Classical $N$-Reflection Equation and Gaudin Models

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 $B{C}_L$-type .

12 pages. Final authors version as published in Lett. Math. Phys (online, open access)

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Cited by in corpus (1)

• 3-dimensional mixed BF theory and Hitchin's integrable system