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On the R-Matrix Realization of Yangians and their Representations

Authors:	Daniel Arnaudon Alexander Molev Eric Ragoucy

Institution:	(1) LAPTH, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux cedex, France;(2) School of Mathematics and Statistics, University of Sydney, NSW, 2006 Sydney, Australia

Abstract:	We study the Yangians ${\text{Y}}(\mathfrak{a})$ associated with the simple Lie algebras $\mathfrak{a}$ of type B, C or D. The algebra ${\text{Y}}(\mathfrak{a})$ can be regarded as a quotient of the extended Yangian ${\text{X}}(\mathfrak{a})$ whose defining relations are written in an R-matrix form. In this paper we are concerned with the algebraic structure and representations of the algebra ${\text{X}}(\mathfrak{a})$ . We prove an analog of the Poincaré–Birkhoff–Witt theorem for ${\text{X}}(\mathfrak{a})$ and show that the Yangian ${\text{Y}}(\mathfrak{a})$ can be realized as a subalgebra of ${\text{X}}(\mathfrak{a})$ . Furthermore, we give an independent proof of the classification theorem for the finite-dimensional irreducible representations of ${\text{X}}(\mathfrak{a})$ which implies the corresponding theorem of Drinfeld for the Yangians ${\text{Y}}(\mathfrak{a})$ . We also give explicit constructions for all fundamental representation of the Yangians. Communicated by Petr Kulish Dedicated to Daniel Arnaudon Submitted: November 22, 2005; Accepted: February 1, 2006

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