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On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization

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Abstract

We study classical twists of Lie bialgebra structures on the polynomial current algebra \({\mathfrak{g}[u]}\), where \({\mathfrak{g}}\) is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. It turns out that quasi-trigonometric r-matrices fall into classes labelled by the vertices of the extended Dynkin diagram of \({\mathfrak{g}}\). We give the complete classification of quasi-trigonometric r-matrices belonging to multiplicity free simple roots (which have coefficient 1 in the decomposition of the maximal root). We quantize solutions corresponding to the first root of \({\mathfrak{sl}(n)}\).

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Authors and Affiliations

  1. Institute for Theoretical and Experimental Physics, Moscow, Russia

    S. M. Khoroshkin

  2. Department of Mathematics, Gothenburg University, Gothenburg, Sweden

    I. I. Pop & A. A. Stolin

  3. Dublin Institute for Advanced Studies, Dublin, Ireland

    M. E. Samsonov

  4. Institute of Nuclear Physics, Moscow State University, Moscow, Russia

    V. N. Tolstoy

Authors
  1. S. M. Khoroshkin
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  2. I. I. Pop
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  3. M. E. Samsonov
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  4. A. A. Stolin
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  5. V. N. Tolstoy
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Correspondence to I. I. Pop.

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Communicated by N.A. Nekrasov

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Khoroshkin, S.M., Pop, I.I., Samsonov, M.E. et al. On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization. Commun. Math. Phys. 282, 625–662 (2008). https://doi.org/10.1007/s00220-008-0554-x

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  • DOI: https://doi.org/10.1007/s00220-008-0554-x

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