) for $\stackrel{\circ }{𝔤}$ and let

$eθ∈𝔤∘θ, gθ∈𝔤∘-θ, hθ∨=[eθ,fθ],$

Then, as explained in [Kac1104219, §7.4], the Lie algebra $𝔤$ is a Kac-Moody Lie algebra with Chevalley generators

$e0=-fθt, f0=eθt-1, h0=-hθ∨+ 2(θ|θ)Km (1.12)$

and ${e}_{1},\dots ,{e}_{\ell },$ ${f}_{1},\dots ,{f}_{\ell },$ ${h}_{1},\dots ,{h}_{\ell },$ and Cartan matrix

$A= (⟨αi∨,αj⟩) 0≤i,j≤ℓ ,whereα0∨= h0,α0=-θ+δ. (1.13)$

This means that $𝔤$ is presented by generators ${e}_{0},{e}_{1},\dots ,{e}_{\ell },$ ${f}_{0},{f}_{1},\dots ,{f}_{\ell },$ ${h}_{0},{h}_{1},\dots ,{h}_{\ell },d$ and "Serre relations", where the relations are determined completely by the information in the Cartan matrix.

The affine Cartan matrix $A$ has rank $\ell \text{.}$ Define ${a}_{0},\dots ,{a}_{\ell }$ and ${a}_{0}^{\vee },\dots ,{a}_{\ell }^{\vee }$ by

$δ=a0α0+a1 α1+…+aℓ αℓandK= a0∨h0+ a1∨h1+…+ aℓ∨hℓ (1.14)$

so that [Kac1104219, Theorem 4.8(c) and §6.1]

$Aδ=0,KA=0, and the a0,…,aℓ are positive relatively prime integers.$

It is useful to note that

$a0∨=1anda0= { 2, if the Cartan matrix is type A2ℓ(2), 2, otherwise. (1.15)$

If ${\omega }_{1},\dots ,{\omega }_{\ell }$ are the fundamental weights of finite dimensional Lie algebra $\stackrel{\circ }{𝔤}$ then the fundamental weights of the affine Lie algebra $𝔤$ are

$Λ0,Λ1 ,…,Λℓ, withΛj= ωj+aj∨Λ0 ,for j=1,2,…,ℓ, (1.16)$

(see [Kac1104219, (12.4.3)]) so that

$Λi(hj)= δij,for i,j∈{0,1,2,…,ℓ}.$

Since $\left\{{h}_{0},{h}_{1},\dots ,{h}_{\ell },d\right\}$ is a basis of ${𝔥}_{ℂ}$ and ${\Lambda }_{i}\left(d\right)=0$ for $i=0,1,\dots ,\ell ,$ it follows that ${𝔥}_{ℂ}^{*}$ has basis $\left\{{\Lambda }_{0},{\Lambda }_{1},\dots ,{\Lambda }_{\ell },\delta \right\}\text{.}$

Define

$P = { λ∈𝔥ℂ* | ⟨λ,αi∨⟩ ∈ℤ for i=0,1,…,ℓ } P+ = { λ∈𝔥ℂ* | ⟨λ,αi∨⟩ ∈{ℤ}_{\ge 0} for i=0,1,…,ℓ } P++ = { λ∈𝔥ℂ* | ⟨λ,αi∨⟩ ∈ℤ>0 for i=0,1,…,ℓ }$

so that

$P=ℂδ+∑i=0ℓ ℤΛi,P+ =ℂδ+∑i=0ℓ ℤ≥0Λi, P++=ℂδ+ ∑i=0ℓℤ>0 Λi.$

Recall, from (1.3), that $\rho \in {𝔥}^{*}$ should be chosen such that $⟨\rho ,{h}_{i}⟩=1,$ for $i=0,1,\dots ,n\text{.}$ If $\stackrel{‾}{\rho }$ is the element $\rho$ for the finite dimensional Lie algebra $\stackrel{\circ }{𝔤},$ then a good choice of $\rho$ for $𝔤$ is

$ρ=ρ‾+h∨ Λ0,where h∨=ρ(K)= a0∨+a1∨+…+ aℓ∨, (1.17)$

is the dual Coxeter number (see [Kac1104219, (6.2.8) or (12.4.2)]). This $\rho$ is characterized by $\rho \left(d\right)=0$ and $\rho \left({h}_{i}\right)=1,$ for $i=0,1,\dots ,\ell \text{.}$ Note that

$ρ=Λ0+Λ1 +…+Λℓ.$

There is a nondegenerate symmetric bilinear form $\left(|\right):{𝔥}_{ℂ}×{𝔥}_{ℂ}\to ℂ$ given by

$(K|K)=0, (α0∨|K)=0, (αi∨|K)=0, for i∈{1,…,ℓ}, (d|K)=a0, (αi∨|αj∨)= ajaj∨αj (αi∨),for i,j∈{0,1,…,ℓ}, (K|d)=0, (α0∨|d)=a0, (αi∨|d)=0, for i∈{1,…,ℓ}, (d|d)=0, (1.18)$

where the $\left(i,j\right)\text{-entry}$ of the Cartan matrix is ${\alpha }_{j}\left({\alpha }_{i}^{\vee }\right)$ (see [Kac1104219, §6.2.1] and [Kac1104219, (1.1.2)]). The form $\left(|\right)$ provides an isomorphism

$ν: 𝔥 ⟶ 𝔥* h ⟼ (h|·) withν(αi∨) =aiai∨αi, ν(K)=δ,ν (d)=a0Λ0, (1.19)$

see [Kac1104219, §6.2.3]. The resulting nondegenerate symmetric bilinear form on ${𝔥}_{ℂ}^{*}$ is

$( | ):𝔥ℂ* ×𝔥ℂ*⟶ℂgiven by (λ|μ)= (ν-1(λ)|ν-1(μ)),$

so that

$(Λ0|Λ0)=0, (α0|Λ0)=a0-1, (Λ0|αi)=0, for i∈{1,…,ℓ}, (Λ0|δ)=1, (αi|αj)= ai∨aiαj(αi∨), for i,j∈{0,1,…,ℓ}, (δ|Λ0)=1, (δ|αi)=0, for i∈{0,1,…,ℓ}, (δ|δ)=0, (1.20)$

see [Kac1104219, §6.2.2 and §6.2.4].