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NOETHERIAN HOPF ALGEBRAS

Published online by Cambridge University Press:  01 October 2013

K. R. GOODEARL*
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA e-mail: goodearl@math.ucsb.edu
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Abstract

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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.

Keywords

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

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), 4764.Google Scholar
2.Andruskiewitsch, N. and Cuadra, J., On the structure of (co-Frobenius) Hopf algebras, J. Noncomm. Geom. 7 (2013), 83104.Google Scholar
3.Andruskiewitsch, N., Etingof, P. and Gelaki, S., Triangular Hopf algebras with the Chevalley property, Michigan Math. J. 49 (2001), 277298.CrossRefGoogle Scholar
4.Andruskiewitsch, N. and Schneider, H.-J., Finite quantum groups and Cartan matrices, Adv. Math. 154 (2000), 145.Google Scholar
5.Andruskiewitsch, N. and Schneider, H.-J., A characterization of quantum groups, J. Reine Angew. Math. 577 (2004), 81104.Google Scholar
6.Angiono, I. E., On Nichols algebras of diagonal type, J. Reine Angew. Math. (in press) (arXiv:1104.0268).Google Scholar
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), 4979.Google Scholar
8.Brown, K. A., Noetherian Hopf algebras, Turkish J. Math. 31 (2007), suppl., 723.Google Scholar
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), 240265.Google Scholar
10.Brown, K. A. and Goodearl, K. R., Lectures on algebraic quantum groups, (Advanced Courses in Mathematics CRM Barcelona) (Birkhäuser Verlag, Basel, 2002).CrossRefGoogle Scholar
11.Brown, K. A. and Zhang, J. J., Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras, J. Algebra 320 (2008), 18141850.CrossRefGoogle Scholar
12.Brown, K. A. and Zhang, J. J., Prime regular Hopf algebras of GK-dimension one, Proc. London Math. Soc. (3) 101 (2010), 260302.CrossRefGoogle Scholar
13.Drinfel'd, V., Quantum groups, in Proceedings of the International Congress of Mathematicians (Berkeley 1986), Vol. 1, (AMS, Providence, RI, 1987), 798820.Google Scholar
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), 26732679.Google Scholar
15.Goodearl, K. R. and Zhang, J. J., Noetherian Hopf algebra domains of Gelfand-Kirillov dimension two, J. Algebra 324 (2010), 31313168.Google Scholar
16.Larson, R. G. and Sweedler, M. E., An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 7594.Google Scholar
17.Lin, B. I.-P., Semiperfect coalgebras, J. Algebra 49 (1977) 357373.Google Scholar
18.Liu, C.-H. and Zhang, J. J., Artinian Hopf algebras are finite dimensional, Proc. Amer. Math. Soc. 135 (2007), 16791680.CrossRefGoogle Scholar
19.Liu, G., On noetherian affine prime regular Hopf algebras of Gelfand–Kirillov dimension 1, Proc. Amer. Math. Soc. 137 (2009), 777785.Google Scholar
20.Lorenz, M. E. and Lorenz, M., On crossed products of Hopf algebras, Proc. Amer. Math. Soc. 123 (1995), 3338.Google Scholar
21.Lu, D.-M., Wu, Q.-S. and Zhang, J. J., Homological integral of Hopf algebras, Trans. Amer. Math. Soc. 359 (2007), 49454975.Google Scholar
22.Lu, D.-M., Wu, Q.-S. and Zhang, J. J., Hopf algebras with rigid dualizing complexes, Israel J. Math. 169 (2009), 89108.CrossRefGoogle Scholar
23.Molnar, R. K., A commutative Noetherian Hopf algebra over a field is finitely generated, Proc. Amer. Math. Soc. 51 (1975), 501502.Google Scholar
24.Montgomery, S., Hopf Algebras and their Actions on Rings, CBMS Regional Conference Series in Mathematics 82 (AMS, Providence, 1993).Google Scholar
25.Radford, D. E., The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. Math. 98 (1976), 333355.Google Scholar
26.Radford, D. E., Finiteness conditions for a Hopf algebra with a nonzero integral, J. Algebra 46 (1977), 189195.Google Scholar
27.Skryabin, S., New results on the bijectivity of antipode of a Hopf algebra, J. Algebra 306 (2006), 622633.CrossRefGoogle Scholar
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), 407414.CrossRefGoogle Scholar
29.Sweedler, M. E., Hopf algebras, (Benjamin, New York, 1969).Google Scholar
30.Takeuchi, M., Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), 561582.Google Scholar
31.Takeuchi, M., There exists a Hopf algebra whose antipode is not injective, Sci. Papers College Gen. Ed. Univ. Tokyo 21 (1971), 127130.Google Scholar
32.Wang, D.-G., Zhang, J. J. and Zhuang, G., Hopf algebras of GK-dimension two with vanishing Ext-group, J. Algebra 388 (2013), 219247.Google Scholar
33.Wu, Q.-S. and Zhang, J. J., Regularity of involutory PI Hopf algebras, J. Algebra 256 (2002), 599610.Google Scholar
34.Wu, Q.-S. and Zhang, J. J., Noetherian PI Hopf algebras are Gorenstein, Trans. Amer. Math. Soc. 355 (2003), 10431066.Google Scholar