Algebras and triangle relations Original Research Article

In this paper the new concept of an n-algebra is introduced, which embodies the combinatorial properties of an n-tensor, in an analogous manner to the way ordinary algebras embody the properties of compositions of maps. The work of Turaev and Viro on 3-manifold invariants is seen to fit naturally into the context of 3-algebras. A new higher dimensional version of Yang-Baxterʹs equation, distinct from Zamolodchikovʹs equation, which resides naturally in these structures, is proposed. A higher dimensional analogue of the relationship between the Yang-Baxter equation and braid groups is then seen to exhibit a similar relationship with Manin and Schechtmanʹs higher braid groups.