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With the help of computer symbolic manipulation (Mathematica) the spectral parameter free solutions of the quantum three-simplex equation, which is a multidimensional generalization of the Yang-Baxter equation by Frenkel and Moore [Commun. Math. Phys. 138 (1991) 259], have been obtained. These solutions depend on the solutions of the 1+ 1 Yang-Baxter equation and help to understand the underlying algebra that is related to a higher braid group [Yu.I. Manin and V.V. Schechtman, Adv. Stud. Pure Math. 17 (1989) 289] which may have a relation with some special kind of three-dimensional lattice spin systems.
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