The universal quantum group of a noncommutative projective space

For any Koszul, Artin-Schelter (AS) regular algebra A, we consider the universal Hopf algebra aut(A) coacting on A, as introduced by Manin. To study the representations (i.e. finite dimensional comodules) of this Hopf algebra, we use the Tannaka-Krein formalism. This allows one to show that aut(A) has a highest weight theory and that the Hopf algebras associated to different AS-regular algebras of the same global dimension are (co)Morita equivalent. If time permits, we will give a more conceptual proof of this last result and also discuss the role of Borel categories. This includes joint work with Michel Van den Bergh.