The affine Lie algebra as an extended loop Lie algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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

$$\AA ={\left(⟨{\alpha }_i^{\vee },{\alpha }_j⟩\right)}_{1\le i,j\le \ell },\text{with}\hspace{0.17em}⟨{\alpha }_i^{\vee },{\alpha }_j⟩={\alpha }_j\left(h_i\right)\text{.}$$

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

$${\stackrel{\circ }{𝔥}}_{ℂ}=ℂ\text{-span}\{h_1,\dots ,h_{\ell }\}\text{.}$$

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

$$⟨,⟩:{\stackrel{\circ }{𝔥}}_{ℂ}\otimes {\stackrel{\circ }{𝔥}}_{ℂ}⟶ℂ\text{and}\begin{array}{cccc}\nu :& {\stackrel{\circ }{𝔥}}_{ℂ}& \stackrel{\sim }{⟶}& {\stackrel{\circ }{𝔥}}_{ℂ}^{*}\\ & h& ⟼& ⟨h,·⟩\end{array}$$

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

$$𝔤=\hat{L}\left(\stackrel{\circ }{𝔤}\right)=\stackrel{\circ }{𝔤}[t,t^{-1}]\oplus ℂK\oplus ℂd,$$

so that $𝔤$ is the $ℂ\text{-span}$ of $\{x{t}^m,K,d\hspace{0.17em}|\hspace{0.17em}x\in \stackrel{\circ }{𝔤},m\in \mathbb{Z}\}$ with bracket given by

$$[x{t}^m,y{t}^n]=[x,y]t^{m+n}+{\delta }_{m,-n}⟨x,y⟩K,K\in Z\left(𝔤\right),[d,x{t}^m]=mx{t}^m,$$

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

$$\begin{array}{cc}{𝔥}_{ℂ}=ℂK\oplus {\stackrel{\circ }{𝔥}}_{ℂ}\oplus ℂd\text{and write}{𝔥}_{ℂ}^{*}=ℂ\delta \oplus {\stackrel{\circ }{𝔥}}_{ℂ}^{*}\oplus ℂ{\Lambda }_0& \text{(1.8)}\end{array}$$

with

$$\begin{array}{cc}\begin{array}{ccc}\delta \left(K\right)=0,& \delta \left(h\right)=0,& \delta \left(d\right)=1,\\ \bar{\lambda }\left(K\right)=0,& & \bar{\lambda }\left(d\right)=0,\\ {\Lambda }_0\left(K\right)=1,& {\Lambda }_0\left(h\right)=0,& {\Lambda }_0\left(d\right)=0,\end{array}& \text{(1.9)}\end{array}$$

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

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

$$\begin{array}{cc}\stackrel{\circ }{𝔤}={\stackrel{\circ }{𝔥}}_{ℂ}\oplus \left(\underset{\alpha \in \stackrel{\circ }{R}}{⨁}{𝔤}_{\alpha }\right)=\left(\underset{\alpha \in {\stackrel{\circ }{R}}^{-}}{⨁}{𝔤}_{\alpha }\right)\oplus {\stackrel{\circ }{𝔥}}_{ℂ}\oplus \left(\underset{\alpha \in {\stackrel{\circ }{R}}^{+}}{⨁}{𝔤}_{\alpha }\right),& \text{(1.10)}\end{array}$$

then the root decomposition of $𝔤$ is given by

$$𝔤={𝔥}_{ℂ}\oplus \left(\underset{k\in \mathbb{Z}}{⨁}{𝔤}_{k\delta }\right)\oplus \left(\underset{\underset{k\in \mathbb{Z}}{\alpha \in \stackrel{\circ }{R}}}{⨁}{𝔤}_{\alpha +k\delta }\right),\text{where}{𝔤}_{\alpha +k\delta }={𝔤}_{\alpha }{t}^k\text{and}{𝔤}_{k\delta }={\stackrel{\circ }{𝔥}}_{ℂ}{t}^k,$$

