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The R-Matrix Presentation for the Yangian of a Simple Lie Algebra

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Abstract

Starting from a finite-dimensional representation of the Yangian \({Y (\mathfrak{g})}\) for a simple Lie algebra \( \mathfrak{g}\) in Drinfeld’s original presentation, we construct a Hopf algebra \({{X}_\mathcal{I}(\mathfrak{g})}\), called the extended Yangian, whose defining relations are encoded in a ternary matrix relation built from a specific R-matrix R\({\mathcal ({u})}\). We prove that there is a surjective Hopf algebra morphism \({{X}_ \mathcal{I} (\mathfrak{g})\twoheadrightarrow Y (\mathfrak{g})}\) whose kernel is generated as an ideal by the coefficients of a central matrix \({\mathcal{Z} \mathcal({u})}\). When the underlying representation is irreducible, we show that this matrix becomes a grouplike central series, thereby making available a proof of a well-known theorem stated by Drinfeld in the 1980s. We then study in detail the algebraic structure of the extended Yangian and prove several generalizations of results which are known to hold for Yangians associated to classical Lie algebras in their R-matrix presentations.

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Acknowledgements

The author gratefully acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada provided via the Alexander Graham Bell Canada Graduate Scholarship (CGS D). He would also like to thank Nicolas Guay and the anonymous reviewers for several helpful comments.

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  1. Department of Mathematical and Statistical Sciences, University of Alberta, CAB 632, Edmonton, AB, T6G 2G1, Canada

    Curtis Wendlandt

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  1. Curtis Wendlandt
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Correspondence to Curtis Wendlandt.

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Communicated by Y. Kawahigashi

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Wendlandt, C. The R-Matrix Presentation for the Yangian of a Simple Lie Algebra. Commun. Math. Phys. 363, 289–332 (2018). https://doi.org/10.1007/s00220-018-3227-4

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