"BGG resolution"의 두 판 사이의 차이

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==introduction==
 
==introduction==
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* The BGG resolution (cf. [5, 31]) resolves a finite-dimensional (simple) g-module (V(λ)) by direct sums of Verma modules indexed by weights “of the same length”in the twisted Weyl orbit (W•λ).
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* <math>L(\cdot)</math> : simple module, <math>V(\cdot)</math> : Verma module
−
* This is used to compute the cohomologies of n+.
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* Weyl character formula. For <math>\lambda\in \Lambda^{+}</math>,
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;thm (Bernstein-Gelfand-Gelfand Resolution).  
+
:<math>
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There is an exact sequence of Verma modules
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\operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda)
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$$
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</math>
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0 \to M_{w_0\cdot \lambda}\to \cdots \bigoplus_{w\in W, \ell(w)=k}M_{w\cdot \lambda}\to \cdots M_{\lambda}\to L_{\lambda}\to 0
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* goal : realize this formula as an Euler characteristic
−
$$
+
* The BGG resolution resolves a finite-dimensional simple <math>\mathfrak{g}</math>-module <math>L(\lambda)</math> by direct sums of Verma modules indexed by weights "of the same length" in the orbit <math>W\cdot \lambda</math>
−
where $\ell(w)$ is the length of the Weyl group element $w$, $w_0$ is the Weyl group element
+
;thm (Bernstein-Gelfand-Gelfand Resolution).
−
of maximal length. Here $\rho$ is half the sum of the positive roots.
+
Let <math>\lambda\in \Lambda^{+}</math>. There is an exact sequence of Verma modules
 +
:<math>
 +
0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0
 +
</math>
 +
where <math>\ell(w)</math> is the length of the Weyl group element <math>w</math>, <math>w_0</math> is the Weyl group element
 +
of maximal length. Here <math>\rho</math> is half the sum of the positive roots.
  
 +
==applications==
 +
* This is used to compute the cohomologies of <math>\mathfrak{n}^+</math>.
 +
* see [[Kostant theorem on Lie algebra cohomology of nilpotent subalgebra]]
  
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==example of BGG resolution==
 
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===$\mathfrak{sl}_2$===
 
  
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* <math>L_{\lambda}</math> : irreducible highest weight module
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==generalization of BGG resolution==
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* <math>M_{\lambda}</math> : Verma modules
 
−
** note that the Verma modules are free modules of rank 1 over <math>\mathbb{C}[F]</math> where $F$ is the annihilation operator of $\mathfrak{sl}_2$
 
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* <math>\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots</math>
 
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* <math>L_{\lambda}=M_{\lambda}/M_{-\lambda-2}</math>
 
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*  BGG resolution
 
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:<math>0\to M_{-\lambda-2}\to M_{\lambda}\to L_{\lambda}\to 0</math>
 
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* number of modules = 2 (=order of Weyl group in general)
 
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*  character of $L_{\lambda}$ = alternating sum of characters of Verma modules
 
−
:<math>\chi_{L_{\lambda}}=\chi_{M_{\lambda}}-\chi_{M_{-\lambda-2}}=\frac{q^{\lambda}}{1-q^{-2}}-\frac{q^{-\lambda-2}}{1-q^{-2}}</math>
 
−
*  comparison with [[Weyl-Kac character formula]]
 
−
:<math>ch(L_{\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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===$\mathfrak{sl}_3$===
 
−
* http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf
 
−
 
 
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==maps between Verma modules==
 
−
 
 
−
*  2 conditions to have non-zero homomorphisms <math>V_{\lambda}\to V_{\mu}</math> between two Verma modules
 
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** <math>\lambda+\rho, \mu+\rho</math> are in the same orbit of Weyl group
 
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** <math>V_{\lambda}\leq V_{\mu}</math>, i.e. <math>\lambda = \mu -\sum \alpha</math>, where the sum is over some positive roots.
 
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===example===
 
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* SL2
 
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** <math>\lambda = \mu -2n</math>, <math>n=0,1,2,\cdots</math>
 
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** <math>(\lambda+1)^2 = (\mu+1)^2</math>
 
−
 
 
−
 
−
==composition series of Verma modules==
 
−
;thm
 
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The Verma module $M(\lambda)$ has a finite composition series
 
−
$$
 
−
M(\lambda)=N_0\supset N_1\supset N_2\supset \cdots N_{r}=O
 
−
$$
 
−
where each $N_i$ is a submodule of $M(\lambda)$ and $N_{i+1}$ is a maximal submodule of $N_i$. Moreover, $N_i/N_{i+1}$ is isomorphic to $L(w\cdot \lambda)$ for some $w\in W$.
 
