The geometry library is here to leverage geometry for programming problems involving control, state estimation, perception, signal processing and physics. My contributions here are really based on just random interest in these topological problems and how they apply to some projects I am working on for fun.
Lie algebra is an abstract algebra that's core is built around group theory. This description of lie algebra is based on this paper: A micro Lie theory for state estimation in robotics.
The core of group theory is based on this definition:
Every group satisfies these axioms:
Closure under 'o' : 
Closure under 
: 
Inverse 
: 
Associativity: 
where 
is the group action.