Sutherland-type Trigonometric Models, Trigonometric Invariants and Multivariate Polynomials

TL;DR

This paper investigates the algebraic structure of trigonometric Olshanetsky-Perelomov Hamiltonians expressed in Fundamental Trigonometric Invariants, confirming a conjecture that such forms preserve polynomial spaces across various Lie algebra root systems.

Contribution

It demonstrates that FTI variables lead to algebraic forms of Hamiltonians for classical and exceptional Lie algebra root systems, confirming a key conjecture.

Findings

FTI variables yield algebraic Hamiltonian forms for classical root systems

The algebraic form is confirmed for the exceptional $E_6$ system

The Hamiltonians preserve polynomial spaces in FTI coordinates

Abstract

It is conjectured that any trigonometric Olshanetsky-Perelomov Hamiltonian written in Fundamental Trigonometric Invariants (FTI) as coordinates takes an algebraic form and preserves an infinite flag of spaces of polynomials. It is shown that try-and-guess variables which led to the algebraic form of trigonometric Olshanetsky-Perelomov Hamiltonians related to root spaces of the classical $A_N, B_N, C_N, D_N, BC_N$ and exceptional $G_2, F_4$ Lie algebras are FTI. This conjecture is also confirmed for the trigonometric $E_6$ Olshanetsky-Perelomov Hamiltonian whose algebraic form is found with the use of FTI.