On the R-Matrix Realization of Yangians and their Representations

By D. Arnaudon, Alexander I. Molev, É. Ragoucy. 2005

72 citations recorded by the source corpus.

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

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1a12f80f26d51050117a2ee8067ab06f53142a0c