The Connes spectrum of group actions and group gradings for certain quotient rings. (English) Zbl 0834.16038
Let \(H\) be a finite-dimensional, semisimple Hopf algebra over an algebraically closed field \(K\), and assume that \(H\) is either commutative or cocommutative. Let \(A\) be an \(H\)-module algebra which is semiprime right Goldie, and let \(Q(A)\) denote its classical right ring of quotients. The main result of this paper is that the Connes spectrum \(\text{CS}(A,H)\) is equal to \(\text{CS}(Q(A),H)\). In the same vein, if \(B\) is an \(H\)-semiprime, \(H\)-module algebra and if \(Q_H(B)\) denotes its \(H\)-symmetric ring of quotients, then it is shown here that \(\text{CS}(B,H)=\text{CS}(Q_H(B),H)\). These results are then applied to finite group actions and group gradings.
Reviewer: D.S.Passman (Madison)
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 16W20 | Automorphisms and endomorphisms |
| 16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 16S40 | Smash products of general Hopf actions |
| 16S60 | Associative rings of functions, subdirect products, sheaves of rings |
Keywords:
symmetric ring of quotients; semisimple Hopf algebras; classical right rings of quotients; Connes spectrum; finite group actions; group gradingsReferences:
| [1] | Jeffrey Bergen and Miriam Cohen, Actions of commutative Hopf algebras, Bull. London Math. Soc. 18 (1986), no. 2, 159 – 164. · Zbl 0563.16003 · doi:10.1112/blms/18.2.159 |
| [2] | William Chin, Crossed products of semisimple cocommutative Hopf algebras, Proc. Amer. Math. Soc. 116 (1992), no. 2, 321 – 327. · Zbl 0762.16016 |
| [3] | Miriam Cohen, Smash products, inner actions and quotient rings, Pacific J. Math. 125 (1986), no. 1, 45 – 66. · Zbl 0553.16005 |
| [4] | Miriam Cohen and Louis H. Rowen, Group graded rings, Comm. Algebra 11 (1983), no. 11, 1253 – 1270. · Zbl 0522.16001 · doi:10.1080/00927878308822904 |
| [5] | 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. · Zbl 0679.16001 |
| [6] | 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. · Zbl 0756.16008 · doi:10.1090/conm/124/1144032 |
| [7] | S. Montgomery, Bi-invertible actions of Hopf algebras, Israel J. Math. 83 (1993), no. 1-2, 45 – 71. · Zbl 0804.16041 · doi:10.1007/BF02764636 |
| [8] | -, Hopf algebras and their actions, CBMS Regional Conf. Ser. in Math., no. 82, Amer. Math. Soc., Providence, RI, 1993. |
| [9] | S. Montgomery and H.-J. Schneider, Hopf crossed products, rings of quotients, and prime ideals, Adv. Math. 112 (1995), no. 1, 1 – 55. · Zbl 0829.16029 · doi:10.1006/aima.1995.1027 |
| [10] | 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. · Zbl 0742.16011 · doi:10.1090/conm/124/1144035 |
| [11] | James Osterburg and D. S. Passman, Computing the Connes spectrum of a Hopf algebra, Israel J. Math. 80 (1992), no. 1-2, 225 – 253. · Zbl 0794.16020 · doi:10.1007/BF02808164 |
| [12] | 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. · Zbl 0782.16021 · doi:10.1090/conm/130/1176129 |
| [13] | James Osterburg and Declan Quinn, Cocommutative Hopf algebra actions and the Connes spectrum, J. Algebra 165 (1994), no. 3, 465 – 475. , https://doi.org/10.1006/jabr.1994.1124 James Osterburg and Xue Yao, The Connes spectrum of group actions and group gradings for certain quotient rings, Trans. Amer. Math. Soc. 347 (1995), no. 6, 2263 – 2275. · Zbl 0809.16034 |
| [14] | D. S. Passman, Algebraic structure of group rings, Academic Press, Boston, MA, 1989. |
| [15] | Donald S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press, Inc., Boston, MA, 1989. · Zbl 0662.16001 |
| [16] | N. Christopher Phillips, Equivariant \?-theory and freeness of group actions on \?*-algebras, Lecture Notes in Mathematics, vol. 1274, Springer-Verlag, Berlin, 1987. · Zbl 0632.46062 |
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