"BGG resolution"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
* BGG resolution http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf | * BGG resolution http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf | ||
| + | ;thm (Bernstein-Gelfand-Gelfand Resolution). | ||
| + | There is an exact sequence of Verma modules | ||
| + | $$ | ||
| + | 0 \to V(w_0\cdot \lambda)\to \cdots \oplus_{w\in W, \ell(w)=k}V(w\cdot \lambda)\to \cdots V(\lambda)\to W_{\lambda}\to 0 | ||
| + | $$ | ||
| + | where $\ell(w)$ is the length of the Weyl group element $w$, $w_0$ is the Weyl group element | ||
| + | of maximal length. Here $\rho$ is half the sum of the positive roots. | ||
2015년 3월 9일 (월) 01:21 판
introduction
- thm (Bernstein-Gelfand-Gelfand Resolution).
There is an exact sequence of Verma modules $$ 0 \to V(w_0\cdot \lambda)\to \cdots \oplus_{w\in W, \ell(w)=k}V(w\cdot \lambda)\to \cdots V(\lambda)\to W_{\lambda}\to 0 $$ where $\ell(w)$ is the length of the Weyl group element $w$, $w_0$ is the Weyl group element of maximal length. Here $\rho$ is half the sum of the positive roots.
example of BGG resolution : $\mathfrak{sl}_2$
- \(W_{\lambda}\) : irreducible highest weight module
- \(V_{\lambda}\) : Verma modules
- note that the Verma modules are free modules of rank 1 over \(\mathbb{C}[F]\) where $F$ is the annihilation operator of $\mathfrak{sl}_2$
- \(\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots\)
- \(W_{\lambda}=V_{\lambda}/V_{-\lambda-2}\)
- BGG resolution
\[0\to V_{-\lambda-2}\to V_{\lambda}\to W_{\lambda}\to 0\]
- 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.
expositions
- Wang, Jing Ping. “Representations of sl(2,C) in the BGG Category O and Master Symmetries.” arXiv:1408.3437 [nlin], August 14, 2014. http://arxiv.org/abs/1408.3437.
- http://stanford.edu/~khare/EoM-BGG-O.pdf
articles
- Khare, Apoorva. ‘Axiomatic Framework for the BGG Category O’. arXiv:1502.06706 [math], 24 February 2015. http://arxiv.org/abs/1502.06706.
- J. Bernstein, I. Gel'fand, and S. Gel'fand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87-92
- Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Structure of Representations That Are Generated by Vectors of Highest Weight’. Akademija Nauk SSSR. Funkcional\cprime Nyi Analiz I Ego Priloženija 5, no. 1 (1971): 1–9.