Smash products, inner actions and quotient rings

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36 Scopus citations


Let if be a Hopf algebra over a field k, and A an H-module algebra over k. Let AH = (a ∈ A|h · a = ε(h)a, all h ∈ H). This paper is mainly concerned with inner actions. We prove the existence of a "symmetric" quotient ring Q of A, which is also an H-module algebra, and consider Q-inner actions, an analogue of X-inner automorphisms. Under certain conditions on A and H we show that Q contains B, a finite-dimensional separable algebra over its center C, a field. Moreover, the centralizer of B in Q is QH. This is used to prove that if AH is P.I. then so is A, and that A is fully integral over AHC of bounded degree. We also consider connections between the A, AH and A#H module structures.

Original languageEnglish
Pages (from-to)45-66
Number of pages22
JournalPacific Journal of Mathematics
Issue number1
StatePublished - 1 Jan 1986

ASJC Scopus subject areas

  • General Mathematics


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