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
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Dedicated to Daniel Arnaudon
Submitted: November 22, 2005; Accepted: February 1, 2006
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Arnaudon, D., Molev, A. & Ragoucy, E. On the R-Matrix Realization of Yangians and their Representations. Ann. Henri Poincaré 7, 1269–1325 (2006). https://doi.org/10.1007/s00023-006-0281-9
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DOI: https://doi.org/10.1007/s00023-006-0281-9