Representations of the $S_N$-Extended Heisenberg Algebra and Relations Between Knizhnik-Zamolodchikov Equations and Quantum Calogero Model

TL;DR

This paper explores the algebraic structures underlying the quantum Calogero model, establishing new connections between solutions of the Knizhnik-Zamolodchikov equations and Calogero wave functions through novel algebraic representations.

Contribution

It introduces lowest-weight representations of the $S_N$-Extended Heisenberg Algebra and constructs flat derivatives linking KZ and Dunkl derivatives, revealing new relations between these mathematical frameworks.

Findings

Constructed flat derivatives interpolating between KZ and Dunkl derivatives.

Established potential new links between KZ solutions and Calogero wave functions.

Provided algebraic foundations for understanding the Calogero model's symmetries.

Abstract

We discuss lowest-weight representations of the $S_N$-Extended Heisenberg Algebras underlying the $N$-body quantum-mechanical Calogero model. Our construction leads to flat derivatives interpolating between Knizhnik-Zamolodchikov and Dunkl derivatives. It is argued that based on these results one can establish new links between solutions of the Knizhnik-Zamolodchikov equations and wave functions of the Calogero model.