en>777sms at 03:29, 14 April 2012

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'''Verma modules''', named after [[Daya-Nand Verma]], are objects in the [[representation theory]] of [[Lie algebra]]s, a branch of [[mathematics]].

Verma modules can be used to prove that an irreducible [[highest weight module]] with [[highest weight]] <math>\lambda</math> is [[dimension (vector space)|finite dimensional]], if and only if the weight <math>\lambda</math> is dominant and integral. Their homomorphisms correspond to [[invariant differential operator]]s over [[flag manifold]]s.

==Definition of Verma modules==
The definition relies on a stack of relatively dense notation. Let <math>F</math> be a field and denote the following:
* <math>\mathfrak{g}</math>, a [[semisimple Lie algebra]] over <math>F</math>, with [[universal enveloping algebra]] <math>\mathcal{U}(\mathfrak{g})</math>.
* <math>\mathfrak{b}</math>, a [[Borel subalgebra]] of <math>\mathfrak{g}</math>, with universal enveloping algebra <math>\mathcal{U}(\mathfrak{b})</math>.
* <math>\mathfrak{h}</math>, a [[Cartan subalgebra]] of <math>\mathfrak{g}</math>. We do '''not''' consider its universal enveloping algebra.
* <math>\lambda \in \mathfrak{h}^*</math>, a fixed [[weight (representation theory)|weight]].
To define the Verma module, we begin by defining some other modules:
* <math>F_\lambda</math>, the one-dimensional <math>F</math>-vector space (i.e. whose underlying set is <math>F</math> itself) together with a <math>\mathfrak{b}</math>-[[module (mathematics)|module]] structure such that <math>\mathfrak{h}</math> acts as multiplication by <math>\lambda</math> and the [[root system|positive root spaces]] act trivially. As <math>F_\lambda</math> is a left <math>\mathfrak{b}</math>-module, it is consequently a left <math>\mathcal{U}(\mathfrak{b})</math>-module.
* Using the [[Poincaré-Birkhoff-Witt theorem]], there is a natural right <math>\mathcal{U}(\mathfrak{b})</math>-module structure on <math>\mathcal{U}(\mathfrak{g})</math> by right multiplication of a subalgebra. <math>\mathcal{U}(\mathfrak{g})</math> is naturally a left <math>\mathfrak{g}</math>-module, and together with this structure, it is a <math>(\mathfrak{g}, \mathcal{U}(\mathfrak{b}))</math>-[[bimodule]].
Now we can define the Verma module (with respect to <math>\lambda</math>) as
: <math>M_\lambda = \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{b})} F_\lambda</math>
which is naturally a left <math>\mathfrak{g}</math>-module (i.e. a [[Lie algebra representation|representation]] of <math>\mathfrak{g}</math>). The Poincaré-Birkhoff-Witt theorem implies that the underlying vector space of <math>M_\lambda</math> is isomorphic to
: <math>\mathcal{U}(\mathfrak{g}_-) \otimes_F F_\lambda</math>
where <math>\mathfrak{g}_-</math> is the Lie subalgebra generated by the negative root spaces of <math>\mathfrak{g}</math>.

==Basic properties==
Verma modules, considered as <math>\mathfrak{g}</math>-[[module (mathematics)|modules]], are [[highest weight module]]s, i.e. they are generated by a [[highest weight vector]]. This highest weight vector is <math>1\otimes 1</math> (the first <math>1</math> is the unit in <math>\mathcal{U}(\mathfrak{g})</math> and the second is
the unit in the field <math>F</math>, considered as the <math>\mathfrak{b}</math>-[[module (mathematics)|module]]
<math>F_\lambda</math>) and it has weight <math>\lambda</math>.

Verma modules are [[weight modules]], i.e. <math>M_\lambda</math> is a [[direct sum of modules|direct sum]] of all its [[weight space]]s. Each weight space in <math>M_\lambda</math> is finite dimensional and the dimension of the <math>\mu</math>-weight space <math>M_\mu</math> is the number of possibilities how to obtain <math>\lambda-\mu</math> as a sum of [[positive root]]s (this is closely related to the so-called [[Kostant partition function]]).

Verma modules have a very important property: If <math>V</math> is any representation generated by a highest weight vector of weight <math>\lambda</math>, there is a [[surjective]] <math>\mathfrak{g}</math>-[[homomorphism]] <math>M_\lambda\to V.</math> That is, all representations with highest weight <math>\lambda</math> that are generated by the highest weight vector (so called [[highest weight module]]s) are [[quotient group|quotients]] of <math>M_\lambda.</math>

<math>M_\lambda</math> contains a unique maximal [[submodule]], and its quotient is the unique (up to [[isomorphism]]) [[irreducible representation]] with highest weight <math>\lambda.</math>

The Verma module <math>M_\lambda</math> itself is irreducible if and only if none of the coordinates of <math>\lambda</math> in the basis of [[fundamental weight]]s is from the set <math>\{0,1,2,\ldots\}</math>.

The Verma module <math>M_\lambda</math> is called ''regular'', if its highest weight λ is on the affine Weyl orbit of a dominant weight <math>\tilde\lambda</math>. In other word, there exist an element w of the Weyl group W such that
:<math>\lambda=w\cdot\tilde\lambda</math>
where <math>\cdot</math> is the [[affine action]] of the Weyl group.

The Verma module <math>M_\lambda</math> is called ''singular'', if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight <math>\tilde\lambda</math> so that <math>\tilde\lambda+\delta</math> is on the wall of the [[fundamental Weyl chamber]] (δ is the sum of all [[fundamental weight]]s).

==Homomorphisms of Verma modules==
For any two weights <math>\lambda, \mu</math> a non-trivial [[homomorphism]]

:<math>M_\mu\rightarrow M_\lambda</math>

may exist only if <math>\mu</math> and <math>\lambda</math> are linked with an [[affine action]] of the [[Weyl group]] <math>W</math> of the Lie algebra <math>\mathfrak{g}</math>. This follows easily from the [[Harish-Chandra theorem]]{{Disambiguation needed|date=March 2012}} on [[infinitesimal central character]]s.

Each homomorphism of Verma modules is injective and the [[dimension]]

:<math>\dim(\operatorname{Hom}(M_\mu, M_\lambda))\leq 1</math>

for any <math>\mu, \lambda</math>. So, there exists a nonzero <math>M_\mu\rightarrow M_\lambda</math> if and only if <math>M_\mu</math> is [[isomorphic]] to a (unique) submodule of <math>M_\lambda</math>.

The full classification of Verma module homomorphisms was done by Bernstein-Gelfand-Gelfand<ref>Bernstein I.N., Gelfand I.M., Gelfand S.I., Structure of Representations that are generated by vectors of highest weight, Functional. Anal. Appl. 5 (1971)</ref> and Verma<ref>Verma N., Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968)</ref> and can be summed up in the following statement:

<blockquote> There exists a nonzero homomorphism <math>M_\mu\rightarrow M_\lambda</math> if and only if there exists
a sequence of weights

::<math>\mu=\nu_0\leq\nu_1\leq\ldots\leq\nu_k=\lambda</math>

such that <math>\nu_{i-1}+\delta=s_{\gamma_i}(\nu_i+\delta)</math> for some positive roots <math>\gamma_i</math> (and <math>s_{\gamma_i}</math> is the corresponding [[root reflection]] and <math>\delta</math> is the sum of all [[fundamental weight]]s) and for each <math>1\leq i\leq k, (\nu_i+\delta)(H_{\gamma_i})</math> is a natural number (<math>H_{\gamma_i}</math> is the [[coroot]] associated to the root <math>\gamma_i</math>).</blockquote>

If the Verma modules <math>M_\mu</math> and <math>M_\lambda</math> are [[Verma module#Basic properties|regular]], then there exists a unique [[dominant weight]] <math>\tilde\lambda</math> and unique elements ''w'', ''w''′ of the [[Weyl group]] ''W'' such that

:P<math>\mu=w'\cdot\tilde\lambda</math>

and

:<math>\lambda=w\cdot\tilde\lambda,</math>

where <math>\cdot</math> is the [[affine action]] of the Weyl group. If the weights are further [[weight (representation theory)#Integral weight|integral]], then there exists a nonzero homomorphism

:<math>M_\mu\to M_\lambda</math>

if and only if

:<math>w \leq w'</math>

in the [[Bruhat ordering]] of the Weyl group.

==Jordan–Hölder series==
Let
:<math>0\subset A\subset B\subset M_\lambda</math>
be a sequence of <math>\mathfrak{g}</math>-modules so that the quotient B/A is irreducible with [[highest weight]] μ. Then there exists a nonzero homomorphism <math>M_\mu\to M_\lambda</math>.

