Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras

Christian Kassel[1]

  • [1] Institut de Recherche Mathématique Avancée, CNRS & Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg, France

Annales mathématiques Blaise Pascal (2013)

  • Volume: 20, Issue: 2, page 175-191
  • ISSN: 1259-1734

Abstract

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We define polynomial H -identities for comodule algebras over a Hopf algebra  H and establish general properties for the corresponding T -ideals. In the case  H is a Taft algebra or the Hopf algebra  E ( n ) , we exhibit a finite set of polynomial H -identities which distinguish the Galois objects over  H up to isomorphism.

How to cite

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Kassel, Christian. "Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras." Annales mathématiques Blaise Pascal 20.2 (2013): 175-191. <http://eudml.org/doc/275545>.

@article{Kassel2013,
abstract = {We define polynomial $H$-identities for comodule algebras over a Hopf algebra $H$ and establish general properties for the corresponding $T$-ideals. In the case $H$ is a Taft algebra or the Hopf algebra $E(n)$, we exhibit a finite set of polynomial $H$-identities which distinguish the Galois objects over $H$ up to isomorphism.},
affiliation = {Institut de Recherche Mathématique Avancée, CNRS & Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg, France},
author = {Kassel, Christian},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Hopf algebra; comodule algebra; polynomial identity; Hopf algebras; comodule algebras; polynomial identities; T-ideals; Galois objects; Taft algebras; algebras of non-commutative polynomials; tensor algebras},
language = {eng},
month = {7},
number = {2},
pages = {175-191},
publisher = {Annales mathématiques Blaise Pascal},
title = {Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras},
url = {http://eudml.org/doc/275545},
volume = {20},
year = {2013},
}

TY - JOUR
AU - Kassel, Christian
TI - Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras
JO - Annales mathématiques Blaise Pascal
DA - 2013/7//
PB - Annales mathématiques Blaise Pascal
VL - 20
IS - 2
SP - 175
EP - 191
AB - We define polynomial $H$-identities for comodule algebras over a Hopf algebra $H$ and establish general properties for the corresponding $T$-ideals. In the case $H$ is a Taft algebra or the Hopf algebra $E(n)$, we exhibit a finite set of polynomial $H$-identities which distinguish the Galois objects over $H$ up to isomorphism.
LA - eng
KW - Hopf algebra; comodule algebra; polynomial identity; Hopf algebras; comodule algebras; polynomial identities; T-ideals; Galois objects; Taft algebras; algebras of non-commutative polynomials; tensor algebras
UR - http://eudml.org/doc/275545
ER -

References

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  1. E. Aljadeff, D. Haile, Simple G -graded algebras and their polynomial identities, Trans. Amer. Math. Soc., to appear Zbl1297.16019
  2. E. Aljadeff, D. Haile, M. Natapov, Graded identities of matrix algebras and the universal graded algebra, Trans. Amer. Math. Soc. 362 (2010), 3125-3147 Zbl1203.16020MR2592949
  3. E. Aljadeff, C. Kassel, Polynomial identities and noncommutative versal torsors, Adv. Math. 218 (2008), 1453-1495 Zbl1152.16026MR2419929
  4. A. S. Amitsur, J. Levitzki, Minimal identities for algebras, Proc. Amer. Math. Soc. 1 (1950), 449-463 Zbl0040.01101MR36751
  5. Y. A. Bahturin, V. Linchenko, Identities of algebras with actions of Hopf algebras, J. Algebra 202 (1998), 634-654 Zbl0912.16009MR1617671
  6. Y. A. Bahturin, M. Zaicev, Identities of graded algebras, J. Algebra 205 (1998), 1-12 Zbl0920.16011MR1631298
  7. M. Beattie, S. Dăscălescu, L. Grünenfelder, Constructing pointed Hopf algebras by Ore extensions, J. Algebra 225 (2000), 743-770 Zbl0948.16026MR1741560
  8. A. Berele, Cocharacter sequences for algebras with Hopf algebra actions, J. Algebra 185 (1996), 869-885 Zbl0864.16019MR1419727
  9. J. Bichon, Galois and bigalois objects over monomial non-semisimple Hopf algebras, J. Algebra Appl. 5 (2006), 653-680 Zbl1117.16025MR2269410
  10. J. Bichon, G. Carnovale, Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras, J. Pure Appl. Algebra 204 (2006), 627-665 Zbl1084.16028MR2185622
  11. X.-W. Chen, H.-L. Huang, Y. Ye, P. Zhang, Monomial Hopf algebras, J. Algebra 275 (2004), 212-232 Zbl1071.16030MR2047446
  12. Y. Doi, M. Takeuchi, Quaternion algebras and Hopf crossed products, Comm. Algebra 23 (1995), 3291-3325 Zbl0833.16036MR1335303
  13. P. Koshlukov, M. Zaicev, Identities and isomorphisms of graded simple algebras, Linear Algebra Appl. 432 (2010), 3141-3148 Zbl1194.16016MR2639274
  14. A. Masuoka, Cleft extensions for a Hopf algebra generated by a nearly primitive element, Comm. Algebra 22 (1994), 4537-4559 Zbl0809.16046MR1284344
  15. S. Montgomery, Hopf algebras and their actions on rings, (1993), Amer. Math. Soc., Providence Zbl0793.16029MR1243637
  16. A. Nenciu, Cleft extensions for a class of pointed Hopf algebras constructed by Ore extensions, Comm. Algebra 29 (2001), 1959-1981 Zbl0991.16034MR1837954
  17. F. Panaite, F. van Oystaeyen, Quasitriangular structures for some pointed Hopf algebras of dimension  2 n , Comm. Algebra 27 (1999), 4929-4942 Zbl0943.16019MR1701714
  18. L. Rowen, Polynomial identities in ring theory, (1980), Academic Press, Inc., New York–London Zbl0461.16001MR576061
  19. M. Sweedler, Hopf algebras, (1969), W. A. Benjamin, Inc., New York Zbl0194.32901MR252485
  20. E. J. Taft, The order of the antipode of finite-dimensional Hopf algebra, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 2631-2633 Zbl0222.16012MR286868

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