  Quantum groups were invented largely to provide solutions of the Yang-Baxter
equation and hence solvable models in 2-dimensional statistical mechanics and
one-dimensional quantum mechanics. They have been hugely successful. But not
all Yang-Baxter solutions fit into the framework of quantum groups. We shall
explain how other mathematical structures, especially subfactors, provide a
language and examples for solvable models. The prevalence of the Connes tensor
product of Hilbert spaces over von Neumann algebras leads us to speculate
concerning its potential role in describing entangled or interacting quantum
systems.
