Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-23T11:11:48.215Z Has data issue: false hasContentIssue false
  • English
  • Français

CLASSIFICATION OF QUIVER HOPF ALGEBRAS AND POINTED HOPF ALGEBRAS OF TYPE ONE

Published online by Cambridge University Press:  06 August 2012

SHOUCHUAN ZHANG ,
HUI-XIANG CHEN  and
YAO-ZHONG ZHANG
SHOUCHUAN ZHANG
Affiliation:
Department of Mathematics, Hunan University, Changsha 410082, PR China School of Mathematics and Physics, The University of Queensland, Brisbane 4072, Australia
HUI-XIANG CHEN
Affiliation:
Department of Mathematics, Yangzhou University, Yangzhou 225002, PR China
YAO-ZHONG ZHANG*
Affiliation:
School of Mathematics and Physics, The University of Queensland, Brisbane 4072, Australia (email: yzz@maths.uq.edu.au)
*
✉For correspondence; e-mail: yzz@maths.uq.edu.au
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.

Quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.

Keywords

quiversHopf algebrasHopf bimodulesNichols algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 2 , April 2013 , pp. 216 - 237
Copyright
©2012 Australian Mathematical Publishing Association Inc.

References

[1]Andruskiewitsch, N. & Schneider, H. J., ‘Lifting of quantum linear spaces and pointed Hopf algebras of order p 3’, J. Algebra 209 (1998), 645–691.CrossRefGoogle Scholar
[2]Andruskiewitsch, N. & Schneider, H. J., ‘Pointed Hopf algebras’, in: New Directions in Hopf Algebras, Mathematical Sciences Research Institute Publications, 43 (Cambridge University Press, Cambridge, 2002), pp. 1–68.Google Scholar
[3]Andruskiewitsch, N. & Schneider, H. J., ‘On the classification of finite-dimensional pointed Hopf algebras’, Ann. of Math. (2) 171 (2010), 375–417.CrossRefGoogle Scholar
[4]Andruskiewitsch, N. & Zhang, S., ‘On pointed Hopf algebras associated to some conjugacy classes in S n’, Proc. Amer. Math. Soc. 135 (2007), 2723–2731.CrossRefGoogle Scholar
[5]Bespalov, Y. & Drabant, B., ‘Hopf (bi-)modules and crossed modules in braided monoidal categories’, J. Pure Appl. Algebra 123 (1998), 105–129.CrossRefGoogle Scholar
[6]Cibils, C. & Rosso, M., ‘Algebres des chemins quantiques’, Adv. Math. 125 (1997), 171–199.CrossRefGoogle Scholar
[7]Cibils, C. & Rosso, M., ‘Hopf quivers’, J. Algebra 254 (2002), 241–251.CrossRefGoogle Scholar
[8]Dijkgraaf, R., Pasquier, V. & Roche, P., ‘Quasi Hopf algebras, group cohomology and orbifold models’, Nucl. Phys. B Proc. Suppl. 18 (1991), 60–72.CrossRefGoogle Scholar
[9]Gaberdiel, M. R., ‘An algebraic approach to logarithmic conformal field theory’, Int. J. Mod. Phys. A 18 (2003), 4593–4638.CrossRefGoogle Scholar
[10]Heckenberger, I., ‘The Weyl groupoid of a Nichols algebra of diagonal type’, Invent. Math. 164 (2006), 175–188.CrossRefGoogle Scholar
[11]Majid, S, ‘Quasi-triangular Hopf algebras and Yang-Baxter equations’, Int. J. Mod. Phys. A 5 (1990), 1–91.CrossRefGoogle Scholar
[12]Nichols, W., ‘Bialgebras of type one’, Comm. Algebra 6 (1978), 1521–1552.CrossRefGoogle Scholar
[13]Sweedler, M. E., Hopf Algebras (Benjamin, New York, 1969).Google Scholar
[14]Van Oystaeyen, F. & Zhang, P., ‘Quiver Hopf algebras’, J. Algebra 280 (2004), 577–589.CrossRefGoogle Scholar
[15]Woronowicz, S. L., ‘Differential calculus on compact matrix pseudogroups (quantum groups)’, Comm. Math. Phys. 122 (1989), 125–170.CrossRefGoogle Scholar
[16]Zhang, S., Braided Hopf Algebras (Hunan Normal University Press, Chungsha, 1999).Google Scholar
[17]Zhang, S., ‘The double bicrossproducts of braided Hopf algebras’, Comm. Algebra 29 (2001), 31–66.CrossRefGoogle Scholar
[18]Zhang, S., Wu, M. & Wang, H., ‘Classification of ramification systems for symmetric groups’, Acta Math. Sin. 51 (2008), 253–264.Google Scholar
[19]Zhang, S. & Zhang, Y.-Z., ‘Structures and representations of generalized path algebras’, Algebr. Represent. Theor. 10 (2007), 117–134.CrossRefGoogle Scholar
[20]Zhang, S., Zhang, Y.-Z. & Chen, H. X., ‘Classification of PM quiver Hopf algebras’, J. Algebra Appl. 6 (2007), 919–950.CrossRefGoogle Scholar