**Algebras and triangle relations**

**R.J. Lawrence**

**Abstract: **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-Baxter's equation,
distinct from Zamolodchikov's equation, which resides naturally in these
structures, is proposed. A higher dimensional analogue of the relationship
betweeen the Yang-Baxter equation and braid groups is then seen to exhibit
a similar relationship with Manin and Schechtman's higher braid groups.

**Keywords: **higher algebra structures, polyhedral decompositions,
caegory theory, Yang-Baxter equation, quantum groups, Turaev-Viro invariants

**AMS subject classification: **57N10 17B37 57M25 57Q05 81R50 82B23

**Length: **30 pages

**Reference: *** J. Pure Appl. Alg. ***100 **(1995) 43-72.
MR1344843 (96i:57017) (review by * J. Stasheff *.)

*Last updated on September 4th, 1996.*

ruthel@math.huji.ac.il