Solutions for the quasi-Yang-Baxter equation. Diagrammatics, axioms and semi-classical approximations

We generalize Nichita, Popovici and Tanasa solutions of the Braid equation to quasi-Yang-Baxter equation. We define quasi-braided Lie algebras in an additive monoidal category as a natural generalization of Majid's braided Lie algebra concept. Quasi-braided Lie algebras provide solutions for the quasi-Yang-Baxter equation. Examples came from Lie algebras in additive monoidal categories with non-strict associativity and from the theory of quasi-triangular quasi-Hopf algebras.