Abstract
We introduce the notion of N-reflection equation which provides a 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 with the rational and trigonometric r-matrices. A central result is the construction of a Poisson algebra associated with 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.
Highlights
Classical integrable systems have been formulated in terms of the classical r-matrix in [1,2,3] for which the central equation is called the classical Yang–Baxter equation
We introduce a new equation which generalizes the classical reflection equation (2) as well as algebraic structures related to ZN models
We show that the Hamiltonian equations of motion generated by the elements in involution in the Poisson subalgebra can be written in Lax form and give an explicit formula for the second matrix of the Lax pair
Summary
This equation appears naturally in classical integrable systems based on the BCL root system and can be interpreted via a Z2 action on the A2L root system. This point of view on integrable boundary conditions, sometimes called “folding” for short, has been used extensively, for example, in [5,6,7]. 4, we show how the N -reflection equation allows one to define a certain Poisson subalgebra of a linear Poisson algebra defined by a classical r -matrix and use this to obtain new integrable Gaudin models. We show that the Hamiltonian equations of motion generated by the elements in involution in the Poisson subalgebra can be written in Lax form and give an explicit formula for the second matrix of the Lax pair
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