%0 Journal Article
%T Classical $N$-Reflection Equation and Gaudin Models
%+ City University London
%+ Laboratoire Charles Coulomb (L2C)
%A Caudrelier, V.
%A Crampé, Nicolas
%Z 12 pages. References added. Explicit relation between our non-skew symmetric r-matrices and standard rational r-matrix given in the Gaudin models section
%< avec comité de lecture
%Z L2C:18-087
%@ 0377-9017
%J Letters in Mathematical Physics
%I Springer Verlag
%P 1–14
%8 2018-10-09
%D 2018
%Z 1803.09931
%R 10.1007/s11005-018-1128-2
%K Classical Yang–Baxter equation
%K Classical reflection equation
%K Gaudin models
%K Non-skew-symmetric r-matrices
%Z Physics [physics]/Mathematical Physics [math-ph]
%Z Mathematics [math]/Mathematical Physics [math-ph]
%Z Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI]Journal articles
%X 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 .
%G English
%L hal-01820521
%U https://hal.science/hal-01820521
%~ CNRS
%~ L2C
%~ TDS-MACS
%~ MIPS
%~ UNIV-MONTPELLIER
%~ UM-2015-2021