On the R-matrix realization of yangians and their representations

Abstract

We study the Yangians Y(a) associated with the simple Lie algebras of type B, C or D. The algebra Y(a) can be regarded as a quotient of the extended Yangian X(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 X(a). We prove an analog of the Poincaré-Birkhoff-Witt theorem for X(a) and show that the Yangian Y(a) can be realized as a subalgebra of Y(a). Furthermore, we give an independent proof of the classification theorem for the finite-dimensional irreducible representations of Y(a) which implies the corresponding theorem of Drinfeld for the Yangians Y(a). We also give explicit constructions for all fundamental representation of the Yangians. Communicated by Petr Kulish. © Birkhäuser Verlag, Basel/Switzerland 2007.