On the R-Matrix Realization of Yangians and their Representations

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 Poincare–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.