−
 
 
−
 
 
−
==weak BGG resolution==
 
−
===standard filtration===
 
−
* We say that $M \in O$ has a standard filtration (also sometimes called a Verma flag) if there is a sequence of submodules
 
−
$$0 = M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_n = M$$
 
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for which each $M^i := M_i/M_{i−1}\, (1 \le i \le n)$ is isomorphic to a Verma module.
 
−
* $D_{k}^{\lambda}$ has a standard filtration involving exactly once each of the Verma modules $M(w\cdot \lambda)$ with $\ell(w)=k$
 
−
 
 
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===standard resolution of trivial module===
 
−
* free $U(\mathfrak{g})$-modules $U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{k}(\mathfrak{g})$
 
−
* standard resolution of trivial module
 
−
$$\cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{p}(\mathfrak{g})\to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{p-1}(\mathfrak{g})\to \cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{0}(\mathfrak{g})\to L(0)$$
 
−
* $D_k$ are free only over $U(\mathfrak{n}^{−})$
 
−
 
 
−
==generalization==
 
 
* There exist generalizations to symmetrizable Kac-Moody algebras, cf. [34].
 
* There exist generalizations to symmetrizable Kac-Moody algebras, cf. [34].
 
* Kempf obtained a resolution of finite-dimensional L(λ) in terms of the Grothendieck-Cousin complex in [26], which is dual to the BGG resolution.
 
* Kempf obtained a resolution of finite-dimensional L(λ) in terms of the Grothendieck-Cousin complex in [26], which is dual to the BGG resolution.
 
* This was extended by Kumar to arbitrary Kac-Moody algebras; he thus obtained the BGG resolution here, and computed the Weyl-Kac character formula and the cohomologies of n+ (cf. [30,§9.3]).
 
* This was extended by Kumar to arbitrary Kac-Moody algebras; he thus obtained the BGG resolution here, and computed the Weyl-Kac character formula and the cohomologies of n+ (cf. [30,§9.3]).
−
 
  
 
==related items==
 
==related items==
 +
* [[Talk on BGG resolution]]
 
* [[Verma modules]]
 
* [[Verma modules]]
 
* [[BGG reciprocity]]
 
* [[BGG reciprocity]]
76번째 줄: 32번째 줄:
 
* [[Kostant theorem on Lie algebra cohomology of nilpotent radical]]
 
* [[Kostant theorem on Lie algebra cohomology of nilpotent radical]]
 
* [[Bott-Borel-Weil Theorem]]
 
* [[Bott-Borel-Weil Theorem]]
 +
* [[Koszul complex]]
  
 
==books==
 
==books==
89번째 줄: 46번째 줄:
  
 
==articles==
 
==articles==
 +
* Pierre Julg, The Bernstein-Gelfand-Gelfand complex for rank one semi simple Lie groups as a Kasparov module, arXiv:1605.07408 [math.OA], May 24 2016, http://arxiv.org/abs/1605.07408
 
* Griffeth, Stephen, and Emily Norton. “Character Formulas and Bernstein-Gelfand-Gelfand Resolutions for Cherednik Algebra Modules.” arXiv:1511.00748 [math], November 2, 2015. http://arxiv.org/abs/1511.00748.
 
* Griffeth, Stephen, and Emily Norton. “Character Formulas and Bernstein-Gelfand-Gelfand Resolutions for Cherednik Algebra Modules.” arXiv:1511.00748 [math], November 2, 2015. http://arxiv.org/abs/1511.00748.
−
* [34] A. Rocha-Caridi, Splitting Criteria for $\mathfrak{g}$-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional, irreducible $\mathfrak{g}$-module, Trans. Amer. Math. Soc.262 (1980), no. 2, 335–366
+
* Zelevinski, Resolvents, dual pairs, and character formulas http://www.ms.unimelb.edu.au/~ram/Resources/ResolventsDualPairsAndCharacterFormulas.html
 +
* [34] A. Rocha-Caridi, Splitting Criteria for <math>\mathfrak{g}</math>-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional, irreducible <math>\mathfrak{g}</math>-module, Trans. Amer. Math. Soc.262 (1980), no. 2, 335–366
 
* [26] G. Kempf, The Grothendieck-Cousin complex of an induced representation , Advances in Mathematics 29 (1978), 310–396
 
* [26] G. Kempf, The Grothendieck-Cousin complex of an induced representation , Advances in Mathematics 29 (1978), 310–396
 
* [31] Lepowsky, J. “A Generalization of the Bernstein-Gelfand-Gelfand Resolution.” Journal of Algebra 49, no. 2 (1977): 496–511.
 
