Affine Kac-Moody algebras as central extensions of loop algebras
Pythagoras0 (토론 | 기여)님의 2020년 11월 16일 (월) 10:47 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 $d(\alpha(n))=n\alpha(n), d(c)=0$
Chevalley generators
- simple Lie algebra \(\mathfrak{g}\)
- l : rank of \(\mathfrak{g}\)
- \((a_{ij})\) : extended Cartan matrix
- $\theta$ : highest root
- generators \(e_i,h_i,f_i , (i=0,1,2,\cdots, l)\)
- Serre relations
- \(\left[h,h'\right]=0\)
- \(\left[e_i,f_j\right]=\delta _{i,j}h_i\)
- \(\left[h,e_j\right]=\alpha_{j}(h)e_j\)
- \(\left[h,f_j\right]=-\alpha_{j}(h)f_j\)
- \(\left(\text{ad} e_i\right){}^{1-a_{i,j}}\left(e_j\right)=0\) (\(i\neq j\))
- \(\left(\text{ad} f_i\right){}^{1-a_{i,j}}\left(f_j\right)=0\) (\(i\neq j\))
isomorphism
- $e_0=f_{\theta}\otimes x, f_0=e_{\theta}\otimes x^{-1}, h_0=-h_{\theta}\otimes 1+c$
- we choose $e_{\theta}$ and $f_{\theta}$ so that
$$ (e_{\theta},f_{\theta})=1 $$
- Affine Kac-Moody algebra
- Central extension of groups and Lie algebras
- Heisenberg group and Heisenberg algebra