The Connes spectrum of group actions and group gradings for certain quotient rings
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- by James Osterburg and Xue Yao PDF
- Trans. Amer. Math. Soc. 347 (1995), 2263-2275 Request permission
Abstract:
Let $H$ be a finite-dimensional, semisimple Hopf algebra over an algebraically closed field $K$ where $H$ is either commutative or cocommutative. We let $A$ be an $H$-module algebra which is semiprime right Goldie. We show that the Connes spectrum of $H$ acting on $A$ is the Connes spectrum of $H$ acting on the classical quotient ring of $A$. In our last section, we define a symmetric quotient ring and show that the Connes spectrum of the ring and its quotient ring are the same. Finally, we apply our results to finite group actions and group gradings.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
- William Chin, Crossed products of semisimple cocommutative Hopf algebras, Proc. Amer. Math. Soc. 116 (1992), no. 2, 321–327. MR 1100646, DOI 10.1090/S0002-9939-1992-1100646-7
- Miriam Cohen, Smash products, inner actions and quotient rings, Pacific J. Math. 125 (1986), no. 1, 45–66. MR 860749, DOI 10.2140/pjm.1986.125.45
- Miriam Cohen and Louis H. Rowen, Group graded rings, Comm. Algebra 11 (1983), no. 11, 1253–1270. MR 696990, DOI 10.1080/00927878308822904
- K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 16, Cambridge University Press, Cambridge, 1989. MR 1020298
- S. Montgomery, Hopf Galois extensions, Azumaya algebras, actions, and modules (Bloomington, IN, 1990) Contemp. Math., vol. 124, Amer. Math. Soc., Providence, RI, 1992, pp. 129–140. MR 1144032, DOI 10.1090/conm/124/1144032
- S. Montgomery, Bi-invertible actions of Hopf algebras, Israel J. Math. 83 (1993), no. 1-2, 45–71. MR 1239716, DOI 10.1007/BF02764636 —, Hopf algebras and their actions, CBMS Regional Conf. Ser. in Math., no. 82, Amer. Math. Soc., Providence, RI, 1993.
- S. Montgomery and H.-J. Schneider, Hopf crossed products, rings of quotients, and prime ideals, Adv. Math. 112 (1995), no. 1, 1–55. MR 1321668, DOI 10.1006/aima.1995.1027
- James Osterburg and D. S. Passman, What makes a skew group ring prime?, Azumaya algebras, actions, and modules (Bloomington, IN, 1990) Contemp. Math., vol. 124, Amer. Math. Soc., Providence, RI, 1992, pp. 165–177. MR 1144035, DOI 10.1090/conm/124/1144035
- James Osterburg and D. S. Passman, Computing the Connes spectrum of a Hopf algebra, Israel J. Math. 80 (1992), no. 1-2, 225–253. MR 1248937, DOI 10.1007/BF02808164
- James Osterburg, D. S. Passman, and Declan Quinn, A Connes spectrum for Hopf algebras, Abelian groups and noncommutative rings, Contemp. Math., vol. 130, Amer. Math. Soc., Providence, RI, 1992, pp. 311–334. MR 1176129, DOI 10.1090/conm/130/1176129
- James Osterburg and Declan Quinn, Cocommutative Hopf algebra actions and the Connes spectrum, J. Algebra 165 (1994), no. 3, 465–475. MR 1275914, DOI 10.1006/jabr.1994.1124 D. S. Passman, Algebraic structure of group rings, Academic Press, Boston, MA, 1989.
- Donald S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press, Inc., Boston, MA, 1989. MR 979094
- N. Christopher Phillips, Equivariant $K$-theory and freeness of group actions on $C^*$-algebras, Lecture Notes in Mathematics, vol. 1274, Springer-Verlag, Berlin, 1987. MR 911880, DOI 10.1007/BFb0078657
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 2263-2275
- MSC: Primary 16W30; Secondary 16S35, 16W50
- DOI: https://doi.org/10.1090/S0002-9947-1995-1273514-5
- MathSciNet review: 1273514