"Affine Kac-Moody algebras as central extensions of loop algebras"의 두 판 사이의 차이

2020년 11월 16일 (월) 11:01 기준 최신판

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

http://mathoverflow.net/questions/24845/explicit-cocycle-for-the-central-extension-of-the-algebraic-loop-group-gct

@@ 6번째 줄: / 6번째 줄: @@
 ==2-cocycle of loop algebra==
-* $L\mathfrak{g}$ : loop algebra
+* <math>L\mathfrak{g}</math> : 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 $dg$
+* <math>c(f,g) = \operatorname{Res}_0 \langle f dg \rangle</math> Here, <math>\langle \cdot \rangle : \mathfrak{g}\otimes \mathfrak{g}\to \mathbb{C}</math> denotes some invariant bilinear form on <math>\mathfrak{g}</math>, and <math>f dg</math> is the <math>\mathfrak{g}\otimes \mathfrak{g}</math>-valued differential given by multiplying <math>f</math> and <math>dg</math>
 * in other words,
-$$
+:<math>
 c(\gamma_1,\gamma_2) = \int \langle \gamma_1(\theta), \gamma'_2(\theta)\rangle d\theta
-$$
+</math>
 ==derivarion==
-* adding $d$ gives $\hat{\mathfrak{g}}$ a $\mathbb{Z}$-grading
+* adding <math>d</math> gives <math>\hat{\mathfrak{g}}</math> a <math>\mathbb{Z}</math>-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 $
+* start with a semisimple Lie algebra <math>\mathfrak{g}</math> with invariant form <math>\langle \cdot,\cdot\rangle </math>
 * make a vector space from it
 * construct the loop algbera
-$$\hat{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$$
+:<math>\hat{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]</math>
-$$\alpha(m)=\alpha\otimes t^m$$
+:<math>\alpha(m)=\alpha\otimes t^m</math>
 * Add a central element to get a central extension and give a bracket
-$$\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c$$
+:<math>\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c</math>
-$$[\alpha(m),\beta(n)]=[\alpha,\beta]\otimes t^{m+n}+m\delta_{m,-n}\langle \alpha,\beta\rangle c$$
+:<math>[\alpha(m),\beta(n)]=[\alpha,\beta]\otimes t^{m+n}+m\delta_{m,-n}\langle \alpha,\beta\rangle c</math>
-$$[c,x] =0, x\in \hat{\mathfrak{g}}$$
+:<math>[c,x] =0, x\in \hat{\mathfrak{g}}</math>
-* add a derivation $d=t\frac{d}{dt}$ to get
+* add a derivation <math>d=t\frac{d}{dt}</math> to get
-$$\tilde{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d$$
+:<math>\tilde{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d</math>
 * define a Lie bracket
-$$[d,x]:=d(x)$$
+:<math>[d,x]:=d(x)</math>
-where $d(\alpha(n))=n\alpha(n), d(c)=0$
+where <math>d(\alpha(n))=n\alpha(n), d(c)=0</math>
@@ 40번째 줄: / 40번째 줄: @@
 * l : rank of <math>\mathfrak{g}</math>
 * <math>(a_{ij})</math> : extended Cartan matrix
-* $\theta$ : highest root
+* <math>\theta</math> : highest root
 * generators <math>e_i,h_i,f_i , (i=0,1,2,\cdots, l)</math>
 * Serre relations
@@ 50번째 줄: / 50번째 줄: @@
 ** <math>\left(\text{ad} f_i\right){}^{1-a_{i,j}}\left(f_j\right)=0</math> (<math>i\neq j</math>)
 ===isomorphism===
-* $e_0=f_{\theta}\otimes x, f_0=e_{\theta}\otimes x^{-1}, h_0=-h_{\theta}\otimes 1+c$
+* <math>e_0=f_{\theta}\otimes x, f_0=e_{\theta}\otimes x^{-1}, h_0=-h_{\theta}\otimes 1+c</math>
+* we choose <math>e_{\theta}</math> and <math>f_{\theta}</math> so that
+:<math>
+(e_{\theta},f_{\theta})=1
+</math>
 ==related items==
+* [[Affine Kac-Moody algebra]]
 * [[Central extension of groups and Lie algebras]]
 * [[Heisenberg group and Heisenberg algebra]]
 ==expositions==