Affine Kac-Moody algebras as central extensions of loop algebras
imported>Pythagoras0님의 2015년 3월 19일 (목) 05:32 판
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,
$$ c(\gamma_1,\gamma_2) = \int \langle \gamma_1(\theta), \gamma'_2(\theta)\rangle d\theta $$
derivarion
- adding $d$ gives $\hat{\mathfrak{g}}$ a $\mathbb{Z}$-grading
- it makes the each root space finite-dimensional
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 $d(\alpha(n))=n\alpha(n), d(c)=0$