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A New Approach to Leibniz Bialgebras

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Abstract

A study of Leibniz bialgebras arising naturally through the double of Leibniz algebras analogue to the classical Drinfeld’s double is presented. A key ingredient of our work is the fact that the underline vector space of a Leibniz algebra becomes a Lie algebra and also a commutative associative algebra, when provided with appropriate new products. A special class of them, the coboundary Leibniz bialgebras, gives us the natural framework for studying the Yang-Baxter equation (YBE) in our context, inspired in the classical Yang-Baxter equation as well as in the associative Yang-Baxter equation. Results of the existence of coboundary Leibniz bialgebra on a symmetric Leibniz algebra under certain conditions are obtained. Some interesting examples of coboundary Leibniz bialgebras are also included. The final part of the paper is dedicated to coboundary Leibniz bialgebra structures on quadratic Leibniz algebras.

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Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, 3001-454, Coimbra, Portugal

    Elisabete Barreiro

  2. Université de Lorraine, Laboratoire IECL, CNRS UMR 7502, Metz cedex 01, Ile du Saulcy, 57045, France

    Saïd Benayadi

Authors
  1. Elisabete Barreiro
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  2. Saïd Benayadi
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Correspondence to Elisabete Barreiro.

Additional information

Presented by Peter Littelmann.

This work was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

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Barreiro, E., Benayadi, S. A New Approach to Leibniz Bialgebras. Algebr Represent Theor 19, 71–101 (2016). https://doi.org/10.1007/s10468-015-9563-6

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  • DOI: https://doi.org/10.1007/s10468-015-9563-6

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