"Affine sl(2)"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
imported>Pythagoras0 |
||
| 74번째 줄: | 74번째 줄: | ||
==explicit construction== | ==explicit construction== | ||
| + | * start with a semisimple Lie algebra $\mathfrak{g}$ with invariant form $\langle \cdot,\cdot\rangle $, | ||
| + | * make a vector space from it, | ||
| + | * Construct a Loop algbera $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$ | ||
| + | * Let $\alpha(m)=\alpha\otimes t^m$, | ||
| + | * Add a central element to get a central extension $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c$, and give a bracket $$[E(m),F(n)]=H\otimes t^{m+n}+m\delta_{m,-n}c$$ | ||
| + | $$[H(m),E(n)]=2E\otimes t^{m+n}$$ | ||
| + | $$[H(m),F(n)]=-2F\otimes t^{m+n}$$ | ||
| + | $$[E(m),E(n)]=[F(m),F(n)]=0$$ | ||
| + | $$\langle c,\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\rangle =0$$ | ||
| + | * Add a derivation $d$, $d=t\frac{d}{dt}$ to get $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d$ | ||
| + | $$d(\alpha(n))=n\alpha(n)$$ | ||
| + | $$d(c)=0$$ | ||
| + | $$\langle c,d\rangle =0$, | ||
| + | * Define a Lie bracket,$[d,x]=d(x)$, | ||
| − | + | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
2013년 5월 20일 (월) 13:33 판
introduction
- affine sl(2) \(A^{(1)}_1\)
- 틀:수학노트
construction from semisimple Lie algebra
- this is borrowed from affine Kac-Moody algebra entry
- Let \(\mathfrak{g}\) be a semisimple Lie algebra with root system \(\Phi\) and the invariant form \(<\cdot,\cdot>\)
- say \(\mathfrak{g}=A_1\), \(\Phi=\{\alpha,-\alpha\}\)
- Cartan matrix
\(\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}\) - Find the highest root \(\alpha\)
- Add another simple root \(\alpha_0\) to the root system \(\Phi\) which is \(\alpha_0=-\alpha\), but we regard this as an independent one now.
- Construct a new Cartan matrix
\(A' = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}\) - Note that this matrix has rank 1 since \((1,1)\) belongs to the null space
- construct a Lie algebra from the new Cartan matrix \(A'\)
- Add a outer derivation\(d=-l_0\) to compensate the degeneracy of the Cartan matrix
\(\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0 \end{pmatrix}\)
basic quantities
- $a_i=1$
- $c_i=a_i^{\vee}=1$
- $a_{ij}$
- coxeter number 2
- dual Coxeter number 2
- Weyl vector
root systems
- \(\Phi=\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\cup \{n\delta|n\in\mathbb{Z},n\neq 0\}\)
- real roots
- \(\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\)
- \(\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\)
- imaginary roots
- \(\{n\delta|n\in\mathbb{Z},n\neq 0\}\)
- \(\delta=\alpha_0+\alpha_1\)
- simple roots
- \(\alpha_0,\alpha_1\)
- \(\alpha_0,\alpha_1\)
- positive roots
- \(\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}\)
- \(\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}\)
fixing a Cartan subalgebra and its dual
- H is a 3-dimensional space
- basis of the Cartan subalgebra H (this defines C and l_0 also)
\(h_0=C-h_1\)
\(h_1\)
\(d=-l_0\) - basis of dual Cartan algebra
\(\omega_0,\alpha_0,\alpha_1\) - dual basis for H
\(\omega_0,\omega_1=\omega_0+\frac{1}{2}\alpha_1,\delta=\alpha_0+\alpha_1\) - Weyl vector
\(\rho=\omega_0+\omega_1\)
- pairing
\(\alpha_0(h_0)=2\)
\(\alpha_0(h_1)=-2\)
\(\alpha_0(d)=1\)
\(\alpha_1(h_0)=-2\)
\(\alpha_1(h_1)=2\)
\(\alpha_1(d)=0\)
\(\omega_0(h_0)=1\)
\(\omega_0(h_1)=0\)
\(\omega_0(d)=0\)
killing form
- invariant symmetric non-deg bilinear forms, $\langle h_i,h_j\rangle =A_{ij}$, $\langle h_0,d\rangle =1$, $\langle h_1,d\rangle =0$, $\langle d,d\rangle =0$,
- with centers (note that $C=h_0+h_1$), $\langle C,h_0\rangle =0$, $\langle C,h_1\rangle =0$, $\langle C,d\rangle =1$,
explicit construction
- start with a semisimple Lie algebra $\mathfrak{g}$ with invariant form $\langle \cdot,\cdot\rangle $,
- make a vector space from it,
- Construct a Loop algbera $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$
- Let $\alpha(m)=\alpha\otimes t^m$,
- Add a central element to get a central extension $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c$, and give a bracket $$[E(m),F(n)]=H\otimes t^{m+n}+m\delta_{m,-n}c$$
$$[H(m),E(n)]=2E\otimes t^{m+n}$$ $$[H(m),F(n)]=-2F\otimes t^{m+n}$$ $$[E(m),E(n)]=[F(m),F(n)]=0$$ $$\langle c,\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\rangle =0$$
- Add a derivation $d$, $d=t\frac{d}{dt}$ to get $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d$
$$d(\alpha(n))=n\alpha(n)$$ $$d(c)=0$$ $$\langle c,d\rangle =0$,
- Define a Lie bracket,$[d,x]=d(x)$,
denominator formula
- Weyl-Kac character formula
\({\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}\)
level k highest weight representation
- integrable highest weight
\(\lambda=\lambda_{0}\omega_0+\lambda_{1}\omega_1\), \(\lambda_{i}\in\mathbb{N}\) - level
\(k=\lambda_{0}+\lambda_{1}\in\mathbb{N}\) - therefore \(\lambda_{0}\in\{0,1,\cdots,k\}\)
central charge
- central charge (depends on the level only)
\(c_{\lambda}=\frac{k}{k+h^{\vee}}\text{dim }\mathfrak{\bar{g}}\) - conformal weight
\(h_{\lambda}=\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}\) - definition of conformal anomaly
\(m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}\)
- strange formula
\(\frac{<\rho,\rho>}{2h^{\vee}}=\frac{\operatorname{dim}\mathfrak{g}}{24}\) - very strange formula
- conformal anomaly
\(m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c_{\lambda}}{24}\)
vertex operator construction
computational resource
books
- Gannon 190p, 193p, 196p,371p
articles
- Lepowsky, James, and Robert Lee Wilson. 1978. “Construction of the affine Lie algebraA 1 (1)”. Communications in Mathematical Physics 62 (1): 43-53. doi:10.1007/BF01940329.