TY - JOUR

T1 - Crossed products and inner actions of hopf algebras

AU - Blattner, Robert J.

AU - Cohen, Miriam

AU - Montgomery, Susan

PY - 1986/1/1

Y1 - 1986/1/1

N2 - 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 p: H ® H is a Hopf algebra epimorphism which is split as a coalgebra map, then H is algebra isomorphic to A #s H, a crossed product of H with the left Hopf kernel A of p. Given any crossed product A #s H with H (weakly) inner on A, then Ar H is isomorphic to a twisted product H [if] with trivial action. Finally, if H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A implies that of A #s H; in particular this is true if the (weak) action of H is inner.

AB - 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 p: H ® H is a Hopf algebra epimorphism which is split as a coalgebra map, then H is algebra isomorphic to A #s H, a crossed product of H with the left Hopf kernel A of p. Given any crossed product A #s H with H (weakly) inner on A, then Ar H is isomorphic to a twisted product H [if] with trivial action. Finally, if H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A implies that of A #s H; in particular this is true if the (weak) action of H is inner.

UR - http://www.scopus.com/inward/record.url?scp=84883435991&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1986-0860387-X

DO - 10.1090/S0002-9947-1986-0860387-X

M3 - Article

AN - SCOPUS:84883435991

VL - 298

SP - 671

EP - 711

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -