A Category for the Adjoint Representation

We construct an abelian category and exact functors in which on the Grothendieck group descend to the action of a simply laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an action in the derived category of . The category is the direct sum of a semisimple category and the category of modules over a certain algebra associated with a Dynkin diagram. In the second half of the paper we show how these algebras appear in the modular representation theory and in the McKay correspondence and explore their relationship with root systems