Finite-dimensional Representations of Yangians

Abstract

In this thesis, we study the cyclicity condition for an ordered tensor product of fundamental representations and the local Weyl modules of Yangians. We provide a sufficient condition for the cyclicity of an ordered tensor product $L=V_{a_1}(ω_{b_1})\otimes V_{a_2}(ω_{b_2})\otimes...\otimes V_{a_k}(ω_{b_k})$ of fundamental representations of the Yangian $Y(\mathfrak{g})$. When $\mathfrak{g}$ is a classical simple Lie algebra or the simple Lie algebra of type $G$, we make the cyclicity condition concrete, which leads to an irreducibility criterion for the ordered tensor product $L$. In the case when $\mathfrak{g}=\mathfrak{sl}_{l+1}$, a sufficient and necessary condition for the irreducibility of the ordered tensor product $L$ is obtained. The cyclicity of the ordered tensor product $L$ is closely related to the structure of the local Weyl modules of $Y(\mathfrak{g})$. We show that every local Weyl module is isomorphic to an ordered tensor product of fundamental representations of $Y(\mathfrak{g})$.