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On Nichols algebras over basic Hopf algebras

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Abstract

This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field \(\mathbb {k}\) of characteristic 0. Concretely, we show that a finite-dimensional Hopf algebra whose Hopf coradical is basic is a lifting of a Nichols algebra of a semisimple Yetter–Drinfeld module and we explain how to classify Nichols algebras of this kind. We provide along the way new examples of Nichols algebras and Hopf algebras with finite Gelfand–Kirillov dimension.

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Acknowledgements

This paper grew from conversations following a talk by Oscar Márquez on joint work in progress with Dirceu Bagio and Gastón A. García at the Colloquium Quantum 17 hosted by the University of Talca (Chile). We thank them for sharing their results as well as María Ronco and María Inés Icaza for hospitality. We also thank Hiroyuki Yamane for pointing out to us the reference [48]. We are grateful to C. D. Ward and H. West (University of Miskatonic, Arkham) for pointing out to us a mistake in the proof of Lemma 3.6. The main results of this paper were communicated at the XXII Coloquio Latinoamericano de Álgebra (Quito, August 2017); the Reunión Anual de la Unión Matemática Argentina (Buenos Aires, December 2017); the Workshop Métodos Categóricos en Álgebras de Hopf (Maldonado, December 2017); the Workshop Tensor categories, Hopf algebras and quantum groups (Marburg, January 2018).

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  1. Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM-CONICET, Medina Allende s/n, Ciudad Universitaria, 5000, Córdoba, Argentina

    Nicolás Andruskiewitsch & Iván Angiono

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  1. Nicolás Andruskiewitsch
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Correspondence to Iván Angiono.

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The work of N. A. and I. A. was partially supported by CONICET, Secyt (UNC), the MathAmSud project GR2HOPF.

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Andruskiewitsch, N., Angiono, I. On Nichols algebras over basic Hopf algebras. Math. Z. 296, 1429–1469 (2020). https://doi.org/10.1007/s00209-020-02493-w

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