The gl_2 Bethe algebra associated with a nilpotent element

TL;DR

This paper investigates the relationships between Bethe algebras associated with the zero matrix and a nilpotent matrix within the universal enveloping algebra of gl_2[t], revealing structural connections in algebraic integrable systems.

Contribution

It provides a detailed description of the relations between Bethe algebras linked to different matrices, specifically zero and nilpotent matrices, in the context of gl_2[t].

Findings

Identifies relations between Bethe algebras for zero and nilpotent matrices

Describes the algebraic structure of Bethe algebras in these cases

Enhances understanding of Bethe algebra interactions in integrable models

Abstract

To any 2x2-matrix K one assigns a commutative subalgebra B^{K}\subset U(gl_2[t]) called a Bethe algebra. We describe relations between the Bethe algebras, associated with the zero matrix and a nilpotent matrix.