Affine Kac-Moody algebras as central extensions of loop algebras

introduction

• Construct the loop algebra from a finite dimensional Lie algebra
• Make a central extension
• Add a outer derivation to compensate the degeneracy of the Cartan matrix

2-cocycle of loop algebra

• $L\mathfrak{g}$ : loop algebra
• $c(f,g) = \operatorname{Res}_0 \langle f dg \rangle$ Here, $\langle \cdot \rangle : \mathfrak{g}\otimes \mathfrak{g}\to \mathbb{C}$ denotes some invariant bilinear form on $\mathfrak{g}$, and $f dg$ is the $\mathfrak{g}\otimes \mathfrak{g}$-valued differential given by multiplying $f$ and $dg$
• in other words,

$$ c(\gamma_1,\gamma_2) = \int \langle \gamma_1(\theta), \gamma'_2(\theta)\rangle d\theta $$

explicit construction

• start with a semisimple Lie algebra $\mathfrak{g}$ with invariant form $\langle \cdot,\cdot\rangle $
• make a vector space from it
• construct the loop algbera

$$\hat{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$$ $$\alpha(m)=\alpha\otimes t^m$$

• Add a central element to get a central extension and give a bracket

$$\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c$$ $$[\alpha(m),\beta(n)]=[\alpha,\beta]\otimes t^{m+n}+m\delta_{m,-n}\langle \alpha,\beta\rangle c$$ $$[c,x] =0, x\in \hat{\mathfrak{g}}$$

• add a derivation $d=t\frac{d}{dt}$ to get

$$\tilde{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d$$

• define a Lie bracket

$$[d,x]:=d(x)$$ where $d(\alpha(n))=n\alpha(n), d(c)=0$

related items

• Central extension of groups and Lie algebras
• Heisenberg group and Heisenberg algebra