Affine Kac-Moody algebras as central extensions of loop algebras
imported>Pythagoras0님의 2015년 3월 18일 (수) 21:22 판 (새 문서: ==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 matr...)
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
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 $d(\alpha(n))=n\alpha(n), d(c)=0$