Crossed products and inner actions of Hopf algebras

Authors:
Robert J. Blattner, Miriam Cohen and Susan Montgomery

Journal:
Trans. Amer. Math. Soc. **298** (1986), 671-711

MSC:
Primary 16A24; Secondary 16A03, 16A72, 46L40

DOI:
https://doi.org/10.1090/S0002-9947-1986-0860387-X

MathSciNet review:
860387

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Abstract: This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if is a Hopf algebra epimorphism which is split as a coalgebra map, then is algebra isomorphic to , a crossed product of with the left Hopf kernel of . Given any crossed product with (weakly) inner on , then is isomorphic to a twisted product with trivial action. Finally, if is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of implies that of ; in particular this is true if the (weak) action of is inner.

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0860387-X

Article copyright:
© Copyright 1986
American Mathematical Society