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 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.^[1] Their homomorphisms correspond to invariant differential operators over flag manifolds.

Definition of Verma modules

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

• ${\mathfrak {g}}$, a semisimple Lie algebra over $F$, with universal enveloping algebra ${\mathcal {U}}({\mathfrak {g}})$.
• ${\mathfrak {b}}$, a Borel subalgebra of ${\mathfrak {g}}$, with universal enveloping algebra ${\mathcal {U}}({\mathfrak {b}})$.
• ${\mathfrak {h}}$, a Cartan subalgebra of ${\mathfrak {g}}$. We do not consider its universal enveloping algebra.
• $\lambda \in {\mathfrak {h}}^{*}$, a fixed weight.

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

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

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^[2] and Verma^[3] 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

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

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 Bernstein–Gelfand–Gelfand in 1975^[4]) :

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).^[5]

See also

• Generalized Verma module
• Weyl module

Notes

1. E.g., Hall 2015 Chapter 9
2. 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)
3. Verma N., Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968)
4. 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.
5. 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 . Cap A., Slovak J., Soucek V., Bernstein-Gelfand-Gelfand sequences, Arxiv preprint math.DG/0001164, 2000 - arxiv.org

References

• Carter, R. (2005), Lie Algebras of Finite and Affine Type, Cambridge University Press, ISBN 0-521-85138-6.
• Dixmier, J. (1977), Enveloping Algebras, Amsterdam, New York, Oxford: North-Holland, ISBN 0-444-11077-1.
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer
• Humphreys, J. (1980), Introduction to Lie Algebras and Representation Theory, Springer Verlag, ISBN 3-540-90052-7.
• Knapp, A. W. (2002), Lie Groups Beyond an introduction (2nd ed.), Birkhäuser, p. 285, ISBN 978-0-8176-3926-6.
• Rocha, Alvany (2001), "BGG resolution", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Roggenkamp, K.; Stefanescu, M. (2002), Algebra - Representation Theory, Springer, ISBN 0-7923-7114-3.

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