Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-19T03:46:47.629Z Has data issue: false hasContentIssue false
  • English
  • Français

NON-AFFINE HOPF ALGEBRA DOMAINS OF GELFAND–KIRILLOV DIMENSION TWO

Published online by Cambridge University Press:  20 March 2017

K. R. GOODEARL  and
J. J. ZHANG
K. R. GOODEARL
Affiliation:
Department of Mathematics University of California at Santa Barbara, Santa Barbara, CA 93106, USA e-mail: goodearl@math.ucsb.edu
J. J. ZHANG
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA e-mail: zhang@math.washington.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We classify all non-affine Hopf algebras H over an algebraically closed field k of characteristic zero that are integral domains of Gelfand–Kirillov dimension two and satisfy the condition Ext1H(k, k) ≠ 0. The affine ones were classified by the authors in 2010 (Goodearl and Zhang, J. Algebra324 (2010), 3131–3168).

Keywords

16T0557T0516P9016E65
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 3 , September 2017 , pp. 563 - 593
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Berstein, I., On the dimension of modules and algebras. IX. Direct limits, Nagoya Math. J. 13 (1958), 83–84.CrossRefGoogle Scholar
2. Brown, K. A. and Gilmartin, P., Hopf algebras under finiteness conditions, Palest. J. Math. 3 (2014), 356–365, Special issue.Google Scholar
3. Brown, K. A., O'Hagan, S., Zhang, J. J. and Zhuang, G., Connected Hopf algebras and iterated Ore extensions, J. Pure Appl. Algebra, 219 (2015), 2405–2433.CrossRefGoogle Scholar
4. Chirvasitu, A., Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras, Algebra Number Theory 8 (5) (2014), 1179–1199.CrossRefGoogle Scholar
5. Goodearl, K. R., Noetherian Hopf algebras, Glasg. Math. J. 55 (A) (2013), 75–87.CrossRefGoogle Scholar
6. Goodearl, K. R. and Zhang, J. J., Noetherian Hopf algebra domains of Gelfand-Kirillov dimension two, J. Algebra 324 (2010), 3131–3168.CrossRefGoogle Scholar
7. Gromov, M., Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73.CrossRefGoogle Scholar
8. Huh, C. and Kim, C. O., Gelfand-Kirillov dimension of skew polynomial rings of automorphism type, Commun. Algebra 24 (1996), 2317–2323.CrossRefGoogle Scholar
9. Lu, D.-M., Wu, Q.-S. and Zhang, J. J., Homological integral of Hopf algebras, Trans. Amer. Math. Soc. 359 (2007), 4945–4975.CrossRefGoogle Scholar
10. McConnell, J. C. and Robson, J. C., Noncommutative noetherian rings (Wiley, Chichester, 1987).Google Scholar
11. Montgomery, S., Hopf algebras and their actions on rings, CBMS regional conference series in mathematics, vol. 82 (AMS, Providence, R1, 1993).CrossRefGoogle Scholar
12. Nichols, W. D. and Zoeller, M. B., A Hopf algebra freeness theorem, Amer. J. Math. 111 (1989), 381–385.CrossRefGoogle Scholar
13. Osofsky, B. L., Upper bounds on homological dimensions, Nagoya Math. J. 32 (1968), 315–322.CrossRefGoogle Scholar
14. Panov, A. N., Ore extensions of Hopf algebras, Mat. Zametki 74 (2003), 425–434.Google Scholar
15. Radford, D. E., Operators on Hopf algebras, Amer. J. Math. 99 (1977), 139–158.CrossRefGoogle Scholar
16. Radford, D. E., Pointed Hopf algebras are free over Hopf subalgebras, J. Algebra 45 (2) (1977), 266–273.CrossRefGoogle Scholar
17. Schauenburg, P., Faithful flatness over Hopf subalgebras: Counterexamples, in Interactions between ring theory and representations of algebras (Murcia, Spain 1998) (F. Van Oystaeyen and M. Saorí n, Editors) (New York, Dekker, 2000), 331–344.Google Scholar
18. Skryabin, S., New results on the bijectivity of antipode of a Hopf algebra, J. Algebra 306 (2) (2006), 622–633.CrossRefGoogle Scholar
19. Takeuchi, M., A correspondence between Hopf ideals and sub-Hopf algebras, Manuscripta Math. 7 (1972), 251–270.CrossRefGoogle Scholar
20. Takeuchi, M., Relative Hopf modules–-Equivalences and freeness criteria, J. Algebra 60 (1979), 452–471.CrossRefGoogle Scholar
21. 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.CrossRefGoogle Scholar
22. Wang, D.-G., Zhang, J. J. and Zhuang, G., Lower bounds of growth of Hopf algebras, Trans. Amer. Math. Soc. 365 (9) (2013), 4963–4986.CrossRefGoogle Scholar
23. Wang, D.-G., Zhang, J. J. and Zhuang, G., Primitive cohomology of Hopf algebras, J. Algebra 464 (2016), 36–96.CrossRefGoogle Scholar
24. Wu, Q.-S. and Zhang, J. J., Noetherian PI Hopf algebras are Gorenstein, Trans. Amer. Math. Soc. 355 (2002), 1043–1066.CrossRefGoogle Scholar
25. Zhuang, G., Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension, J. London Math. Soc. (2) 87 (3) (2013), 877–898.CrossRefGoogle Scholar