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.