Generalized Gaudin spin chains, nonskew symmetric r-matrices, and reflection equation algebras

Abstract

Using a one-parametric family of automorphisms σ(u) of finite-dimensional simple Lie algebras g and classical skew-symmetric r-matrices r12(u−v), we construct “twisted” nonskew symmetric classical r-matrices r12σ(u,v) with spectral parameters. We show that in the special case σ(u)=AdK(u) and classical matrix Lie algebras g, they are connected with the quasiclassical limit of the reflection equation algebras. Using them and results of our previous papers, we construct new integrable quantum spin chains of the “generalized Gaudin” type with an arbitrary value of spin living at each site.