Solutions of the Quantum-Yang-Baxter-Equation from Symmetric Spaces

Abstract

We show that for each semi-Riemannian locally symmetric space the curvature tensor gives rise to a rational solution $r$ of the classical Yang-Baxter equation with spectral parameter. For several Riemannian globally symmetric spaces $M$ such as real, complex and quaternionic Grassmann manifolds we explicitly compute solutions $R$ of the quantum Yang Baxter equations (represented in the tangent spaces of $M$) which generalize the quantum $R$-matrix found by Zamolodchikov and Zamolodchikov in 1979.