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

$$\begin{array}{cc}{R}^{+}=\{k\delta ,\alpha +k\delta ,-\alpha +(k-1)\delta \hspace{0.17em}|\hspace{0.17em}\alpha \in {\stackrel{\circ }{R}}^{+},k\in {\mathbb{Z}}_{>0}\}\text{.}& \text{(1.11)}\end{array}$$

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

$$e_{\theta }\in {\stackrel{\circ }{𝔤}}_{\theta },g_{\theta }\in {\stackrel{\circ }{𝔤}}_{-\theta },h_{{\theta }^{\vee }}=[e_{\theta },f_{\theta }],$$

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

$$\begin{array}{cc}{e}_0=-f_{\theta }t,f_0=e_{\theta }{t}^{-1},h_0=-h_{{\theta }^{\vee }}+\frac{2}{\left(\theta \right|\theta )}Km& \text{(1.12)}\end{array}$$

and $e_1,\dots ,e_{\ell },$ $f_1,\dots ,f_{\ell },$ $h_1,\dots ,h_{\ell },$ and Cartan matrix

$$\begin{array}{cc}A={\left(⟨{\alpha }_i^{\vee },{\alpha }_j⟩\right)}_{0\le i,j\le \ell },\text{where}{\alpha }_0^{\vee }=h_0,{\alpha }_0=-\theta +\delta \text{.}& \text{(1.13)}\end{array}$$

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

$$\begin{array}{cc}\delta =a_0{\alpha }_0+a_1{\alpha }_1+\dots +a_{\ell }{\alpha }_{\ell }\text{and}K=a_0^{\vee }{h}_0+a_1^{\vee }{h}_1+\dots +a_{\ell }^{\vee }{h}_{\ell }& \text{(1.14)}\end{array}$$

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

$$A\delta =0,KA=0,\text{and the}\hspace{0.17em}{a}_0,\dots ,a_{\ell }\hspace{0.17em}\text{are positive relatively prime integers.}$$

It is useful to note that

$$\begin{array}{cc}{a}_0^{\vee }=1\text{and}{a}_0=\{\begin{array}{cc}2,& \text{if the Cartan matrix is type}\hspace{0.17em}{A}_{2\ell }^{\left(2\right)},\\ 2,& \text{otherwise.}\end{array}& \text{(1.15)}\end{array}$$

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

$$\begin{array}{cc}{\Lambda }_0,{\Lambda }_1,\dots ,{\Lambda }_{\ell },\text{with}{\Lambda }_j={\omega }_j+a_j^{\vee }{\Lambda }_0,\text{for}\hspace{0.17em}j=1,2,\dots ,\ell ,& \text{(1.16)}\end{array}$$

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

$${\Lambda }_i\left(h_j\right)={\delta }_{ij},\text{for}\hspace{0.17em}i,j\in \{0,1,2,\dots ,\ell \}\text{.}$$

Since $\{h_0,h_1,\dots ,h_{\ell },d\}$ 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 $\{{\Lambda }_0,{\Lambda }_1,\dots ,{\Lambda }_{\ell },\delta \}\text{.}$

Define

$$\begin{array}{ccc}P& =& \{\lambda \in {𝔥}_{ℂ}^{*}\hspace{0.17em}|\hspace{0.17em}⟨\lambda ,{\alpha }_i^{\vee }⟩\in \mathbb{Z}\hspace{0.17em}\text{for}\hspace{0.17em}i=0,1,\dots ,\ell \}\\ P^{+}& =& \{\lambda \in {𝔥}_{ℂ}^{*}\hspace{0.17em}|\hspace{0.17em}⟨\lambda ,{\alpha }_i^{\vee }⟩\in $ {\mathbb{Z}}_{\ge 0}$\hspace{0.17em}\text{for}\hspace{0.17em}i=0,1,\dots ,\ell \}\\ P^{++}& =& \{\lambda \in {𝔥}_{ℂ}^{*}\hspace{0.17em}|\hspace{0.17em}⟨\lambda ,{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{>0}\hspace{0.17em}\text{for}\hspace{0.17em}i=0,1,\dots ,\ell \}\end{array}$$

so that

$$P=ℂ\delta +\sum _{i=0}^{\ell }\mathbb{Z}{\Lambda }_i,P^{+}=ℂ\delta +\sum _{i=0}^{\ell }{\mathbb{Z}}_{\ge 0}{\Lambda }_i,P^{++}=ℂ\delta +\sum _{i=0}^{\ell }{\mathbb{Z}}_{>0}{\Lambda }_i\text{.}$$

