lie theory – Mathtuition88

Differentiable Manifold

Differentiable manifold

An $n$ -dimensional (differentiable) manifold $M^n$ is a Hausdorff topological space with a countable (topological) basis, together with a maximal differentiable atlas.

This atlas consists of a family of charts $\displaystyle h_\lambda: U_\lambda\to U'_\lambda\subset\mathbb{R}^n,$ where the domains of the charts, $\{U_\lambda\}$ , form an open cover of $M^n$ , the $U'_\lambda$ are open in $\mathbb{R}^n$ , the charts (local coordinates) $h_\lambda$ are homeomorphisms, and every change of coordinates $\displaystyle h_{\lambda\mu}=h_\mu\circ h_\lambda^{-1}$ is differentiable on its domain of definition $h_\lambda(U_\lambda\cap U_\mu)$ .

Source: Representations of Compact Lie Groups (Graduate Texts in Mathematics)

Lie Groups

One of the best books on Lie Groups is said to be Representations of Compact Lie Groups (Graduate Texts in Mathematics). It is one of the rarer books from the geometric approach, as opposed to the algebraic approach.

Lie group

A Lie group is a differentiable manifold $G$ which is also a group such that the group multiplication $\displaystyle \mu:G\times G\to G$ (and the map sending $g$ to $g^{-1}$ ) is a differentiable map.