Iterative construction of $U_q (s\ell (n+1)) $ representations and Lax matrix factorisation

TL;DR

This paper presents a method for constructing representations of quantum groups using Lax matrices, demonstrating how these matrices can be factorized into simpler components related to lower-rank representations.

Contribution

It introduces an iterative approach to build $U_q (s ext{ extlangle} ext{n+1} ext{ extrangle})$ representations via Lax matrix factorization, extending algebraic induction techniques.

Findings

Lax matrix of the constructed representation factorizes into lower-rank components.

The approach simplifies the construction of quantum group representations.

Provides a new perspective on algebraic induction using Lax matrices.

Abstract

The construction of a generic representation of $g\ell(n+1)$ or of the trigonomentric deformation of its enveloping algebra known as algebraic induction is conveniently formulated in term of Lax matrices. The Lax matrix of the constructed representation factorises into parts determined by the Lax matrix of a generic representation of the algebra with reduced rank and others appearing in the factorised expression of the Lax matrix of the special Jordan-Schwinger representation.