Irreducible actions and faithful actions of hopf algebras
- Cite this article as:
- Bergen, J., Cohen, M. & Fischman, D. Israel J. Math. (1990) 72: 5. doi:10.1007/BF02764609
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LetH be a Hopf algebra acting on an algebraA. We will examine the relationship betweenA, the ring of invariantsAH, and the smash productA # H. We begin by studying the situation whereA is an irreducibleA # H module and, as an application of our first main theorem, show that ifD is a division ring then [D : DH]≦dimH. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness ofA # H is strongly related to the faithfulness of the left and right actions ofA # H onA.