NOETHERIAN HOPF ALGEBRAS
Glasgow mathematical journal, Tome 55 (2013) no. A, pp. 75-87

Voir la notice de l'article provenant de la source Cambridge University Press

A brief survey of some aspects of noetherian Hopf algebras is given, concentrating on structure, homology, and classification, and accompanied by a panoply of open problems.
DOI : 10.1017/S0017089513000517
Mots-clés : 16T05, 16E65, 16P40
GOODEARL, K. R. NOETHERIAN HOPF ALGEBRAS. Glasgow mathematical journal, Tome 55 (2013) no. A, pp. 75-87. doi: 10.1017/S0017089513000517
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[1] 1.Andruskiewitsch, N. and Angiono, I. E., On Nichols algebras with generic braiding, in Modules and Comodules (Porto 2006) (Brzeziński, T., et al., Editors) (Birkhäuser Verlag, Basel, 2008), 47–64. Google Scholar

[2] 2.Andruskiewitsch, N. and Cuadra, J., On the structure of (co-Frobenius) Hopf algebras, J. Noncomm. Geom. 7 (2013), 83–104. Google Scholar

[3] 3.Andruskiewitsch, N., Etingof, P. and Gelaki, S., Triangular Hopf algebras with the Chevalley property, Michigan Math. J. 49 (2001), 277–298. Google Scholar | DOI

[4] 4.Andruskiewitsch, N. and Schneider, H.-J., Finite quantum groups and Cartan matrices, Adv. Math. 154 (2000), 1–45. Google Scholar

[5] 5.Andruskiewitsch, N. and Schneider, H.-J., A characterization of quantum groups, J. Reine Angew. Math. 577 (2004), 81–104. Google Scholar

[6] 6.Angiono, I. E., On Nichols algebras of diagonal type, J. Reine Angew. Math. (in press) (arXiv:1104.0268). Google Scholar

[7] 7.Brown, K. A., Representation theory of Noetherian Hopf algebras satisfying a polynomial identity, in Trends in the representation theory of finite dimensional algebras (Seattle 1997) (Green, E. L. and Huisgen-Zimmermann, B., Editors), Contemporary Mathematics, Vol. 229 (AMS, Providence, RI, 1998), 49–79. Google Scholar

[8] 8.Brown, K. A., Noetherian Hopf algebras, Turkish J. Math. 31 (2007), suppl., 7–23. Google Scholar

[9] 9.Brown, K. A. and Goodearl, K. R., Homological aspects of Noetherian PI Hopf algebras and irreducible modules of maximal dimension, J. Algebra 198 (1997), 240–265. Google Scholar

[10] 10.Brown, K. A. and Goodearl, K. R., Lectures on algebraic quantum groups, (Advanced Courses in Mathematics CRM Barcelona) (Birkhäuser Verlag, Basel, 2002). Google Scholar | DOI

[11] 11.Brown, K. A. and Zhang, J. J., Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras, J. Algebra 320 (2008), 1814–1850. Google Scholar | DOI

[12] 12.Brown, K. A. and Zhang, J. J., Prime regular Hopf algebras of GK-dimension one, Proc. London Math. Soc. (3) 101 (2010), 260–302. Google Scholar | DOI

[13] 13.Drinfel'D, V., Quantum groups, in Proceedings of the International Congress of Mathematicians (Berkeley 1986), Vol. 1, (AMS, Providence, RI, 1987), 798–820. Google Scholar

[14] 14.Gelaki, S. and Letzter, E. S., An affine PI Hopf algebra not finite over a normal commutative Hopf subalgebra, Proc. Amer. Math. Soc. 131 (2003), 2673–2679. Google Scholar

[15] 15.Goodearl, K. R. and Zhang, J. J., Noetherian Hopf algebra domains of Gelfand-Kirillov dimension two, J. Algebra 324 (2010), 3131–3168. Google Scholar

[16] 16.Larson, R. G. and Sweedler, M. E., An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75–94. Google Scholar

[17] 17.Lin, B. I.-P., Semiperfect coalgebras, J. Algebra 49 (1977) 357–373. Google Scholar

[18] 18.Liu, C.-H. and Zhang, J. J., Artinian Hopf algebras are finite dimensional, Proc. Amer. Math. Soc. 135 (2007), 1679–1680. Google Scholar | DOI

[19] 19.Liu, G., On noetherian affine prime regular Hopf algebras of Gelfand–Kirillov dimension 1, Proc. Amer. Math. Soc. 137 (2009), 777–785. Google Scholar

[20] 20.Lorenz, M. E. and Lorenz, M., On crossed products of Hopf algebras, Proc. Amer. Math. Soc. 123 (1995), 33–38. Google Scholar

[21] 21.Lu, D.-M., Wu, Q.-S. and Zhang, J. J., Homological integral of Hopf algebras, Trans. Amer. Math. Soc. 359 (2007), 4945–4975. Google Scholar

[22] 22.Lu, D.-M., Wu, Q.-S. and Zhang, J. J., Hopf algebras with rigid dualizing complexes, Israel J. Math. 169 (2009), 89–108. Google Scholar | DOI

[23] 23.Molnar, R. K., A commutative Noetherian Hopf algebra over a field is finitely generated, Proc. Amer. Math. Soc. 51 (1975), 501–502. Google Scholar

[24] 24.Montgomery, S., Hopf Algebras and their Actions on Rings, CBMS Regional Conference Series in Mathematics 82 (AMS, Providence, 1993). Google Scholar

[25] 25.Radford, D. E., The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. Math. 98 (1976), 333–355. Google Scholar

[26] 26.Radford, D. E., Finiteness conditions for a Hopf algebra with a nonzero integral, J. Algebra 46 (1977), 189–195. Google Scholar

[27] 27.Skryabin, S., New results on the bijectivity of antipode of a Hopf algebra, J. Algebra 306 (2006), 622–633. Google Scholar | DOI

[28] 28.Small, L. W., Stafford, J. T. and Warfield, R. B. Jr., Affine algebras of Gelfand–Kirillov dimension one are PI, Math. Proc. Cambridge Phil. Soc. 97 (1985), 407–414. Google Scholar | DOI

[29] 29.Sweedler, M. E., Hopf algebras, (Benjamin, New York, 1969). Google Scholar

[30] 30.Takeuchi, M., Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), 561–582. Google Scholar

[31] 31.Takeuchi, M., There exists a Hopf algebra whose antipode is not injective, Sci. Papers College Gen. Ed. Univ. Tokyo 21 (1971), 127–130. Google Scholar

[32] 32.Wang, D.-G., Zhang, J. J. and Zhuang, G., Hopf algebras of GK-dimension two with vanishing Ext-group, J. Algebra 388 (2013), 219–247. Google Scholar

[33] 33.Wu, Q.-S. and Zhang, J. J., Regularity of involutory PI Hopf algebras, J. Algebra 256 (2002), 599–610. Google Scholar

[34] 34.Wu, Q.-S. and Zhang, J. J., Noetherian PI Hopf algebras are Gorenstein, Trans. Amer. Math. Soc. 355 (2003), 1043–1066. Google Scholar

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