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Almost involutive Hopf algebras

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Abstract

We define the concept of companion automorphism of a Hopf algebra H as an automorphism \(\sigma :H\rightarrow H\) such that \(\sigma ^2=\mathcal {S}^2\), where \(\mathcal {S}\) denotes the antipode. This automorphism can be viewed as a special additional symmetry. A Hopf algebra is said to be almost involutive (AI) if it admits a companion automorphism. We present examples and study some of the basic properties and constructions of AI-Hopf algebras centering the attention in the finite dimensional case. In particular we show that within the family of Hopf algebras of dimension smaller or equal than 15, only in dimension eight and twelve, there are non-almost involutive Hopf algebras.

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Authors and Affiliations

  1. Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400, Montevideo, Uruguay

    Andrés Abella & Walter Ferrer Santos

Authors
  1. Andrés Abella
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  2. Walter Ferrer Santos
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Correspondence to Andrés Abella.

Additional information

Dedicated to Professor Cesar Polcino in his 70th birthday.

The second author would like to thank Csic-UDELAR, Conycit-MEC, Uruguay and Anii, Uruguay.

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Abella, A., Ferrer Santos, W. Almost involutive Hopf algebras. São Paulo J. Math. Sci. 10, 273–285 (2016). https://doi.org/10.1007/s40863-015-0028-y

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  • DOI: https://doi.org/10.1007/s40863-015-0028-y

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