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Braided racks, Hurwitz actions and Nichols algebras with many cubic relations

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Abstract

We classify Nichols algebras of irreducible Yetter–Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption turns out to be equivalent to a factorization assumption on the Hilbert series. Besides the known Nichols algebras we obtain a new example. Our method is based on a combinatorial invariant of the Hurwitz orbits with respect to the action of the braid group on three strands.

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References

  1. A. Ardizzoni, M. Beattie, C. Menini, Cocycle deformations for Hopf algebras with a coalgebra projection, J. Algebra 324 (2010), no. 4, 673–705.

    Article  MathSciNet  MATH  Google Scholar 

  2. N. Andruskiewitsch, F. Fantino, G. A. Garcia, L. Vendramin, On Nichols algebras associated to simple racks, Contemp. Math. 537 (2011) 31–56.

    Article  MathSciNet  Google Scholar 

  3. N. Andruskiewitsch, F. Fantino, M. Graña, L. Vendramin, Finite-dimensional pointed Hopf algebras with alternating groups are trivial, Ann. Mat. Pura Appl. (4) 190 (2011), no. 2, 225–245.

    MathSciNet  MATH  Google Scholar 

  4. N. Andruskiewitsch, F. Fantino, M. Graña, L. Vendramin, Pointed Hopf algebras over the sporadic simple groups, J. Algebra 325 (2011), no. 1, 305–320.

    Article  MathSciNet  MATH  Google Scholar 

  5. N. Andruskiewitsch, M. Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), no. 2, 177–243.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Aschbacher, M. Hall, Jr., Groups generated by a class of elements of order 3, J. Algebra 24 (1973), 591–612.

    Article  MathSciNet  MATH  Google Scholar 

  7. N. Andruskiewitsch, I. Heckenberger, H.-J. Schneider, The Nichols algebra of a semisimple Yetter–Drinfeld module, Amer. J. Math. 132 (2010), no. 6, 1493–1547.

    MathSciNet  MATH  Google Scholar 

  8. N. Andruskiewitsch, About finite dimensional Hopf algebras, in: Quantum Symmetries in Theoretical Physics and Mathematics (Bariloche, 2000), Contemp. Math., Vol. 294, Amer. Math. Soc., Providence, RI, 2002, pp. 1–57.

    Chapter  Google Scholar 

  9. I. Angiono, A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems, preprint: arXiv:1008.4144.

  10. I. Angiono, Nichols algebras with standard braiding, Algebra Number Theory 3 (2009), no. 1, 35–106.

    Article  MathSciNet  MATH  Google Scholar 

  11. N. Andruskiewitsch, D. Radford, H.-J. Schneider, Complete reducibility theorems for modules over pointed Hopf algebras, J. Algebra 324 (2010), no. 11, 2932–2970.

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Andruskiewitsch, H.-J. Schneider, Lifting of quantum linear spaces and pointed Hopf algebras of order p 3, J. Algebra 209 (1998), no. 2, 658–691.

    Article  MathSciNet  MATH  Google Scholar 

  13. N. Andruskiewitsch, H.-J. Schneider, Finite quantum groups and Cartan matrices, Adv. Math. 154 (2000), no. 1, 1–45.

    Article  MathSciNet  MATH  Google Scholar 

  14. N. Andruskiewitsch, H.-J. Schneider, Pointed Hopf algebras, in: New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ., Vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 1–68.

    Google Scholar 

  15. N. Andruskiewitsch, H.-J. Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. of Math. (2) 171 (2010), no. 1, 375–417.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Aschbacher, 3-Transposition Groups, Cambridge Tracts in Mathematics, Vol. 124, Cambridge University Press, Cambridge, 1997.

    MATH  Google Scholar 

  17. Y. Bazlov, Nichols–Woronowicz algebra model for Schubert calculus on Coxeter groups, J. Algebra 297 (2006), no. 2, 372–399.

