Representation theory for line operators

Work that started nearly 20 years ago, involving complex Chern-Simons theory, ideal triangulations and my first collaborations with Stavros, has brought me recently to ask increasingly general questions about extended operators (the things that form knots or other interesting topological/geometric structures) in quantum field theory. In particular, can line operators in a QFT always be described as modules for a quantum-group-like algebra? And, if so, where (physically) does the "quantum group" actually appear in the QFT? I'll give some answers and some examples, including a "topologization" of the Drinfeld-double construction in /any /3d topological QFT with certain properties; and some derived analogues of Yang-Baxter equations for shifted r-matrices.
(Parts of this are based on collaborations with W. Niu and V. Py.)