Crossed products and inner actions of Hopf algebras

Blattner, Robert J.; Cohen, Miriam; Montgomery, Susan

doi:10.1090/S0002-9947-1986-0860387-X

Crossed products and inner actions of Hopf algebras
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by Robert J. Blattner, Miriam Cohen and Susan Montgomery PDF

Trans. Amer. Math. Soc. 298 (1986), 671-711 Request permission

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 $\pi :H \to \overline H$ is a Hopf algebra epimorphism which is split as a coalgebra map, then $H$ is algebra isomorphic to $A{\# _\sigma }H$, a crossed product of $H$ with the left Hopf kernel $A$ of $\pi$. Given any crossed product $A{\# _\sigma }H$ with $H$ (weakly) inner on $A$, then $A{\# _\sigma }H$ is isomorphic to a twisted product ${A_\tau }[H]$ with trivial action. Finally, if $H$ is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of $A$ implies that of $A{\# _\sigma }H$; in particular this is true if the (weak) action of $H$ is inner.

References

Jeffrey Bergen and Miriam Cohen, Actions of commutative Hopf algebras, Bull. London Math. Soc. 18 (1986), no. 2, 159–164. MR 818820, DOI 10.1112/blms/18.2.159
Robert J. Blattner and Susan Montgomery, A duality theorem for Hopf module algebras, J. Algebra 95 (1985), no. 1, 153–172. MR 797661, DOI 10.1016/0021-8693(85)90099-7
Miriam Cohen and Davida Fishman, Hopf algebra actions, J. Algebra 100 (1986), no. 2, 363–379. MR 840582, DOI 10.1016/0021-8693(86)90082-7
M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), no. 1, 237–258. MR 728711, DOI 10.1090/S0002-9947-1984-0728711-4
Everett C. Dade, Group-graded rings and modules, Math. Z. 174 (1980), no. 3, 241–262. MR 593823, DOI 10.1007/BF01161413
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
Harish-Chandra, On representations of Lie algebras, Ann. of Math. (2) 50 (1949), 900–915. MR 30945, DOI 10.2307/1969586

Hopf algebras

V. K. Harčenko, Groups and Lie algebras acting on noncommutative rings, Uspekhi Mat. Nauk 35 (1980), no. 2(212), 67–90, 279 (Russian). MR 571647
M. B. Landstad, J. Phillips, I. Raeburn, and C. E. Sutherland, Representations of crossed products by coactions and principal bundles, Trans. Amer. Math. Soc. 299 (1987), no. 2, 747–784. MR 869232, DOI 10.1090/S0002-9947-1987-0869232-0
Richard Gustavus Larson and Moss Eisenberg Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75–94. MR 240169, DOI 10.2307/2373270
Martin Lorenz, Primitive ideals in crossed products and rings with finite group actions, Math. Z. 158 (1978), no. 3, 285–294. MR 480612, DOI 10.1007/BF01214799
Martin Lorenz and D. S. Passman, Two applications of Maschke’s theorem, Comm. Algebra 8 (1980), no. 19, 1853–1866. MR 588448, DOI 10.1080/00927878008822549
Martin Lorenz and D. S. Passman, Prime ideals in group algebras of polycyclic-by-finite groups, Proc. London Math. Soc. (3) 43 (1981), no. 3, 520–543. MR 635568, DOI 10.1112/plms/s3-43.3.520

Combinatorial group theory

Susan Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. MR 590245
Susan Montgomery, $X$-inner automorphisms of filtered algebras, Proc. Amer. Math. Soc. 83 (1981), no. 2, 263–268. MR 624911, DOI 10.1090/S0002-9939-1981-0624911-5
Susan Montgomery and D. S. Passman, Crossed products over prime rings, Israel J. Math. 31 (1978), no. 3-4, 224–256. MR 516150, DOI 10.1007/BF02761494
Susan Montgomery and D. S. Passman, $X$-inner automorphisms of group rings, Houston J. Math. 7 (1981), no. 3, 395–402. MR 640981

inner automorphisms of crossed products and semiinvariants of Hopf algebras

Yoshiomi Nakagami and Masamichi Takesaki, Duality for crossed products of von Neumann algebras, Lecture Notes in Mathematics, vol. 731, Springer, Berlin, 1979. MR 546058
Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
Robert Steinberg, Complete sets of representations of algebras, Proc. Amer. Math. Soc. 13 (1962), 746–747. MR 141710, DOI 10.1090/S0002-9939-1962-0141710-X
Moss Eisenberg Sweedler, Cohomology of algebras over Hopf algebras, Trans. Amer. Math. Soc. 133 (1968), 205–239. MR 224684, DOI 10.1090/S0002-9947-1968-0224684-2
Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485