Verma module

From Wikipedia, the free encyclopedia
Jump to: navigation, search

Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.

Verma modules can be used to prove that an irreducible highest weight module with highest weight \lambda is finite-dimensional, if and only if the weight \lambda is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds.

Definition of Verma modules[edit]

The definition relies on a stack of relatively dense notation. Let F be a field and denote the following:

To define the Verma module, we begin by defining some other modules:

  • F_\lambda, the one-dimensional F-vector space (i.e. whose underlying set is F itself) together with a \mathfrak{b}-module structure such that \mathfrak{h} acts as multiplication by \lambda and the positive root spaces act trivially. As F_\lambda is a left \mathfrak{b}-module, it is consequently a left \mathcal{U}(\mathfrak{b})-module.
  • Using the Poincaré–Birkhoff–Witt theorem, there is a natural right \mathcal{U}(\mathfrak{b})-module structure on \mathcal{U}(\mathfrak{g}) by right multiplication of a subalgebra. \mathcal{U}(\mathfrak{g}) is naturally a left \mathfrak{g}-module, and together with this structure, it is a (\mathfrak{g}, \mathcal{U}(\mathfrak{b}))-bimodule.

Now we can define the Verma module (with respect to \lambda) as

M_\lambda = \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{b})} F_\lambda

which is naturally a left \mathfrak{g}-module (i.e. a representation of \mathfrak{g}). The Poincaré–Birkhoff–Witt theorem implies that the underlying vector space of M_\lambda is isomorphic to

\mathcal{U}(\mathfrak{g}_-) \otimes_F F_\lambda

where \mathfrak{g}_- is the Lie subalgebra generated by the negative root spaces of \mathfrak{g}.

Basic properties[edit]

Verma modules, considered as \mathfrak{g}-modules, are highest weight modules, i.e. they are generated by a highest weight vector. This highest weight vector is 1\otimes 1 (the first 1 is the unit in \mathcal{U}(\mathfrak{g}) and the second is the unit in the field F, considered as the \mathfrak{b}-module F_\lambda) and it has weight \lambda.

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

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

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

The Verma module M_\lambda itself is irreducible if and only if none of the coordinates of \lambda in the basis of fundamental weights is from the set \{0,1,2,\ldots\}.

The Verma module M_\lambda is called regular, if its highest weight λ is on the affine Weyl orbit of a dominant weight \tilde\lambda. In other word, there exist an element w of the Weyl group W such that

\lambda=w\cdot\tilde\lambda

where \cdot is the affine action of the Weyl group.

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

Homomorphisms of Verma modules[edit]

For any two weights \lambda, \mu a non-trivial homomorphism

M_\mu\rightarrow M_\lambda

may exist only if \mu and \lambda are linked with an affine action of the Weyl group W of the Lie algebra \mathfrak{g}. This follows easily from the Harish-Chandra theorem on infinitesimal central characters.

Each homomorphism of Verma modules is injective and the dimension

\dim(\operatorname{Hom}(M_\mu, M_\lambda))\leq 1

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

The full classification of Verma module homomorphisms was done by Bernstein-Gelfand-Gelfand[1] and Verma[2] and can be summed up in the following statement:

There exists a nonzero homomorphism M_\mu\rightarrow M_\lambda if and only if there exists

a sequence of weights

\mu=\nu_0\leq\nu_1\leq\ldots\leq\nu_k=\lambda
such that \nu_{i-1}+\delta=s_{\gamma_i}(\nu_i+\delta) for some positive roots \gamma_i (and s_{\gamma_i} is the corresponding root reflection and \delta is the sum of all fundamental weights) and for each 1\leq i\leq k, (\nu_i+\delta)(H_{\gamma_i}) is a natural number (H_{\gamma_i} is the coroot associated to the root \gamma_i).

If the Verma modules M_\mu and M_\lambda are regular, then there exists a unique dominant weight \tilde\lambda and unique elements w, w′ of the Weyl group W such that

P\mu=w'\cdot\tilde\lambda

and

\lambda=w\cdot\tilde\lambda,

where \cdot is the affine action of the Weyl group. If the weights are further integral, then there exists a nonzero homomorphism

M_\mu\to M_\lambda

if and only if

w \leq w'

in the Bruhat ordering of the Weyl group.

Jordan–Hölder series[edit]

Let

0\subset A\subset B\subset M_\lambda

be a sequence of \mathfrak{g}-modules so that the quotient B/A is irreducible with highest weight μ. Then there exists a nonzero homomorphism M_\mu\to M_\lambda.

An easy consequence of this is, that for any highest weight modules V_\mu, V_\lambda such that

V_\mu\subset V_\lambda

there exists a nonzero homomorphism M_\mu\to M_\lambda.

Bernstein–Gelfand–Gelfand resolution[edit]

Let V_\lambda be a finite-dimensional irreducible representation of the Lie algebra \mathfrak{g} with highest weight λ. We know from the section about homomorphisms of Verma modules that there exists a homomorphism

M_{w'\cdot\lambda}\to M_{w\cdot\lambda}

if and only if

w\leq w'

in the Bruhat ordering of the Weyl group. The following theorem describes a resolution of V_\lambda in terms of Verma modules (it was proved by BernsteinGelfandGelfand in 1975[3]) :

There exists an exact sequence of \mathfrak{g}-homomorphisms

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

where n is the length of the largest element of the Weyl group.

A similar resolution exists for generalized Verma modules 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 operators in a special type of Cartan geometry, the parabolic geometries. These are Cartan geometries modeled on the pair (G, P) where G is a Lie group and P a parabolic subgroup).[4]

See also[edit]

Notes[edit]

  1. ^ 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)
  2. ^ Verma N., Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968)
  3. ^ 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.
  4. ^ 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 [1]. Cap A., Slovak J., Soucek V., Bernstein-Gelfand-Gelfand sequences, Arxiv preprint math.DG/0001164, 2000 - arxiv.org [2]

References[edit]

This article incorporates material from Verma module on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.