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 of a complex semisimple Lie algebra. Specifically, although Verma modules themselves are infinite dimensional, quotients of them can be used to construct finite dimensional representations with highest weight , where is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds.

There are two standard constructions of the Verma module, both of which involve the concept of universal enveloping algebra. We continue the notation of the previous section: is a complex semisimple Lie algebra, is a fixed Cartan subalgebra, is the associated root system with a fixed set of positive roots. For each , we choose nonzero elements and .

Whichever construction of the Verma module is used, one has to prove that it is nontrivial, i. e., not the zero module. Actually, it is possible to use the Poincaré–Birkhoff–Witt theorem to show that the underlying vector space of is isomorphic to

where is the Lie subalgebra generated by the negative root spaces of (that is, the 's).

Verma modules are weight modules, i. e. is a direct sum of all its weight spaces. Each weight space in is finite dimensional and the dimension of the weight space is the number of ways of expressing as a sum of positive roots (this is closely related to the so called Kostant partition function). This assertion follows from the earlier claim that the Verma module is isomorphic as a vector space to , along with the Poincaré–Birkhoff–Witt theorem for .

contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight If the highest weight is dominant and integral, one then proves that this irreducible quotient is actually finite dimensional. As an example, consider the case discussed above. If the highest weight is "dominant integral"—meaning simply that it is a non negative integer—then and the span of the elements is invariant. The quotient representation is then irreducible with dimension The quotient representation is spanned by linearly independent vectors The action of is the same as in the Verma module, except that in the quotient, as compared to in the Verma module. The Verma module itself is irreducible if and only if is antidominant. Consequently, when is integral, is irreducible if and only if none of the coordinates of in the basis of fundamental weights is from the set , while in general, this condition is necessary but insufficient for to be irreducible.

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 all fundamental weights).