Differential-Dunkl Operators and Nonstandard Solutions to the Classical Yang-Baxter Equation

TL;DR

This paper explores nonstandard solutions to the classical Yang-Baxter equation using differential-Dunkl operators linked to rational Cherednik algebras, providing new insights and partial answers to existing conjectures.

Contribution

It introduces an interpretation of specific r-matrices via differential-Dunkl operators and addresses a conjecture about boundary solutions to the classical Yang-Baxter equation.

Findings

Interpretation of r_{n-2,n} in terms of Dunkl operators

Partial resolution of a conjecture on boundary solutions

Connection between Cherednik algebras and Yang-Baxter solutions

Abstract

For every pair of positive coprime integers, m and n, with m<n, there is an associated generalized Cremmer-Gervais r-matrix r_{m,n} for the Lie algebra sl_n which provides a nonstandard quasitriangular solution to the classical Yang-Baxter equation. We give an interpretation of r_{n-2,n} (for n odd) in terms of differential-Dunkl operators related to the polynomial representation of dihedral-type rational Cherednik algebras. Finally, we use this interpretation to partially answer a conjecture of Gerstenhaber and Giaquinto concerning boundary solutions to the classical Yang-Baxter equation.