Verma module on arvau

Wikipedia / Verma module

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 in the classification of irreducible representations is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds.

Informal construction

We can explain the idea of a Verma module as follows. be a semisimple Lie algebra be a fixed Cartan subalgebra's as "lowering operators."

is dominant integral, however, one can construct a finite-dimensional quotient module of the Verma module. Thus, Verma modules play an important role in the classification of finite-dimensional representations. Specifically, they are an important tool in the hard part of the theorem of the highest weight.

, we certainly want

and

's:

We now impose only by subtracting integer.

, then by the easy part of the Poincaré–Birkhoff–Witt theorem, we can rewrite

as a linear combination, any term with a raising operator is zero, any factors in the Cartan act as scalars, and thus we end up with an element of the original form.

's in a specific order:

's are non-negative integers. Actually, it turns out that such vectors form a basis for the Verma module.

looks like, it still remains to give a rigorous construction of it. In any case, the Verma module gives—for any is dominant and integral, one can construct a finite-dimensional, irreducible quotient of the Verma module.

The case of sl(2; C)

is spanned by linearly independent and the action of the basis elements is as follows:

has to terminate.

is an arbitrary complex number

Definition of Verma modules

There are two standard constructions of the Verma module, both of which involve the concept of universal enveloping algebra.

As a quotient of the enveloping algebra

The first construction-module, by the universal property, there will be a linear map given by

should include all vectors of the form

as the quotient vector space

is the left ideal generated by all elements of the form

and

-module.

By extension of scalars

The "extension of scalars-module, we can form the tensor product:

, uniquely determined by the requirement that

is a "Borel subalgebra as follows:

is the one-dimensional vector space-module and the positive root spaces act trivially:
.

is supposed to act on the highest weight vector in a Verma module.

Now, it follows from the Poincaré–Birkhoff–Witt theorem as follow:

The structure of the Verma module

is isomorphic to

's).

Basic properties

-modules, are highest weight modules, i.e. they are generated by a highest weight vector-module.

Multiplicities

Verma modules are weight modules is a direct sum of all its weight spaces as a sum of positive roots (this is closely related to the so-called Kostant partition function.

Universal property

, there is a surjective-homomorphism that are generated by the highest weight vector (so called highest weight modules) are quotients

Irreducible quotient module

contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation is dominant and integral, one then proves that this irreducible quotient is actually finite dimensional.

is the same as in the Verma module, except in the Verma module.

is antidominant. is irreducible if and only if in the basis of fundamental weights to be irreducible.

Other properties

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

is the affine action of the Weyl group.

is called singular is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights).

Homomorphisms of Verma modules

a non-trivial homomorphism

are linked with an affine action of the Weyl group. This follows easily from the Harish-Chandra theorem on infinitesimal central characters.

Each homomorphism of Verma modules is injective and the dimension

is isomorphic.

The full classification of Verma module homomorphisms was done by Bernstein–Gelfand–Gelfand and Verma and can be summed up in the following statement:

if and only if there exists a sequence of weights

is the corresponding root reflection is the sum of all fundamental weights is the coroot).

are regular, then there exists a unique dominant weight and unique elements w, w′ of the Weyl group W such that

and

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

if and only if

in the Bruhat ordering of the Weyl group.

Jordan–Hölder series

Let

-modules so that the quotient B/A is irreducible with highest weight.

An easy consequence of this is, that for any highest weight modules such that

Bernstein–Gelfand–Gelfand resolution

be a finite-dimensional irreducible representation of the Lie algebra with highest weight λ. We know from the section about homomorphisms of Verma modules that there exists a homomorphism

if and only if

in the Bruhat ordering of the Weyl group. The following theorem describes a projective resolution in terms of Verma modules (it was proved by Bernstein Gelfand Gelfand in 1975) :

-homomorphisms

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.