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imported>Pythagoras0 잔글 (Pythagoras0 사용자가 Affine sl(2) $A^{(1)} 1$ 문서를 Affine sl(2) 문서로 옮겼습니다.) |
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2012년 9월 20일 (목) 08:22 판
introduction
- affine sl(2) \($A^{(1)}_1$\)
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
\(<h_i,h_j>=A_{ij}\)
\(<h_0,d>=1\)
\(<h_1,d>=0\)
\(<d,d>=0\) - with centers (note that C=h_0+h_1)
\(<C,h_0>=0\)
\(<C,h_1>=0\)
\(<C,d>=1\)
explicit construction
- start with a semisimple Lie algebra \(\mathfrak{g}\) with invariant form \(<\cdot,\cdot>\)
- make a vector space from it
- Construct a Loop algbera
\(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\)
\(\alpha(m)=\alpha\otimes t^m\) -
Add a central element to get a central extension and give a bracket
\(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\)
\([E(m),F(n)]=H\otimes t^{m+n}+m\delta_{m,-n}c\)
\([H(m),E(n)]=2E\otimes t^{m+n}\)
\([H(m),F(n)]=-2F\otimes t^{m+n}\)
\([E(m),E(n)]=[F(m),F(n)]=0\)
\(<c,\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c>=0\) - Add a derivation \(d\)
\(d=t\frac{d}{dt}\)
\(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d\)
\(d(\alpha(n))=n\alpha(n)\)
\(d(c)=0\)
\(<c,d>=0\) - 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
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- Gannon 190p, 193p, 196p,371p
- 2010년 books and articles
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
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.
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage
experts on the field