Finite-dimensional Representations of Yangians

TL;DR

This thesis investigates the structure and irreducibility conditions of tensor products of fundamental representations of Yangians, providing concrete criteria for classical Lie algebras and establishing a link with local Weyl modules.

Contribution

It offers new sufficient and necessary conditions for the cyclicity and irreducibility of tensor products of Yangian representations, especially for type A Lie algebras.

Findings

Provided a sufficient condition for cyclicity of tensor products.

Derived an irreducibility criterion for tensor products of type A Yangians.

Showed local Weyl modules are isomorphic to tensor products of fundamental representations.

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 $L=V_{a_1}(\omega_{b_1})\otimes V_{a_2}(\omega_{b_2})\otimes...\otimes V_{a_k}(\omega_{b_k})$ 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 $Y(\mathfrak{g})$.