This is called the exponential map, and it maps the Lie algebra g into the Lie group G . It provides a diffeomorphism between a neighborhood of 0 in g and a neighborhood of e in G . This exponential map is a generalization of the exponential function for real numbers (because R is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because C is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because M ( n , R ) with the regular commutator is the Lie algebra of the Lie group G L ( n , R ) of all invertible matrices).