"BGG resolution"의 두 판 사이의 차이

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==introduction==
 
==introduction==
* Weyl character formula. For $\lambda\in \Lambda^{+}$,
+
* <math>L(\cdot)</math> : simple module, <math>V(\cdot)</math> : Verma module
$$
+
* Weyl character formula. For <math>\lambda\in \Lambda^{+}</math>,
 +
:<math>
 
\operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda)
 
\operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda)
$$
+
</math>
 
* goal : realize this formula as an Euler characteristic
 
* 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 twisted Weyl orbit $W\cdot \lambda$
+
* 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>
 
;thm (Bernstein-Gelfand-Gelfand Resolution).
 
;thm (Bernstein-Gelfand-Gelfand Resolution).
Fix $\lambda\in \Lambda^{+}$. There is an exact sequence of Verma modules
+
Let <math>\lambda\in \Lambda^{+}</math>. There is an exact sequence of Verma modules
$$
+
:<math>
0 \to M({w_0\cdot \lambda})\to \cdots \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots M({\lambda})\to L({\lambda})\to 0
+
0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0
$$
+
</math>
where $\ell(w)$ is the length of the Weyl group element $w$, $w_0$ is the Weyl group element
+
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 $\rho$ is half the sum of the positive roots.
+
of maximal length. Here <math>\rho</math> is half the sum of the positive roots.
 
 
==overview==
 
* Weyl character formula and characters of Verma modules
 
* property of character map on short exact sequences
 
* Euler-Poincare mapping
 
* principal block : filtering through central characters
 
** is a block a $U(\mathfrak{g})$-submodule? yes
 
** how to check that it preserves the exactness : any homomorphism between modules belonging to different blocks will be zero
 
* combinatorial results
 
** consider the set of sum of k distinct roots. Which elements are linked to $0$?
 
** Bruhat ordering
 
* Bruhat ordering and strong linkage relation
 
** let $\lambda \in \Lambda^+$ (which is regular for the dot-action of $W$)
 
**  $w'\cdot \lambda< w \cdot \lambda $ translates into $w < w'$ in the Bruhat ordering
 
* strong linkage relation and extension of Verma modules
 
 
 
==example of BGG resolution==
 
===$\mathfrak{sl}_2$===
 
 
 
* <math>L({\lambda})</math> : irreducible highest weight module
 
* <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 $\mathfrak{sl}_2$
 
* <math>\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots</math>
 
* <math>L({\lambda})=M({\lambda})/M({-\lambda-2})</math>
 
*  BGG resolution
 
:<math>0\to M({-\lambda-2})\to M({\lambda})\to L({\lambda})\to 0</math>
 
* number of modules = 2 (=order of Weyl group in general)
 
*  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>\chi(L({\lambda}))=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\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>
 
 
 
===$\mathfrak{sl}_3$===
 
* http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf
 
 
 
==Euler-Poincare characteristic==
 
* $A$ : abelian category
 
* Euler-Poincare map $\varphi$ on an object in $A$
 
* $\varphi$ turns a short exact sequence into an alternating sum
 
* we can define Euler-Poincare characteristic $\chi_{\varphi}$ of a complex as the alternating sum of Euler-Poincare map on the homology
 
* the main result is that the Euler-Poincre characteristic can be computed from a different resolution and it is independent of the choice of it
 
 
 
==Verma modules==
 
===maps between 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>V_{\lambda}\leq V_{\mu}</math>, i.e. <math>\lambda = \mu -\sum \alpha</math>, where the sum is over some positive roots.
 
====example====
 
* SL2
 
** <math>\lambda = \mu -2n</math>, <math>n=0,1,2,\cdots</math>
 
** <math>(\lambda+1)^2 = (\mu+1)^2</math>
 
 
 
 
===composition series of Verma modules===
 
;thm
 
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 $w\in W$.
 
