On the general solution of the permuted classical Yang–Baxter equation and quasigraded Lie algebras

Abstract

Using the technique of the quasigraded Lie algebras, we construct general spectral-parameter dependent solutions r12( u, v) of the permuted classical Yang–Baxter equation with the values in the tensor square of simple Lie algebra [Formula: see text]. We show that they are connected with infinite-dimensional Lie algebras with Adler–Kostant–Symmes decompositions and are labeled by solutions of a constant quadratic equation on the linear space [Formula: see text], N ≥ 1. We formulate the conditions when the corresponding r-matrices are skew-symmetric, i.e., they are equivalent to the ones described by Belavin–Drinfeld classification. We illustrate the developed theory by the example of the elliptic r-matrix of Sklyanin. We apply the obtained result to the explicit construction of the generalized quantum and classical Gaudin spin chains.