Lie algebras with triangular decompositions, non-skew-symmetric classical rr-matrices and Gaudin-type integrable systems

We consider the non-skew-symmetric g⊗gg⊗g-valued classical rr-matrices r12(u1,u2)r12(u1,u2) with the spectral parameters possessing additional symmetries with respect to a finite-dimensional Lie subalgebra g0g0. Using them and the arbitrary (non-skew-symmetric) solution c12c12 of a modified Yang–Baxter equation on g0g0 we construct new classical non-skew-symmetric rr-matrices r12c(u1,u2). We show that both types of rr-matrices are connected to the Lie algebras with the “triangular” decomposition and re-obtain our result using the corresponding classical RR-operators. We consider “twisted” loop Lie algebras as our main examples and explicitly obtain the corresponding rr-matrices r12(u1,u2)r12(u1,u2) and r12c(u1,u2). We use the constructed non-skew-symmetric classical rr-matrices in order to produce mutually commuting quantum Gaudin-type hamiltonians.