Highest-weight Theory: Verma Modules

We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semi-simple Lie algebra (or, equivalently, of a compact Lie group). It turns out that such representations can be characterized by their “highest-weight”. The first method we’ll consider is purely Lie-algebraic, it begins by constructing a universal representation with a given highest-weight, then the finite-dimensional representation we want is found as a quotient of this. These universal representations are known as “Verma modules”. They are infinite dimensional, not completely reducible, non-unitary and not representations of the corresponding group, but have quotients that do have the properties we want.