Verma module
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
[1] Their homomorphisms correspond to invariant differential operators over flag manifolds.
be an arbitrary linear functional, not necessarily dominant or integral.
Thus, Verma modules play an important role in the classification of finite-dimensional representations of
Specifically, they are an important tool in the hard part of the theorem of the highest weight, namely showing that every dominant integral element actually arises as the highest weight of a finite-dimensional irreducible representation of
We now attempt to understand intuitively what the Verma module with highest weight
A simple re-ordering argument shows that there is only one possible way the full Lie algebra
, then by the easy part of the Poincaré–Birkhoff–Witt theorem, we can rewrite as a linear combination of products of Lie algebra elements with the raising operators
acting first, the elements of the Cartan subalgebra, and last the lowering operators
, 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.
To understand the structure of the Verma module a bit better, we may choose an ordering of the positive roots as
is dominant and integral, one can construct a finite-dimensional, irreducible quotient of the Verma module.
These formulas are motivated by the way the basis elements act in the finite-dimensional representations of
is an arbitrary complex number, not necessarily real or positive or an integer.
There are two standard constructions of the Verma module, both of which involve the concept of universal enveloping algebra.
The first construction[5] of the Verma module is a quotient of the universal enveloping algebra
The previous discussion motivates the following construction of Verma module.
The "extension of scalars" procedure is a method for changing a left module
(not necessarily commutative) into a left module over a larger algebra
-module over itself, the above tensor product carries a left module structure over the larger algebra
is supposed to act on the highest weight vector in a Verma module.
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
is dominant and integral, one then proves that this irreducible quotient is actually finite dimensional.
is called singular, if there is no dominant weight on the affine orbit of λ.
This follows easily from the Harish-Chandra theorem on infinitesimal central characters.
If the weights are further integral, then there exists a nonzero homomorphism if and only if in the Bruhat ordering of the Weyl group.
in terms of Verma modules (it was proved by Bernstein–Gelfand–Gelfand in 1975[12]) : There exists an exact sequence of
A similar resolution exists for generalized Verma modules as well.
