Verma Module - Definition of Verma Modules

Definition of Verma Modules

The definition relies on a stack of relatively dense notation. Let be a field and denote the following:

• , a semisimple Lie algebra over, with universal enveloping algebra .
• , a Borel subalgebra of, with universal enveloping algebra .
• , a Cartan subalgebra of . We do not consider its universal enveloping algebra.
• , a fixed weight.

To define the Verma module, we begin by defining some other modules:

• , the one-dimensional -vector space (i.e. whose underlying set is itself) together with a -module structure such that acts as multiplication by and the positive root spaces act trivially. As is a left -module, it is consequently a left -module.
• Using the Poincaré-Birkhoff-Witt theorem, there is a natural right -module structure on by right multiplication of a subalgebra. is naturally a left -module, and together with this structure, it is a -bimodule.

Now we can define the Verma module (with respect to ) as

which is naturally a left -module (i.e. an infinite-dimensional representation of ). The Poincaré-Birkhoff-Witt theorem implies that the underlying vector space of is isomorphic to

where is the Lie subalgebra generated by the negative root spaces of .