Yang-Baxter Equations, Hopf Algebras and Geometric Interpretations

The Yang-Baxter equations are important tools in mathematics, physics and computer science. Yang-Baxter equations are recognised as the origin of the theory of quantum groups or quasitriangular Hopf algebras. We propose new solutions for Yang-Baxter equations. Connecting celebrated topics, the current paper also deals with a many circles version of the Johnson–Tzitzeica theorem. Taking as a model some results from the Yang-Baxter theory, a self-dual theorem for cones is stated and proved.