Yang-Baxter systematically

Sjoerd Crans, CTRC Seminar 9 March 1999

Recommended reading:

``Braided tensor categories'' by Joyal and Street [Adv. Math. 102 (1993), 20-78],
``2-Categories and Zamolodchikov tetrahedra equations'' by Kapranov and Voevodsky [Proc. Symp. Pure Math. 56 (1994), 177-260],
``Newton polytopes of the classical resultant and discriminant'' by Gelfand, Kapranov and Zelevinsky [Adv. Math. 84 (1990), 237-254].

In a braided monoidal category, the Yang-Baxter hexagon commutes for each triple of objects. There are two different proofs for this, subdividing the hexagon in two different ways. I looked at the converse: given a monoidal category together with Yang-Baxter system, a collection of objects Y and a collection of arrows R_A,B: A tensor

B => B

A, for A, B in Y such that the Yang-Baxter hexagon commutes for each triple of objects from Y, does this give a braided monoidal category? Using the 2-functor st: MCat -> MCat I constructed a braided monoidal category from a monoidal category with a Yang-Baxter system, with naturality of the braiding in each variable being proven via the two subdivisions of the Yang-Baxter hexagon respectively. This construction gives the object part of a biequivalence between the 2-category of braided monoidal categories whose only arrows are composites of R's and the 2-category of Yang-Baxter systems where Y consists of all objects of the monoidal category.

See the paper ``Higher-dimensional Yang-Baxter and Zamolodchikov equations'' (in preparation).

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