Classical $N$-Reflection Equation and Gaudin Models

TL;DR

This paper introduces the N-reflection equation as a generalization of the classical reflection equation, constructs associated Poisson algebras, and derives new Gaudin-type Hamiltonians with potential applications in integrable systems.

Contribution

It defines the N-reflection equation, develops its theoretical framework, and constructs new Gaudin Hamiltonians, extending classical integrable boundary condition models.

Findings

Established the theory of N-reflection equations for rational and trigonometric r-matrices.

Constructed Poisson algebras associated with non skew-symmetric r-matrices.

Derived new Gaudin-type Hamiltonians, including BC_L-type cases.

Abstract

We introduce the notion of $N$-reflection equation which provides a large generalization of the usual classical reflection equation describing integrable boundary conditions. The latter is recovered as a special example of the $N=2$ case. The basic theory is established and illustrated with several examples of solutions of the $N$-reflection equation associated to the rational and trigonometric $r$-matrices. A central result is the construction of a Poisson algebra associated to a non skew-symmetric $r$-matrix whose form is specified by a solution of the $N$-reflection equation. Generating functions of quantities in involution can be identified within this Poisson algebra. As an application, we construct new classical Gaudin-type Hamiltonians, particular cases of which are Gaudin Hamiltonians of $BC_L$-type .