Theorem

Universal Cover of a Lie Group

Lie Groups and Lie Algebras · lie-groups.tex

If $G$ is a connected (Definition~Connectedness and Path-Connectedness) Lie group, its universal cover $\widetilde{G}$ (Definition~Covering Space) carries a unique Lie group structure such that the covering map $p : \widetilde{G} \to G$ is a Lie group homomorphism (Definition~Group Homomorphism). The kernel $\ker p$ (Definition~Kernel and Image) is a discrete central subgroup (Definition~Subgroup and Normal Subgroup) of $\widetilde{G}$.

Depends on

Dependency Graph

flowchart LR classDef current fill:#6366f1,color:#fff,stroke:#4f46e5 ndc4990c2["Connectedness and Path-Connectedness"] nc5e5fa8f["Covering Space"] ne087180c["Group Homomorphism"] n6fd9cebe["Kernel and Image"] n3d67b716["Subgroup and Normal Subgroup"] n738e8542["Universal Cover of a Lie Group"]:::current ndc4990c2 --> n738e8542 nc5e5fa8f --> n738e8542 ne087180c --> n738e8542 n6fd9cebe --> n738e8542 n3d67b716 --> n738e8542 click ndc4990c2 "../objects/dc4990c2.html" "_self" click nc5e5fa8f "../objects/c5e5fa8f.html" "_self" click ne087180c "../objects/e087180c.html" "_self" click n6fd9cebe "../objects/6fd9cebe.html" "_self" click n3d67b716 "../objects/3d67b716.html" "_self"