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 $\bar{\rho }$ is the element $\rho$ for the finite dimensional Lie algebra $\stackrel{\circ }{𝔤},$ then a good choice of $\rho$ for $𝔤$ is

$$\begin{array}{cc}\rho =\bar{\rho }+h^{\vee }{\Lambda }_0,\text{where}{h}^{\vee }=\rho \left(K\right)=a_0^{\vee }+a_1^{\vee }+\dots +a_{\ell }^{\vee },& \text{(1.17)}\end{array}$$

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

$$\rho ={\Lambda }_0+{\Lambda }_1+\dots +{\Lambda }_{\ell }\text{.}$$

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

$$\begin{array}{cc}\begin{array}{c}\left(K\right|K)=0,\left({\alpha }_0^{\vee }\right|K)=0,\left({\alpha }_i^{\vee }\right|K)=0,\hspace{0.17em}\text{for}\hspace{0.17em}i\in \{1,\dots ,\ell \},\left(d\right|K)=a_0,\\ \left({\alpha }_i^{\vee }\right|{\alpha }_j^{\vee })=\frac{a_j}{a_j^{\vee }}{\alpha }_j\left({\alpha }_i^{\vee }\right),\text{for}\hspace{0.17em}i,j\in \{0,1,\dots ,\ell \},\\ \left(K\right|d)=0,\left({\alpha }_0^{\vee }\right|d)=a_0,\left({\alpha }_i^{\vee }\right|d)=0,\hspace{0.17em}\text{for}\hspace{0.17em}i\in \{1,\dots ,\ell \},\left(d\right|d)=0,\end{array}& \text{(1.18)}\end{array}$$

where the $(i,j)\text{-entry}$ 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

$$\begin{array}{cc}\begin{array}{cccc}\nu :& 𝔥& ⟶& {𝔥}^{*}\\ & h& ⟼& \left(h\right|·)\end{array}\text{with}\nu \left({\alpha }_i^{\vee }\right)=\frac{a_i}{a_i^{\vee }}{\alpha }_i,\nu \left(K\right)=\delta ,\nu \left(d\right)=a_0{\Lambda }_0,& \text{(1.19)}\end{array}$$

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

$$\left(\hspace{0.17em}\right|\hspace{0.17em}):{𝔥}_{ℂ}^{*}\times {𝔥}_{ℂ}^{*}⟶ℂ\text{given by}\left(\lambda \right|\mu )=\left({\nu }^{-1}\left(\lambda \right)\right|{\nu }^{-1}\left(\mu \right)),$$

so that

$$\begin{array}{cc}\begin{array}{c}\left({\Lambda }_0\right|{\Lambda }_0)=0,\left({\alpha }_0\right|{\Lambda }_0)=a_0^{-1},\left({\Lambda }_0|{\alpha }_i\right)=0,\hspace{0.17em}\text{for}\hspace{0.17em}i\in \{1,\dots ,\ell \},\left({\Lambda }_0\right|\delta )=1,\\ \left({\alpha }_i\right|{\alpha }_j)=\frac{a_i^{\vee }}{a_i}{\alpha }_j\left({\alpha }_i^{\vee }\right),\text{for}\hspace{0.17em}i,j\in \{0,1,\dots ,\ell \},\\ \left(\delta \right|{\Lambda }_0)=1,\left(\delta \right|{\alpha }_i)=0,\hspace{0.17em}\text{for}\hspace{0.17em}i\in \{0,1,\dots ,\ell \},\left(\delta \right|\delta )=0,\end{array}& \text{(1.20)}\end{array}$$

see [Kac1104219, §6.2.2 and §6.2.4].

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