Affine Kac-Moody algebras as central extensions of loop algebras

수학노트
Pythagoras0 (토론 | 기여)님의 2020년 11월 16일 (월) 10:47 판
둘러보기로 가기 검색하러 가기

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

$$ (e_{\theta},f_{\theta})=1 $$

related items

expositions