Yang-Baxter systematically

Sjoerd Crans, CTRC Seminar 9 March 1999

Recommended reading: 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 RA,B: A tensor B => B tensor 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).

Previous and next abstract.