Affine Kac-Moody algebras as central extensions of loop algebras
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
\[ (e_{\theta},f_{\theta})=1 \]
- Affine Kac-Moody algebra
- Central extension of groups and Lie algebras
- Heisenberg group and Heisenberg algebra