Lie algebras with triangular decompositions, non-skew-symmetric classical [formula omitted]-matrices and Gaudin-type integrable systems

Abstract

We consider the non-skew-symmetric g ⊗ g -valued classical r -matrices r 12 ( u 1 , u 2 ) with the spectral parameters possessing additional symmetries with respect to a finite-dimensional Lie subalgebra g 0 . Using them and the arbitrary (non-skew-symmetric) solution c 12 of a modified Yang–Baxter equation on g 0 we construct new classical non-skew-symmetric r -matrices r 12 c ( u 1 , u 2 ) . We show that both types of r -matrices are connected to the Lie algebras with the “triangular” decomposition and re-obtain our result using the corresponding classical R -operators. We consider “twisted” loop Lie algebras as our main examples and explicitly obtain the corresponding r -matrices r 12 ( u 1 , u 2 ) and r 12 c ( u 1 , u 2 ) . We use the constructed non-skew-symmetric classical r -matrices in order to produce mutually commuting quantum Gaudin-type hamiltonians.