labeled Lie groups

Labeled Lie groups are mathematical structures that combine continuous symmetry concepts with additional specific features called labels. A Lie group itself describes smooth, continuous transformations like rotations or shifts that preserve certain properties. When "labeled," each element (or transformation) of the group has extra information attached, like a tag or label, which helps distinguish or classify them further. Think of it as having a set of symmetrical objects where each is marked with a unique identifier, allowing for more detailed organization and analysis of their properties and behaviors within mathematical or physical systems.