Abstract: Recently it has been proved that any (not necesssarily affine) normal algebraic monoid M is a closed submonoid of the endomophism monoid of a conveniently chosen homogeneous vector bundle over A(M), the Albanese variety of M. This result suggests that the category of homogeneous vector bundles over the abelian variety A(M) is the natural setting for a representation theory of M. Such a theory has not yet been developed, even in the case when M is an algebraic group.

The main goal of this talk is to present the basic facts about the structure of the endomorphism monoid of a homogeneous vector bundle over an abelian variety. These endomorphism monoids play a role similar to the endomorphism monoid of a vector space (the monoid of n by n matrices) in the affine case.

This is a joint work with Leticia Brambila-Paz.