On Nichols algebras over basic Hopf algebras
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Mathematische Zeitschrift
On Nichols algebras over basic Hopf algebras Nicolás Andruskiewitsch1 · Iván Angiono1 Received: 25 June 2019 / Accepted: 14 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field 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. Mathematics Subject Classification 16T05
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The context . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Nichols algebras over basic Hopf algebras . . . . . . . . . . . 1.3 The main result . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 2 Nichols algebras from decomposable braided vector spaces . . . . 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nichols algebras . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The coinvariant Nichols algebra . . . . . . . . . . . . . . . . 2.4 Nichols algebras of semisimple Yetter–Drinfeld modules . . . 3 Semisimplicity and the diagram . . . . . . . . . . . . . . . . . . . 3.1 Nichols algebras of graded or filtered Yetter–Drinfeld modules 3.2 Pointed Hopf algebras . . . . . . . . . . . . . . . . . . . . . . 3.3 Semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Decompositions from diagonal type . . . . . . . . . . . . . . . . . 4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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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Iván Angiono [email protected] Nicolás Andruskiewitsch [email protected]
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Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM-CONICET, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
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N. Andruskiewitsch, I. Angiono 4.2 Dimension W = 2 . . . . . . . . . . . 4.3 Cartan type Aθ ,
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