Somewhat related is noether's theorem (from Emmy Noether) that draws direct correspondence between symmetries and conserved quantities. E.g. conservation of linear momentum corresponds to a system that is invariant to translations. So you can find some of the fundamentals of a system by looking at symmetries and Lie groups/algebras give you tools to look at symmetries.
gsf_emergency_6|2 months ago
Charge is conserved => symmetry (though not capturing exactly the "(non-Noetherian) localization" that is special to it)
GP suggested the opposite thought process-- as you rightly imply:
disagreement between 2 observers whether charge is conserved or not => discovering that _something else_ is conserved