we have produced a left -module . We now apply this construction in the setting of a semisimple Lie algebra. We let be the subalgebra of spanned by and the root vectors with . (Thus, is a "Borel subalgebra" of .) We can form a left module over the universal enveloping algebra as follows: is the one-dimensional vector space spanned by a single vector together with a -module structure such that acts as multiplication by and the positive root spaces act trivially: . The motivation for this formula is that it describes how is supposed to act on the highest weight vector in a Verma module. Now, it follows from the Poincaré–Birkhoff–Witt theorem that is a subalgebra of . Thus, we may apply the extension of scalars technique to convert from a left -module into a left -module as follow: . Since is a left -module, it is, in particular, a module (representation) for . The structure of the Verma module 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). Basic properties Verma modules, considered as -modules, are highest weight modules, i.e. they are generated by a highest weight vector. This highest weight vector is (the first is the unit in and the second is the unit in the field , considered as the -module ) and it has weight . Multiplicities 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 . Universal property 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 (so called highest weight modules) are quotients of Irreducible quotient module