$gl_{n+1}$ algebra of Matrix Differential Operators and Matrix Quasi-exactly-solvable Problems

TL;DR

This paper explicitly constructs the $gl_{n+1}$ algebra using matrix differential operators and applies it to develop algebraic Hamiltonians for matrix versions of Calogero and Sutherland models.

Contribution

It provides explicit forms of $gl_{n+1}$ algebra generators as matrix differential operators and introduces matrix algebraic Hamiltonians for generalized integrable models.

Findings

Explicit $gl_{n+1}$ algebra generators derived as matrix differential operators

Matrix algebraic Hamiltonians for generalized Calogero and Sutherland models presented

Potential applications in solving matrix quasi-exactly-solvable problems

Abstract

The generators of the algebra $gl_{n+1}$ in a form of differential operators of the first order acting on ${\bf R}^n$ with matrix coefficients are explicitly written. The algebraic Hamiltonians for a matrix generalization of $3-$body Calogero and Sutherland models are presented.