An easy consequence of this is, that for any [[highest weight module]]s <math>V_\mu, V_\lambda</math> such that
:<math>V_\mu\subset V_\lambda</math>
there exists a nonzero homomorphism <math>M_\mu\to M_\lambda</math>.

==Bernstein–Gelfand–Gelfand resolution==
Let <math>V_\lambda</math> be a finite dimensional [[irreducible representation]] of the [[Lie algebra]] <math>\mathfrak{g}</math> with [[highest weight]] λ. We know from the section about homomorphisms of Verma modules that there exists a homomorphism

:<math>M_{w'\cdot\lambda}\to M_{w\cdot\lambda}</math>

if and only if

:<math>w\leq w'</math>

in the [[Bruhat order|Bruhat ordering]] of the [[Weyl group]]. The following theorem describes a [[Resolution (algebra)|resolution]] of <math>V_\lambda</math> in terms of Verma modules (it was proved by [[Joseph Bernstein|Bernstein]]-[[Israel Gelfand|Gelfand]]-[[Sergei Gelfand|Gelfand]] in 1975<ref>Bernstein I.N., Gelfand I.M., Gelfand S.I., Differential Operators on the Base Affine Space and a Study of g-Modules, Lie Groups and Their Representations, I. M. Gelfand, Ed., Adam Hilger, London, 1975.}</ref>) :

<blockquote>
There exists an exact sequence of <math>\mathfrak{g}</math>-homomorphisms
: <math>0\to \oplus_{w\in W,\,\, \ell(w)=n} M_{w\cdot \lambda}\to \cdots \to \oplus_{w\in W,\,\, \ell(w)=2} M_{w\cdot \lambda}\to \oplus_{w\in W,\,\, \ell(w)=1} M_{w\cdot \lambda}\to M_\lambda\to V_\lambda\to 0</math>
where ''n'' is the length of the largest element of the Weyl group.
</blockquote>

A similar resolution exists for [[generalized Verma module]]s as well. It is denoted shortly as the ''BGG resolution''.

Recently, these resolutions were studied in special cases, because of their connections to [[invariant differential operator]]s in a special type of [[Cartan geometry]], the [[parabolic geometry (differential geometry)|parabolic geometries]]. These are Cartan geometries modeled on the pair (''G'', ''P'') where ''G'' is a [[Lie group]] and ''P'' a [[parabolic subgroup]]).<ref>For more information, see: Eastwood M., Variations on the de Rham complex, Notices Amer. Math. Soc, 1999 - ams.org. Calderbank D.M., Diemer T., Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, Arxiv preprint math.DG/0001158, 2000 - arxiv.org [http://arxiv.org/abs/math/0001158]. Cap A., Slovak J., Soucek V., Bernstein-Gelfand-Gelfand sequences, Arxiv preprint math.DG/0001164, 2000 - arxiv.org [http://arxiv.org/abs/math/0001164]</ref>

==See also==
*[[Generalized Verma module]]
*[[Weyl module]]

==Notes==
{{reflist}}

==References==
*{{citation|last=Carter|first=R.|title=Lie Algebras of Finite and Affine Type|publisher=Cambridge University Press|year=2005|isbn=0-521-85138-6}}.
*{{citation|last=Knapp|first=A. W.|title=Lie Groups Beyond an introduction|publisher=Birkhäuser|edition=2nd|year=2002|page=285|isbn=978-0-8176-3926-6}}.
*{{citation|last=Dixmier|first=J.|title=Enveloping Algebras|publisher=North-Holland|publication-place=Amsterdam, New York, Oxford|year=1977|isbn=0-444-11077-1}}.
*{{citation|last=Humphreys|first=J.|title=Introduction to Lie Algebras and Representation Theory|publisher=Springer Verlag|year=1980|isbn=3-540-90052-7}}.
*{{springer|title=BGG resolution|id=B/b120210|first=Alvany|last=Rocha|year=2001}}
*{{citation|last1=Roggenkamp|first1=K.|last2=Stefanescu|first2=M.|title=Algebra - Representation Theory|publisher=Springer|year=2002|isbn=0-7923-7114-3}}.

{{PlanetMath attribution|id=3665|title=Verma module}}

[[Category:Representation theory of Lie algebras]]

Dependence relation - Revision history

en>777sms at 03:29, 14 April 2012