On pointed Hopf algebras associated to some conjugacy classes in 𝕊_{𝕟}

Andruskiewitsch, Nicolás; Zhang, Shouchuan

doi:10.1090/S0002-9939-07-08880-6

On pointed Hopf algebras associated to some conjugacy classes in $\mathbb {S}_n$
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by Nicolás Andruskiewitsch and Shouchuan Zhang PDF

Proc. Amer. Math. Soc. 135 (2007), 2723-2731 Request permission

Abstract:

We show that any pointed Hopf algebra with infinitesimal braiding associated to the conjugacy class of $\pi \in \mathbb {S}_n$ is infinite-dimensional, if either the order of $\pi$ is odd, or all cycles in the decomposition of $\pi$ as a product of disjoint cycles have odd order except for exactly two transpositions.

References

Nicolás Andruskiewitsch and Matías Graña, Braided Hopf algebras over non-abelian finite groups, Bol. Acad. Nac. Cienc. (Córdoba) 63 (1999), 45–78 (English, with English and Spanish summaries). Colloquium on Operator Algebras and Quantum Groups (Spanish) (Vaquerías, 1997). MR 1714540
Nicolás Andruskiewitsch and Matías Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), no. 2, 177–243. MR 1994219, DOI 10.1016/S0001-8708(02)00071-3
N. Andruskiewitsch and H.-J. Schneider, Lifting of quantum linear spaces and pointed Hopf algebras of order $p^3$, J. Algebra 209 (1998), no. 2, 658–691. MR 1659895, DOI 10.1006/jabr.1998.7643
Nicolás Andruskiewitsch and Hans-Jürgen Schneider, Finite quantum groups and Cartan matrices, Adv. Math. 154 (2000), no. 1, 1–45. MR 1780094, DOI 10.1006/aima.1999.1880
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, DOI 10.2977/prims/1199403805
—, On the classification of finite-dimensional pointed Hopf algebras, preprint math.QA/0502157, Ann. Math., to appear.
R. Dijkgraaf, V. Pasquier, and P. Roche, Quasi Hopf algebras, group cohomology and orbifold models, Nuclear Phys. B Proc. Suppl. 18B (1990), 60–72 (1991). Recent advances in field theory (Annecy-le-Vieux, 1990). MR 1128130, DOI 10.1016/0920-5632(91)90123-V
Sergey Fomin and Anatol N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 147–182. MR 1667680
Matías Graña, On Nichols algebras of low dimension, New trends in Hopf algebra theory (La Falda, 1999) Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 111–134. MR 1800709, DOI 10.1090/conm/267/04267
I. Heckenberger, The Weyl groupoid of a Nichols algebra of diagonal type, Invent. Math. 164 (2006), no. 1, 175–188. MR 2207786, DOI 10.1007/s00222-005-0474-8
Alexander Milinski and Hans-Jürgen Schneider, Pointed indecomposable Hopf algebras over Coxeter groups, New trends in Hopf algebra theory (La Falda, 1999) Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 215–236. MR 1800714, DOI 10.1090/conm/267/04272
Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
S. J. Witherspoon, The representation ring of the quantum double of a finite group, J. Algebra 179 (1996), no. 1, 305–329. MR 1367852, DOI 10.1006/jabr.1996.0014