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Balister, B. Bollobás, J. R. Johnson, M. Walters, Random majority percolation, Random Structures Algorithms 36 (2010), no. 3, 315–340.

    MathSciNet  MATH  Google Scholar 

  19. M. Beattie, S. Dăscălescu, Ş. Raianu, Lifting of Nichols algebras of type B 2, Israel J. Math. 132 (2002), 1–28. With an appendix by the authors and I. Rutherford.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Brieskorn, Automorphic sets and braids and singularities, in: Braids (Santa Cruz, CA, 1986), Contemp. Math., Vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 45–115.

    Google Scholar 

  21. J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford, M. Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947–3989.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. M. Cohen, J.W. Knopper, GBNP, A GAP Package for Gröbner Bases of Noncommutative Polynomials, available at http://dam02.win.tue.nl/products/gbnp/.

  23. H. Coxeter, Factor groups of the braid group, Proc. of the Fourth Can. Math. Congress, Banff 1957, University of Toronto Press, 1959, pp. 95–122.

  24. D. Didt, Pointed Hopf algebras and quasi-isomorphisms, Algebr. Represent. Theory 8 (2005), no. 3, 347–362. MR 2176141 (2006g:16084)

    Article  MathSciNet  MATH  Google Scholar 

  25. B. Fischer, Finite groups generated by 3-transpositions. I, Invent. Math. 13 (1971), 232–246.

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Fomin, A. N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, in: Advances in Geometry, Progr. Math., Vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 147–182.

    Chapter  Google Scholar 

  27. M. R. Gaberdiel, An algebraic approach to logarithmic conformal field theory, in: Proceedings of the School and Workshop on Logarithmic Conformal Field Theory and its Applications (Tehran, 2001), Vol. 18, 2003, pp. 4593–4638.

  28. The GAP Group, 2006, GAP—Groups, Algorithms, and Programming, Version 4.4.12, available at http://www.gap-system.org.

  29. G. A. García, A. García Iglesias, Finite dimensional pointed Hopf algebras over S 4, Israel J. Math. 183 (2011), 417–444.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Graña, I. Heckenberger, On a factorization of graded Hopf algebras using Lyndon words, J. Algebra 314 (2007), no. 1, 324–343.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Graña, I. Heckenberger, L. Vendramin, Nichols algebras of group type with many quadratic relations, Adv. Math. 227 (2011), no. 5, 1956–1989.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Graña, Nichols algebras of nonabelian group type, zoo of examples, available at http://mate.dm.uba.ar/~matiasg/zoo.html.

  33. M. Graña, Quandle knot invariants are quantum knot invariants, J. Knot Theory Ramifications 11 (2002), no. 5, 673–681.

    Article  MathSciNet  MATH  Google Scholar 

  34. I. Heckenberger, The Weyl groupoid of a Nichols algebra of diagonal type, Invent. Math. 164 (2006), no. 1, 175–188.

    Article  MathSciNet  MATH  Google Scholar 

  35. I. Heckenberger, Classification of arithmetic root systems, Adv. Math. 220 (2009), no. 1, 59–124.

    Article  MathSciNet  MATH  Google Scholar 

  36. I. Heckenberger, H.-J. Schneider, Right coideal subalgebras of Nichols algebras and the Duo order on the Weyl groupoid, preprint: arXiv:0909.0293.

  37. I. Heckenberger, H.-J. Schneider, Root systems and Weyl groupoids for Nichols algebras, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 623–654.

    Article  MathSciNet  MATH  Google Scholar 

  38. J. I. Hall, L. H. Soicher, Presentations of some 3-transposition groups, Comm. Algebra 23 (1995), no. 7, 2517–2559.

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), no. 1, 1–60.

    Article  MathSciNet  MATH  Google Scholar 

  40. V. K. Kharchenko, A quantum analogue of the Poincaré–Birkhoff–Witt theorem, Algebra and Logic 38 (1999), no. 4, 259–276.