 
 
===action of center on Verma modules===
 
* check
 
 
 
 
 
===maximal submodule of Verma modules===
 
* Maximal Submodule of $M(\lambda), \lambda \in \Lambda+$ (see 2.6)
 
 
 
==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
 
$$0 = M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_n = M$$
 
for which each $M^i := M_i/M_{i−1}\, (1 \le i \le n)$ is isomorphic to a Verma module.
 
;thm (Weak BGG resolution)
 
There is an exact sequence
 
$$0 = D_m^{\lambda} \to \subset D_{m-1}^{\lambda} \to \cdots \to D_2^{\lambda} \to D_1^{\lambda} \to L(0) \to 0$$
 
where $D_{k}^{\lambda}$ has a standard filtration involving exactly once each of the Verma modules $M(w\cdot \lambda)$ with $\ell(w)=k$
 
 
 
* we prove this for $\lambda=0$ and apply the translation function to extend it
 
 
 
===strategy of the proof===
 
* The sequence of modules $D_k:=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\Lambda^{k}(\mathfrak{g}/\mathfrak{b})$ is a relative version of the standard resolution of the trivial module in [[Lie algebra cohomology]]
 
 
 
===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 $U(\mathfrak{n}^{−})$
 
===weights of exterior powers===
 
* what are the weights of $\wedge^k (\mathfrak{g}/\mathfrak{b})$ as $\mathfrak{b}$-module?
 
* let $\beta$ be a sum of $k$-distinct negative root. Can we find $w\in W$ such that $w\cdot 0=\beta$?
 
* note that $|W\cdot 0|=|W|$
 
* yes, if and only if $\beta$ is a sum of elements in $w \Phi^+ \cap \Phi^-$
 
 
 
===extensions of Verma modules===
 
* $\mu, \lambda\in \mathfrak{h}^{*}$
 
* $\mu \uparrow \lambda$ if $\mu = \lambda$ or there is a root $\alpha$ such that $\mu=s_{\alpha}\cdot \lambda < \lambda $
 
* we say $\mu$ is strongly linked to $\lambda$ if $\mu = \lambda$ or there exist root $\alpha_1,\cdots, \alpha_r\in \Phi^+$ such that $\mu=(s_{\alpha_1}\cdots s_{\alpha_r})\cdot \lambda \uparrow (s_{\alpha_2}\cdots s_{\alpha_r})\cdot \lambda \uparrow ( s_{\alpha_r})\cdot \lambda \uparrow  \cdots  \uparrow \lambda $
 
;thm
 
Let $\lambda\in \mathfrak{h}^{*}$.
 
 
 
(a) If $\operatorname{Ext}_{\mathcal{O}}(M(\mu),M(\lambda))\neq 0$ for $\mu\in \mathfrak{h}^{*}$, then $\mu \uparrow \lambda$ but $\mu \neq \lambda$
 
 
 
(b) Let $\lambda\in \Lambda^{+}$ and $w,w'\in W$. If $\operatorname{Ext}_{\mathcal{O}}(M(w'\cdot\lambda),M(w\cdot\lambda))\neq 0$, then $w<w'$ in the [[Bruhat ordering]]. In particular, $\ell(w)<\ell(w')$.
 
  
 
==applications==
 
==applications==
* This is used to compute the cohomologies of $\mathfrak{n}^+$.  
+
* This is used to compute the cohomologies of <math>\mathfrak{n}^+</math>.  
 
* see [[Kostant theorem on Lie algebra cohomology of nilpotent subalgebra]]
 
* see [[Kostant theorem on Lie algebra cohomology of nilpotent subalgebra]]
  
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==related items==
 
==related items==
 +
* [[Talk on BGG resolution]]
 
* [[Verma modules]]
 
* [[Verma modules]]
 
* [[BGG reciprocity]]
 
* [[BGG reciprocity]]
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* [[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==
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==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.
 
* Zelevinski, Resolvents, dual pairs, and character formulas http://www.ms.unimelb.edu.au/~ram/Resources/ResolventsDualPairsAndCharacterFormulas.html
 
* 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
+
* [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]]
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[[분류:migrate]]

2020년 11월 16일 (월) 05:25 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.