Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

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
MathSciNet review: 2320655
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16A62, 16W30, 16E70, 20J50

Retrieve articles in all Journals with MSC (2000): 16A62, 16W30, 16E70, 20J50

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.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia