A complete set of intertwiners for arbitrary tensor product representations via current algebras

TL;DR

This paper constructs an explicit and complete set of intertwiners for tensor product representations of reductive Lie algebras using current algebra actions, advancing the understanding of their structure.

Contribution

It provides a novel explicit construction of intertwiners for tensor product modules via current algebra, extending previous work and offering new tools for representation theory.

Findings

Explicit intertwiners constructed for tensor product modules

Complete set of intertwiners described in terms of current algebra

Advances understanding of module structure in Lie algebra representations

Abstract

Let $\mathfrak{g}$ be a reductive Lie algebra and let $\vec{V}(\vec{\lambda})$ be a tensor product of $k$ copies of finite dimensional irreducible $\mathfrak{g}$-modules. Choosing $k$ points in $\mathbb{C}$, $\vec{V}(\vec{\lambda})$ acquires a natural structure of the current algebra $\mathfrak{g}\otimes \mathbb{C}[t]$-module. Following a work of Rao [R], we produce an explicit and complete set of $\mathfrak{g}$-module intertwiners of $\vec{V}(\vec{\lambda})$ in terms of the action of the current algebra.