Triple Products and Yang-Baxter Equation (I): Octonionic and Quaternionic Triple Systems

We can recast the Yang-Baxter equation as a triple product equation. Assuming the triple product to satisfy some algebraic relations, we can find new solutions of the Yang-Baxter equation. This program has been completed here for the simplest triple systems which we call octonionic and quaternionic. The solutions are of rational type.Comment: 29 page

Angular Momentum, Quaternion, Octonion, and Lie-Super Algebra osp(1,2)

We will derive both quaternion and octonion algebras as the Clebsch-Gordan algebras based upon the su(2) Lie algebra by considering angular momentum spaces of spin one and three. If we consider both spin 1 and 1/2 states, then the same method will lead to the Lie-super algebra osp(1,2). Also, the quantum generalization of the method is discussed on the basis of the quantum group $su_q(2)$.Comment: 17 pages, TE

Construction of Lie Superalgebras from Triple Product Systems

Any simple Lie superalgebras over the complex field can be constructed from some triple systems. Examples of Lie superalgebras $D(2,1;\alpha)$, G(3) and F(4) are given by utilizing a general construction method based upon $(-1,-1)$ balanced Freudenthal-Kantor triple system.Comment: 8 pages, no figure