Abstract
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 A H 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 U q ( sl ( n ) ) , n ⩾ 2 , are geometrically reductive for any parameter q ≠ ± 1 .
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