"BGG resolution"의 두 판 사이의 차이
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imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 |
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* <math>W_{\lambda}</math> : irreducible highest weight module | * <math>W_{\lambda}</math> : irreducible highest weight module | ||
| − | * <math>V_{\lambda}</math> : Verma modules | + | * <math>V_{\lambda}</math> : Verma modules |
** note that the Verma modules are free modules of rank 1 over <math>\mathbb{C}[F]</math> | ** note that the Verma modules are free modules of rank 1 over <math>\mathbb{C}[F]</math> | ||
* <math>\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots</math> | * <math>\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots</math> | ||
* <math>W_{\lambda}=V_{\lambda}/V_{-\lambda-2}</math> | * <math>W_{\lambda}=V_{\lambda}/V_{-\lambda-2}</math> | ||
| − | * BGG resolution | + | * BGG resolution<math>0\to V_{-\lambda-2}\to V_{\lambda}\to W\to 0</math> |
| − | * number of | + | * number of modules = 2 (=order of Weyl group in general) |
| − | * character of W = alternating sum of characters of Verma modules | + | * character of W = alternating sum of characters of Verma modules<math>\chi_{W_{\lambda}}=\chi_{V_{\lambda}}-\chi_{V_{-\lambda-2}}=\frac{q^{\lambda}}{1-q^{-2}}-\frac{q^{-\lambda-2}}{1-q^{-2}}</math> |
| − | * comparison with [[Weyl-Kac character formula]] | + | * comparison with [[Weyl-Kac character formula]] |
| + | :<math>ch(W_{\lambda})=\frac{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho})}{e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}=\frac{q^{\lambda+1}-q^{-\lambda-1}}{q^{1}(1-q^{-2})}</math> where I used <math>\rho=1,\alpha=2</math> and <math>w(\lambda+\rho)=-\lambda-\rho</math> | ||
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==maps between Verma modules== | ==maps between Verma modules== | ||
| − | * 2 conditions to have non-zero homomorphisms <math>V_{\lambda}\to V_{\mu}</math> between two Verma modules | + | * 2 conditions to have non-zero homomorphisms <math>V_{\lambda}\to V_{\mu}</math> between two Verma modules |
** <math>\lambda+\rho, \mu+\rho</math> are in the same orbit of Weyl group | ** <math>\lambda+\rho, \mu+\rho</math> are in the same orbit of Weyl group | ||
** <math>V_{\lambda}\leq V_{\mu}</math>, i.e. <math>\lambda = \mu -\sum \alpha</math>, where the sum is over some positive roots. | ** <math>V_{\lambda}\leq V_{\mu}</math>, i.e. <math>\lambda = \mu -\sum \alpha</math>, where the sum is over some positive roots. | ||
| − | * example in SL2 | + | * example in SL2 |
** <math>\lambda = \mu -2n</math>, <math>n=0,1,2,\cdots</math> | ** <math>\lambda = \mu -2n</math>, <math>n=0,1,2,\cdots</math> | ||
** <math>(\lambda+1)^2 = (\mu+1)^2</math> | ** <math>(\lambda+1)^2 = (\mu+1)^2</math> | ||
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==books== | ==books== | ||
* James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008. | * James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008. | ||
2013년 8월 20일 (화) 03:22 판
example of BGG resolution : sl_2
- \(W_{\lambda}\) : irreducible highest weight module
- \(V_{\lambda}\) : Verma modules
- note that the Verma modules are free modules of rank 1 over \(\mathbb{C}[F]\)
- \(\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots\)
- \(W_{\lambda}=V_{\lambda}/V_{-\lambda-2}\)
- BGG resolution\(0\to V_{-\lambda-2}\to V_{\lambda}\to W\to 0\)
- number of modules = 2 (=order of Weyl group in general)
- character of W = alternating sum of characters of Verma modules\(\chi_{W_{\lambda}}=\chi_{V_{\lambda}}-\chi_{V_{-\lambda-2}}=\frac{q^{\lambda}}{1-q^{-2}}-\frac{q^{-\lambda-2}}{1-q^{-2}}\)
- comparison with Weyl-Kac character formula
\[ch(W_{\lambda})=\frac{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho})}{e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}=\frac{q^{\lambda+1}-q^{-\lambda-1}}{q^{1}(1-q^{-2})}\] where I used \(\rho=1,\alpha=2\) and \(w(\lambda+\rho)=-\lambda-\rho\)
maps between Verma modules
- 2 conditions to have non-zero homomorphisms \(V_{\lambda}\to V_{\mu}\) between two Verma modules
- \(\lambda+\rho, \mu+\rho\) are in the same orbit of Weyl group
- \(V_{\lambda}\leq V_{\mu}\), i.e. \(\lambda = \mu -\sum \alpha\), where the sum is over some positive roots.
- example in SL2
- \(\lambda = \mu -2n\), \(n=0,1,2,\cdots\)
- \((\lambda+1)^2 = (\mu+1)^2\)
books
- James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008.