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Homological integral of Hopf algebras

Author(s): D.-M. Lu; Q.-S. Wu; J. J. Zhang
Journal: Trans. Amer. Math. Soc. 359 (2007), 4945-4975.
MSC (2000): Primary 16A62, 16W30; Secondary 16E70, 20J50
Posted: May 16, 2007
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Abstract: The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke's theorem for infinite dimensional Hopf algebras. The generalization of Maschke's theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.

References:

[ASZ]
K. Ajitabh, S.P. Smith and J.J. Zhang, Auslander-Gorenstein rings, Comm. Algebra 26 (1998), no. 7, 2159-2180. MR 1626582 (99g:16010)

[Be]
D.J. Benson, Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, 190. Cambridge University Press, Cambridge, 1993. MR 1249931 (94j:13003)

[Bra]
A. Braun, An additivity principle for p.i. rings. J. Algebra 96 (1985), no. 2, 433-441. MR 0810539 (87f:16011)

[BW]
A. Braun and R.B. Warfield Jr., Symmetry and localization in Noetherian prime PI rings. J. Algebra 118 (1988), no. 2, 322-335. MR 0969675 (89k:16031)

[Bro]
K. A. Brown, Representation theory of Noetherian Hopf algebras satisfying a polynomial identity, Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997), 49-79, Contemp. Math., 229, AMS, Providence, RI, 1998. MR 1676211 (99m:16056)

[BG]
K. A. Brown and K. R. Goodearl, Homological aspects of Noetherian PI Hopf algebras of irreducible modules and maximal dimension, J. Algebra 198 (1997), 240-265. MR 1482982 (99c:16036)

[Co]
P.M. Cohn, Algebra. Vol. 2. With errata to Vol. I. John Wiley & Sons, London-New York-Sydney, 1977. MR 0530404 (58:26625)

[GW]
K.R. Goodearl and R.B. Warfield Jr., ``An introduction to noncommutative Noetherian rings'', Second edition. London Mathematical Society Student Texts, 61. Cambridge University Press, Cambridge, 2004. MR 2080008 (2005b:16001)

[GP]
C. Greither and B. Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), no. 1, 239-258. MR 0878476 (88i:12006)

[IS]
W. Imrich and N. Seifter, A bound for groups of linear growth, Arch. Math. (Basel) 48 (1987), no. 2, 100-104. MR 0878419 (88e:20035)

[LR1]
R.G. Larson and D.E. Radford, Finite-dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. Algebra 117 (1988), no. 2, 267-289. MR 0957441 (89k:16016)

[LR2]
R.G. Larson and D.E. Radford, Semisimple cosemisimple Hopf algebras, Amer. J. Math. 110 (1988), no. 1, 187-195. MR 0926744 (89a:16011)

[LS]
R.G. Larson and M. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969) 75-94. MR 0240169 (39:1523)

[Liu]
G.-X. Liu, On Noetherian affine prime regular Hopf algebras of Gelfand-Kirillov dimension $ 1$, Proc. Amer. Math. Soc. (to appear).

[LL]
M.E. Lorenz and M. Lorenz, On crossed products of Hopf algebras, Proc. Amer. Math. Soc. 123 (1995), no. 1, 33-38. MR 1227522 (95c:16014)

[MR]
J. C. McConnell and J. C . Robson, ``Noncommutative Noetherian Rings,'' Wiley, Chichester, 1987. MR 0934572 (89j:16023)

[Mo]
S. Montgomery, ``Hopf algebras and their actions on rings'', CBMS Regional Conference Series in Mathematics, 82, Providence, RI, 1993. MR 1243637 (94i:16019)

[Par]
B. Pareigis, Forms of Hopf algebras and Galois theory, Topics in algebra, Part 1 (Warsaw, 1988), 75-93, Banach Center Publ., 26, Part 1, PWN, Warsaw, 1990 MR 1171227 (93f:16038)

[Pas]
D.S. Passman, The algebraic structure of group rings, Reprint of the 1977 original. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1985. MR 0798076 (86j:16001)

[Ro]
J.J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, 85. Academic Press, Inc. New York-London, 1979. MR 0538169 (80k:18001)

[Sc]
H.-J. Schneider, Some remarks on exact sequences of quantum groups. Comm. Algebra 21 (1993), no. 9, 3337-3357. MR 1228767 (94e:17026)

[SSW]
L. W. Small, J.T. Stafford and R. B. Warfield Jr., Affine algebras of Gelfand-Kirillov dimension one are PI, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 3, 407-414. MR 0778674 (86g:16025)

[Sp]
T.A. Springer, Linear algebraic groups, 2nd edition. Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713 (99h:20075)

[St]
J. Stallings, Group theory and three-dimensional manifolds, Yale University Press, New Haven, Conn.-London, 1971. MR 0415622 (54:3705)

[SZ]
J. T. Stafford and J. J. Zhang, Homological properties of (graded) Noetherian PI rings, J. Algebra 168 (1994), no. 3, 988-1026. MR 1293638 (95h:16030)

[Ta]
E.J. Taft, The order of the antipode of finite-dimensional Hopf algebra, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 2631-2633. MR 0286868 (44:4075)

[WV]
A.J. Wilkie and L. van den Dries, An effective bound for groups of linear growth, Arch. Math. (Basel) 42 (1984), no. 5, 391-396. MR 0756689 (85k:20107)

[WZ1]
Q.-S. Wu and J.J. Zhang, Noetherian PI Hopf algebras are Gorenstein, Trans. Amer. Math. Soc. 355 (2003), no. 3, 1043-1066. MR 1938745 (2003m:16056)

[WZ2]
Q.-S. Wu and J.J. Zhang, Regularity of Involutory PI Hopf Algebras, J. Algebra 256 (2002), no. 2, 599-610. MR 1939124 (2004g:16042)

[WZ3]
Q.-S. Wu and J.J. Zhang, Homological identities for noncommutative rings, J. Algebra, 242 (2001), 516-535. MR 1848957 (2002k:16011)

[YZ]
A. Yekutieli and J.J. Zhang, Residual complex over noncommutative rings, J. Algebra 259 (2003), no. 2, 451-493. MR 1955528 (2004a:16010)

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Additional Information:

D.-M. Lu
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People's Republic of China
Email: dmlu@zju.edu.cn

Q.-S. Wu
Affiliation: Institute of Mathematics, Fudan University, Shanghai, 200433, People's Republic of China
Email: qswu@fudan.edu.cn

J. J. Zhang
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: zhang@math.washington.edu

DOI: 10.1090/S0002-9947-07-04159-1
PII: S 0002-9947(07)04159-1
Keywords: Hopf algebra, homological integral, Gorenstein property, regularity, Gelfand-Kirillov dimension, integral order, integral quotient, PI degree
Received by editor(s): May 16, 2005
Received by editor(s) in revised form: July 11, 2005
Posted: May 16, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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