* [31] Lepowsky, J. “A Generalization of the Bernstein-Gelfand-Gelfand Resolution.” Journal of Algebra 49, no. 2 (1977): 496–511.
 
*  J. Bernstein, I. Gel'fand, and S. Gel'fand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87-92
 
*  J. Bernstein, I. Gel'fand, and S. Gel'fand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87-92
−
* [5] Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Differential Operators on the Base Affine Space and a Study of $\mathfrak{g}$-Modules’. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 21–64. Halsted, New York, 1975. http://www.ams.org/mathscinet-getitem?mr=0578996.
+
* [5] Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Differential Operators on the Base Affine Space and a Study of <math>\mathfrak{g}</math>-Modules’. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 21–64. Halsted, New York, 1975. http://www.ams.org/mathscinet-getitem?mr=0578996.
 
* 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.
 
* 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.
  
 
[[분류:Lie theory]]
 
[[분류:Lie theory]]
 +
[[분류:abstract concepts]]
 +
[[분류:migrate]]

2020년 11월 16일 (월) 04:25 기준 최신판

introduction

  • \(L(\cdot)\) : simple module, \(V(\cdot)\) : Verma module
  • Weyl character formula. For \(\lambda\in \Lambda^{+}\),

\[ \operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda) \]

  • goal : realize this formula as an Euler characteristic
  • The BGG resolution resolves a finite-dimensional simple \(\mathfrak{g}\)-module \(L(\lambda)\) by direct sums of Verma modules indexed by weights "of the same length" in the orbit \(W\cdot \lambda\)
thm (Bernstein-Gelfand-Gelfand Resolution).

Let \(\lambda\in \Lambda^{+}\). There is an exact sequence of Verma modules \[ 0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0 \] 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.

applications


generalization of BGG resolution

  • There exist generalizations to symmetrizable Kac-Moody algebras, cf. [34].
  • Kempf obtained a resolution of finite-dimensional L(λ) in terms of the Grothendieck-Cousin complex in [26], which is dual to the BGG resolution.
  • This was extended by Kumar to arbitrary Kac-Moody algebras; he thus obtained the BGG resolution here, and computed the Weyl-Kac character formula and the cohomologies of n+ (cf. [30,§9.3]).

related items

books

  • [30] Shrawan Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Birkhauser, Progress in Math. 204, Boston, 2002
  • James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008.


expositions

articles

  • Pierre Julg, The Bernstein-Gelfand-Gelfand complex for rank one semi simple Lie groups as a Kasparov module, arXiv:1605.07408 [math.OA], May 24 2016, http://arxiv.org/abs/1605.07408
  • Griffeth, Stephen, and Emily Norton. “Character Formulas and Bernstein-Gelfand-Gelfand Resolutions for Cherednik Algebra Modules.” arXiv:1511.00748 [math], November 2, 2015. http://arxiv.org/abs/1511.00748.
  • Zelevinski, Resolvents, dual pairs, and character formulas http://www.ms.unimelb.edu.au/~ram/Resources/ResolventsDualPairsAndCharacterFormulas.html
  • [34] A. Rocha-Caridi, Splitting Criteria for \(\mathfrak{g}\)-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional, irreducible \(\mathfrak{g}\)-module, Trans. Amer. Math. Soc.262 (1980), no. 2, 335–366
  • [26] G. Kempf, The Grothendieck-Cousin complex of an induced representation , Advances in Mathematics 29 (1978), 310–396
  • [31] Lepowsky, J. “A Generalization of the Bernstein-Gelfand-Gelfand Resolution.” Journal of Algebra 49, no. 2 (1977): 496–511.
  • J. Bernstein, I. Gel'fand, and S. Gel'fand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87-92
  • [5] Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Differential Operators on the Base Affine Space and a Study of \(\mathfrak{g}\)-Modules’. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 21–64. Halsted, New York, 1975. http://www.ams.org/mathscinet-getitem?mr=0578996.
  • 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.
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