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 g. 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 g⊕N, 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.