Local Weyl modules and cyclicity of tensor products for Yangians

TL;DR

This paper establishes criteria for the cyclicity and irreducibility of tensor products of fundamental Yangian modules, linking these properties to local Weyl modules and providing concrete conditions for classical Lie algebras.

Contribution

It introduces explicit cyclicity and irreducibility conditions for tensor products of fundamental Yangian modules, especially for classical Lie algebras, and relates local Weyl modules to tensor products.

Findings

Cyclic tensor products are characterized by specific conditions.

Irreducibility criteria are explicitly given for  cases.

Every local Weyl module is isomorphic to a tensor product of fundamental representations.

Abstract

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, 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 $Y(\mathfrak{g})$.