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Nichols algebras with many cubic relations


Authors: I. Heckenberger, A. Lochmann and L. Vendramin
Journal: Trans. Amer. Math. Soc. 367 (2015), 6315-6356
MSC (2010): Primary 16T05; Secondary 20F99, 16P90
DOI: https://doi.org/10.1090/S0002-9947-2015-06231-X
Published electronically: January 30, 2015
MathSciNet review: 3356939
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Abstract: Nichols algebras of group type with many cubic relations are classified under a technical assumption on the structure of Hurwitz orbits of the third power of the underlying indecomposable rack. All such Nichols algebras are finite-dimensional, and their Hilbert series have a factorization into quantum integers. Also, all known finite-dimensional elementary Nichols algebras turn out to have many cubic relations. The technical assumption of our theorem can be removed if a conjecture in the theory of cellular automata can be proven.


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  • 1. Nicolás Andruskiewitsch and Matías Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), no. 2, 177-243. MR 1994219 (2004i:16046)
  • 2. Nicolás Andruskiewitsch and Hans-Jürgen Schneider, Pointed Hopf algebras, New directions in Hopf algebras, Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 1-68. MR 1913436 (2003e:16043)
  • 3. E. Brieskorn, Automorphic sets and braids and singularities, Braids (Santa Cruz, CA, 1986), Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 45-115. MR 975077 (90a:32024)
  • 4. Tullio Ceccherini-Silberstein and Michel Coornaert, Cellular automata and groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. MR 2683112 (2011j:37002)
  • 5. M. Graña, I. Heckenberger, and L. Vendramin, Nichols algebras of group type with many quadratic relations, Adv. Math. 227 (2011), no. 5, 1956-1989. MR 2803792 (2012f:16077)
  • 6. I. Heckenberger, A. Lochmann, and L. Vendramin, Braided racks, Hurwitz actions and Nichols algebras with many cubic relations, Transform. Groups 17 (2012), no. 1, 157-194. MR 2891215
  • 7. A. Hurwitz, Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), no. 1, 1-60. MR 1510692
  • 8. Christian Kassel and Vladimir Turaev, Braid groups, with the graphical assistance of Olivier Dodane. Graduate Texts in Mathematics, vol. 247, Springer, New York, 2008. MR 2435235 (2009e:20082)
  • 9. Robert A. Rankin, Modular forms and functions, Cambridge University Press, Cambridge, 1977. MR 0498390 (58:16518)
  • 10. L. Vendramin, On the classification of quandles of low order, J. Knot Theory Ramifications 21 (2012), no. 9, 1250088, 10. MR 2926571

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Additional Information

I. Heckenberger
Affiliation: FB Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein- Straße, 35032 Marburg, Germany
Email: heckenberger@mathematik.uni-marburg.de

A. Lochmann
Affiliation: FB Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein- Straße, 35032 Marburg, Germany
Email: lochmann@mathematik.uni-marburg.de

L. Vendramin
Affiliation: FB Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein- Straße, 35032 Marburg, Germany
Email: lvendramin@dm.uba.ar

DOI: https://doi.org/10.1090/S0002-9947-2015-06231-X
Received by editor(s): February 3, 2013
Received by editor(s) in revised form: June 25, 2013
Published electronically: January 30, 2015
Additional Notes: The first author was supported by the German Research Foundation via a Heisenberg professorship
The third author was supported by CONICET and the Alexander von Humboldt Foundation
Article copyright: © Copyright 2015 American Mathematical Society

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