From what I understand, Hopf algebras are useful algebraic objects for introducing a particular kind of renormalisation.
However, I'm struggling to understand whether this is just a neat high-level algebraic 'trick', or whether it actually has any actual physical interpretation, and whether using Hopf algebras sheds any light on physical theories. How relevant and useful have Hopf algebras been in modern quantum field theory? Is there any visual, 'Feynman-style' intuition of why Hopf algebras are useful, or even necessary? I don't see the point of introducing additional complicated algebra if it doesn't provide any physical insight.
