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Representations of copointed Hopf algebras arising from the tetrahedron rack

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Abstract

We study the copointed Hopf algebras attached to the Nichols algebra of the affine rack \(\mathrm{Aff}({\mathbb F}_4,\omega )\), also known as tetrahedron rack, and the \(2\)-cocycle\(-1\). We investigate the so-called Verma modules and classify all the simple modules. We conclude that these algebras are of wild representation type and not quasitriangular, also we analyze when these are spherical.

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Acknowledgments

The authors thank professor Nicolás Andruskiewitsch for proposing this problem and useful suggestions for this article. The first author also thanks Carolina Renz for her hospitality during her stay in Córdoba. Bárbara Pogorelsky was partially supported by Capes-Brazil. Cristian Vay was partially supported by ANPCyT-Foncyt, CONICET, MinCyT (Córdoba) and Secyt (UNC).

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Authors and Affiliations

  1. Instituto de Matematica, Universidade Federal do Rio Grande do Sul, Av. Bento Goncalves 9500, Porto Alegre, RS, 91509-900, Brazil

    Barbara Pogorelsky

  2. FaMAF-CIEM CONICET, Universidad Nacional de Cordoba, Medina Allende s/n, Ciudad Universitaria, Cordoba, 5000, Republic Argentine

    Cristian Vay

Authors
  1. Barbara Pogorelsky
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  2. Cristian Vay
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Correspondence to Barbara Pogorelsky.

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Appendix

Appendix

The next tables describe the structure of the \(12\)-dimensional simple modules of \(\mathcal {A}_{G,\lambda }\). These were used in Lemma 15.

See Tables 1, 2, 3, 4, 5, 6 and 7

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Pogorelsky, B., Vay, C. Representations of copointed Hopf algebras arising from the tetrahedron rack. Ann Univ Ferrara 60, 407–427 (2014). https://doi.org/10.1007/s11565-013-0197-5

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