Real Lie Groups of Finite Dimension

TL;DR

This survey provides a rigorous overview of finite dimensional real Lie groups, emphasizing the tangent space formalism and its equivalence to other methods, filling a gap in the literature.

Contribution

It offers a comprehensive, formal treatment of tangent spaces in Lie groups, aligning proofs with this approach, which is scarce in existing literature.

Findings

Formalism of tangent spaces is equivalent to curve and derivation methods.

Rigorous proofs of fundamental Lie group facts are provided.

Addresses a gap in literature regarding tangent space formalism.

Abstract

This survey is about the fundamentals of the theory of finite dimensional Lie groups over the field of real numbers. The notion of the tangent space of a manifold at a point is considered to be defined via the well known chart and vector formalism, here, a formalism equivalent to other commonly used ones (namely, the curve and derivation methods). The proofs of all assertions about Lie groups are in alignment with this formalism, here. Dealing with the basic facts of Lie groups rigorously with regard to this formulation of tangent space seems to be scarce in the literature.