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. We can explain the idea of a Verma module as follows. Let be a semisimple Lie algebra (over , for simplicity). Let be a fixed Cartan subalgebra of and let be the associated root system. Let be a fixed set of positive roots. For each , choose a nonzero element for the corresponding root space and a nonzero element in the root space . We think of the 's as "raising operators" and the 's as "lowering operators." Now let be an arbitrary linear functional, not necessarily dominant or integral. Our goal is to construct a representation of with highest weight that is generated by a single nonzero vector with weight . The Verma module is one particular such highest-weight module, one that is maximal in the sense that every other highest-weight module with highest weight is a quotient of the Verma module. It will turn out that Verma modules are always infinite dimensional; if 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 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 should look like.