  We construct an abelian category C and exact functors in C 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 C. The category C is the direct
sum of a semisimple category and the category of modules over a certain algebra
A, associated to 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.
