Yang-Baxter and reflection equations: unifying structures behind quantum and classical integrable field theories

The Yang-Baxter equation (YBE) is central in the theory of quantum integrable systems. For decades, together with its companion for problems with boundaries (the quantum reflection equation), it has been studied and used in the quantum realm. But it was suggested by Drinfeld in 1990 that the general study of the so-called « set-theoretical YBE » is also important. It turns out that classical integrable field theories provide a means to construct solutions to this equation, called Yang-Baxter maps, by looking at soliton collisions. Using the vector nonlinear Schrödinger (NLS) equation as the main example, we will present this notion of "classical solutions of the quantum YBE". Then, we will show how the new concept of set-theoretical reflection equation naturally emerges by studying integrable boundary field theories. Solutions to this equation, which we call reflection maps, arise from the reflection of solitons on the boundary. In the present context, factorization of interactions is the unifying principle behind integrability. This is of course well-know for quantum field theories but is essentially unexplored classically.