"BGG resolution"의 두 판 사이의 차이

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

This is used to compute the cohomologies of \(\mathfrak{n}^+\).
see Kostant theorem on Lie algebra cohomology of nilpotent subalgebra

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

http://rvirk.com/notes/student/catObasics.pdf
BGG resolution http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf
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

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.

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* Weyl character formula. For $\lambda\in \Lambda^{+}$,
+* <math>L(\cdot)</math> : simple module, <math>V(\cdot)</math> : Verma module
-$$
+* Weyl character formula. For <math>\lambda\in \Lambda^{+}</math>,
+:<math>
 \operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda)
-$$
+</math>
-* realize this formula as an Euler characteristic
+* goal : realize this formula as an Euler characteristic
-* The BGG resolution (cf. [5, 31]) resolves a finite-dimensional (simple) 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>
-* This is used to compute the cohomologies of $\mathfrak{n}^+$.
 ;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
-$$
+:<math>
-\to M({w_0\cdot \lambda})\to \cdots \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots M({\lambda})\to L({\lambda})\to 0
+\to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0
-$$
+</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.
+==applications==
+* This is used to compute the cohomologies of <math>\mathfrak{n}^+</math>.
+* see [[Kostant theorem on Lie algebra cohomology of nilpotent subalgebra]]
-==overview==
-* property of character map on short exact sequences
-* Euler-Poincare mapping
-* Weyl character formula
-* principal block : filtering through central characters
-** is a block a $U(\mathfrak{g})$-submodule?
-** 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 extension of Verma modules
-==example of BGG resolution==
+==generalization 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>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>
-===$\mathfrak{sl}_3$===
-* 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 $
-;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, $\ell(w)<\ell(w')$.
-==generalization==
 * 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==
+* [[Talk on BGG resolution]]
 * [[Verma modules]]
 * [[BGG reciprocity]]
@@ 131번째 줄: / 32번째 줄: @@
 * [[Kostant theorem on Lie algebra cohomology of nilpotent radical]]
 * [[Bott-Borel-Weil Theorem]]
+* [[Koszul complex]]
 ==books==
@@ 144번째 줄: / 46번째 줄: @@
 ==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
+* [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
 * [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.
+* [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.
 [[분류:Lie theory]]
+[[분류:abstract concepts]]
+[[분류:migrate]]