Transformation Groups and Erlangen Program - Applications
TheoremLie's Third Theorem
Every finite-dimensional real Lie algebra is the Lie algebra of some Lie group. This establishes a correspondence between infinitesimal symmetries (Lie algebras) and global transformation groups (Lie groups).
Applied to geometry: infinitesimal symmetries (Killing vector fields) integrate to isometries, connecting local and global geometric structure.
Applications include gauge theories in physics, where Lie groups describe symmetries of fundamental interactions. The Standard Model uses as its gauge group, exemplifying Klein's program in modern physics.