    Article  MathSciNet  Google Scholar 

  41. A. N. Kirillov, T. Maeno, Nichols–Woronowicz model of coinvariant algebra of complex reflection groups, J. Pure Appl. Algebra 214 (2010), no. 4, 402–409.

    Article  MathSciNet  MATH  Google Scholar 

  42. L. Krop, D. E. Radford, Representations of pointed Hopf algebras and their Drinfel’d quantum doubles, J. Algebra 321 (2009), no. 9, 2567–2603.

    Article  MathSciNet  MATH  Google Scholar 

  43. C. Kassel, M. Rosso, V. Turaev, Quantum groups and knot invariants, in: Panoramas et Synthèses [Panoramas and Syntheses], Vol. 5, Société Mathématique de France, Paris, 1997.

  44. A. Klimyk, K. Schmüdgen, Quantum Groups and Their Representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.

    Book  MATH  Google Scholar 

  45. S. Majid, Noncommutative differentials and Yang-Mills on permutation groups S n , in: Hopf Algebras in Noncommutative Geometry and Physics, Lecture Notes in Pure and Appl. Math., Vol. 239, Dekker, New York, 2005, pp. 189–213.

  46. A. Masuoka, Hopf ideals of the generalized quantum double associated to skewpaired Nichols algebras, preprint: arXiv:0911.2282.

  47. K. Murasugi, B. I. Kurpita, A Study of Braids, Mathematics and its Applications, Vol. 484, Kluwer Academic Publishers, Dordrecht, 1999.

    MATH  Google Scholar 

  48. M. Mombelli, Module categories over pointed Hopf algebras, Math. Z. 266 (2010), no. 2, 319–344.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Mastnak, J. Pevtsova, P. Schauenburg, S. Witherspoon, Cohomology of finite dimensional pointed Hopf algebras, Proc. Lond. Math. Soc. (3) 100 (2010), no. 2, 377–404.

    Article  MathSciNet  MATH  Google Scholar 

  50. A. Milinski, H.-J. Schneider, Pointed indecomposable Hopf algebras over Coxeter groups, in: New Trends in Hopf algebra Theory (La Falda, 1999), Contemp. Math., Vol. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 215–236.

  51. W. D. Nichols, Bialgebras of type one, Comm. Algebra 6 (1978), no. 15, 1521–1552.

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Rosso, Quantum groups and quantum shuffles, Invent. Math. 133 (1998), no. 2, 399–416.

    Article  MathSciNet  MATH  Google Scholar 

  53. P. Schauenburg, A characterization of the Borel-like subalgebras of quantum enveloping algebras, Comm. Algebra 24 (1996), no. 9, 2811–2823.

    Article  MathSciNet  MATH  Google Scholar 

  54. A. Semikhatov, I. Tipunin, The Nichols algebra of screenings, preprint: arXiv:1101.5810.

  55. F. Sausset, C. Toninelli, G. Biroli, G. Tarjus, Bootstrap percolation and kinetically constrained models on hyperbolic lattices, J. Stat. Phys. 138 (2010), nos. 1–3, 411–430.

    Article  MathSciNet  MATH  Google Scholar 

  56. L. Vendramin, Nichols algebras associated to the transpositions of the symmetric group are twist-equivalent, accepted for publication in Proc. Amer. Math. Soc.

  57. S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), no. 1, 125–170.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Philipps-Universität Marburg, FB Mathematik und Informatik, Hans-Meerwein-Straße, 35032, Marburg, Germany

    I. Heckenberger & A. Lochmann

  2. Depto. de Matemática, FCEyN, Universidad de Buenos Aires, Pab. 1, Ciudad Universitaria (1428), Buenos Aires, Argentina

    L. Vendramin

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  1. I. Heckenberger
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Heckenberger, I., Lochmann, A. & Vendramin, L. Braided racks, Hurwitz actions and Nichols algebras with many cubic relations. Transformation Groups 17, 157–194 (2012). https://doi.org/10.1007/s00031-012-9176-7

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