Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras
Annales mathématiques Blaise Pascal, Volume 20 (2013) no. 2, pp. 175-191.

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.

Nous définissons le concept d’identité polynomiale pour une algèbre-comodule sur une algèbre de Hopf H. Nous présentons des identités polynomiales explicites distinguant à isomorphisme près les objets galoisiens d’une algèbre de Taft ou de l’algèbre de Hopf E(n).

DOI: 10.5802/ambp.325
Classification: 16R50,  16T05,  16T15
Keywords: Hopf algebra, comodule algebra, polynomial identity
Christian Kassel 1

1 Institut de Recherche Mathématique Avancée, CNRS & Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg, France
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Christian Kassel. Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras. Annales mathématiques Blaise Pascal, Volume 20 (2013) no. 2, pp. 175-191. doi : 10.5802/ambp.325. https://ambp.centre-mersenne.org/articles/10.5802/ambp.325/

[1] E. Aljadeff; D. Haile Simple G-graded algebras and their polynomial identities, Trans. Amer. Math. Soc., to appear (arXiv:1107.4713)

[2] E. Aljadeff; D. Haile; M. Natapov Graded identities of matrix algebras and the universal graded algebra, Trans. Amer. Math. Soc., Volume 362 (2010), pp. 3125-3147 | DOI | MR | Zbl

[3] E. Aljadeff; C. Kassel Polynomial identities and noncommutative versal torsors, Adv. Math., Volume 218 (2008), pp. 1453-1495 | DOI | MR | Zbl

[4] A. S. Amitsur; J. Levitzki Minimal identities for algebras, Proc. Amer. Math. Soc., Volume 1 (1950), pp. 449-463 | DOI | MR | Zbl

[5] Y. A. Bahturin; V. Linchenko Identities of algebras with actions of Hopf algebras, J. Algebra, Volume 202 (1998), pp. 634-654 | DOI | MR | Zbl

[6] Y. A. Bahturin; M. Zaicev Identities of graded algebras, J. Algebra, Volume 205 (1998), pp. 1-12 | DOI | MR | Zbl

[7] M. Beattie; S. Dăscălescu; L. Grünenfelder Constructing pointed Hopf algebras by Ore extensions, J. Algebra, Volume 225 (2000) no. 2, pp. 743-770 | DOI | MR | Zbl

[8] A. Berele Cocharacter sequences for algebras with Hopf algebra actions, J. Algebra, Volume 185 (1996), pp. 869-885 | DOI | MR | Zbl

[9] J. Bichon Galois and bigalois objects over monomial non-semisimple Hopf algebras, J. Algebra Appl., Volume 5 (2006), pp. 653-680 | DOI | MR | Zbl

[10] J. Bichon; G. Carnovale Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras, J. Pure Appl. Algebra, Volume 204 (2006), pp. 627-665 | DOI | MR | Zbl

[11] X.-W. Chen; H.-L. Huang; Y. Ye; P. Zhang Monomial Hopf algebras, J. Algebra, Volume 275 (2004), pp. 212-232 | DOI | MR | Zbl

[12] Y. Doi; M. Takeuchi Quaternion algebras and Hopf crossed products, Comm. Algebra, Volume 23 (1995), pp. 3291-3325 | DOI | MR | Zbl

[13] P. Koshlukov; M. Zaicev Identities and isomorphisms of graded simple algebras, Linear Algebra Appl., Volume 432 (2010), pp. 3141-3148 | DOI | MR | Zbl

[14] A. Masuoka Cleft extensions for a Hopf algebra generated by a nearly primitive element, Comm. Algebra, Volume 22 (1994), pp. 4537-4559 | DOI | MR | Zbl

[15] S. Montgomery Hopf algebras and their actions on rings, Amer. Math. Soc., Providence, 1993 | MR | Zbl

[16] A. Nenciu Cleft extensions for a class of pointed Hopf algebras constructed by Ore extensions, Comm. Algebra, Volume 29 (2001), pp. 1959-1981 | DOI | MR | Zbl

[17] F. Panaite; F. van Oystaeyen Quasitriangular structures for some pointed Hopf algebras of dimension 2 n , Comm. Algebra, Volume 27 (1999), pp. 4929-4942 | DOI | MR | Zbl

[18] L. Rowen Polynomial identities in ring theory, Academic Press, Inc., New York–London, 1980 | MR | Zbl

[19] M. Sweedler Hopf algebras, W. A. Benjamin, Inc., New York, 1969 | MR | Zbl

[20] E. J. Taft The order of the antipode of finite-dimensional Hopf algebra, Proc. Nat. Acad. Sci. U.S.A., Volume 68 (1971), pp. 2631-2633 | DOI | MR | Zbl

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