Geometrically reductive Hopf algebras and their invariants

By analogy with the Mumford definition of geometrically reductive algebraic group, we introduce the concept of geometrically reductive Hopf algebra (over a field). Then we prove that if H is a geometrically reductive Hopf algebra and A is a commutative, finitely generated and locally finite H-module algebra, then the algebra of invariants AH is finitely generated. We also prove that in characteristic 0 a Hopf algebra H is geometrically reductive if and only if every finite dimensional H-module is semisimple, and that in positive characteristic every finite dimensional Hopf algebra is geometrically reductive. Finally, we prove that in positive characteristic the quantum enveloping Hopf algebras Uq(sl(n)), n⩾2, are geometrically reductive for any parameter q≠±1.