Title: Current algebras and quivers.
We discuss category of graded representations of the
current algebras of simple Lie algebras along with several
important subcategories. In particular, we prove that these
categories are directed and hence equivalent to a category
of representations of quivers with relations. We show that a
number of intereseting quivers come from our study. Finally,
we relate our constructions to the the Kirillov--Reshetikhin
modules and prove that they are projective objects in a
suitably chosen subcategory and show that the associated
finite-dimensional algebra is Koszul. The talk is based on
joint work with Jacob Greenstein