- Verma module
Verma modules, named after
Daya-Nand Verma , are objects in therepresentation theory ofLie algebra s, a branch ofmathematics .The definition of a Verma module looks complicated, but Verma modules are very natural objects, with useful properties. Their homomorphisms correspond to
invariant differential operators overflag manifold s.Verma modules can be used to prove that an irreducible
highest weight module withhighest weight is finite dimensional, if and only if the weight is dominant and integral.Definition of Verma modules
The definition relies on a stack of relatively dense notation. Let be a field and denote the following:
* , asemisimple Lie algebra over , withuniversal enveloping algebra .
* , aBorel subalgebra of , with universal enveloping algebra .
* , aCartan subalgebra of . We do not consider its universal enveloping algebra.
* , a fixed weight.To define the Verma module, we begin by defining some other modules:
* , the one-dimensional -vector space (i.e. whose underlying set is itself) together with a -module structure such that acts as multiplication by and the positive root spaces act trivially. As is a left -module, it is consequently a left -module.
* Using thePoincaré-Birkhoff-Witt theorem , there is a natural right -module structure on by right multiplication of a subalgebra. is obviously a left -module, and together with this structure, it is a -bimodule .Now we can define the Verma module (with respect to ) as: which is naturally a left -module (i.e. an infinite-dimensional representation of ). The Poincaré-Birkhoff-Witt theorem implies that the underlying vector space of is isomorphic to: where is the Lie subalgebra generated by the negative root spaces of .Basic properties
Verma modules, considered as -modules, are
highest weight module s, i.e. they are generated by ahighest weight vector . This highest weight vector is (the first is the unit in and the second isthe unit in the field , considered as the -module) and it has weight .Verma modules are
weight modules , i.e. is adirect sum of all itsweight space s. Each weight space in is finite dimensional and the dimension of the -weight space is the number of possibilities how to obtain as a sum ofpositive root s (this is closely related to the so-calledKostant partition function ).Verma modules have a very important property: If is any representation generated by a highest weight vector of weight , there is a
surjective -homomorphism That is, all representations with highest weight that are generated by the highest weight vector (s.c.highest weight module s) are quotients ofcontains a unique maximal
submodule , and its quotient is the unique (up toisomorphism )irreducible representation with highest weightThe Verma module itself is irreducible if and only if none of the coordinates of in the basis of
fundamental weight s is from the set .The Verma module is called "regular", if its highest weight λ is on the affine Weyl orbit of a dominant weight . In other word, there exist an element w of the Weyl group W such that: where is the
affine action of the Weyl group.The Verma module is called "singular", if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight so that is on the wall of the
fundamental Weyl chamber (δ is the sum of allfundamental weight s).Homomorphisms of Verma modules
For any two weights a non-trivial
homomorphism :
may exist only if and are linked with an
affine action of theWeyl group of the Lie algebra . This follows easily from theHarish-Chandra theorem oninfinitesimal central character s.Each homomorphism of Verma modules is injective and the
dimension :
for any . So, there exists a nonzero if and only if is
isomorphic to a (unique) submodule of .The full classification of Verma module homomorphisms was done by Bernstein-Gelfand-Gelfand [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)] and Verma [Verma N., Structure of certain induced representations of complex semisimple Lie algebras}, Bull. Amer. Math. Soc. 74 (1968)] and can be summed up in the following statement:
There exists a nonzero homomorphism if and only if there exists a sequence of weights
::
such that for some positive roots (and is the corresponding
root reflection and is the sum of allfundamental weight s) and for each is a natural number ( is thecoroot associated to the root ).If the Verma modules and are regular, then there exists a unique
dominant weight and unique elements "w", "w"′ of theWeyl group "W" such that:P
and
:
where 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-Holder Series
Let :be a sequence of -modules so that the quotient B/A is irreducible with
highest weight μ. Then there exists a nonzero homomorphism .An easy consequence of this is, that for any
highest weight module s such that: there exists a nonzero homomorphism .Bernstein-Gelfand-Gelfand resolution
Let be a finite dimensional
irreducible representation of theLie algebra withhighest 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 theWeyl group . The following theorem describes a resolution of in terms of Verma modules (it was proved by Bernstein-Gelfand-Gelfand in 1975 [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.}] ):There exists an exact sequence of -homomorphisms:where "n" is the length of the largest element of the Weyl group.
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 ofCartan geometry , theparabolic geometries . These are Cartan geometries modeled on the pair ("G", "P") where "G" is aLie group and "P" aparabolic subgroup ). [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] ] .ee also
*
Lie algebra representation
*Universal enveloping algebra
*Generalized Verma module Notes
References
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*citation|last=Humphreys|first=J.|title=Introduction to Lie Algebras and Representation Theory|publisher=Springer Verlag|year=1980|isbn=3540900527.
*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=0792371143.
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