# The universal cover of a Lie group

The relationship between Lie groups and Lie algebras is particularly straightforward for simply connected Lie groups:

• Every Lie algebra corresponds to a unique simply connected Lie group $${G^{*}}$$
• There is a group homomorphism $${\phi}$$ from this unique simply connected Lie group $${G^{*}}$$ to any other connected Lie group $${G}$$ with the same Lie algebra, with $${\textrm{Ker}\phi\cong\pi_{1}\left(G\right)}$$ discrete

This last implies that for any Lie group $${G\cong G^{*}/\pi_{1}\left(G\right)}$$, the simply connected Lie group $${G^{*}}$$ with the same Lie algebra has a fixed number of points that map down to any point in $${G}$$. $${G^{*}}$$ can thus be pictured as “wrapping around” any such $${G}$$ some number of times, and is therefore called the universal covering group. The above depicts the universal covering group $${G^{*}}$$ and its homomorphism to any other Lie group $${G}$$ with the same Lie algebra. A one-dimensional subalgebra and corresponding one-dimensional subgroups are shown as lines.

The idea of a space covering another generalizes to any topological space: a covering space $${C}$$ of a space $${X}$$ has a continuous surjective map to $${X}$$ whose inverse in a neighborhood of any point of $${X}$$ is a union of mutually disjoint open sets homeomorphic to that neighborhood. The points that map to a point $${p\in X}$$ are called the fiber over $${p}$$, and the disjoint open sets over a neighborhood of $${p}$$ are called sheets. Under reasonable connectivity requirements, every space then has a unique simply connected universal covering space that covers all connected covers. The above depicts the infinite-sheeted universal covering space $${\mathbb{R}}$$ of $${S^{1}}$$.