Bibliography:
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) 7.2 "The covering problem" gives some amazing intuition on the subject as usual.
Furthermore, the non-compact part is always isomorphic to , only the non-compact part can have more interesting structure.
The most important example is perhaps and , both of which have the same Lie algebra, but are not isomorphic.
This simply connected is called the universal covering group.
The unique group referred to at: every Lie algebra has a unique single corresponding simply connected Lie group.
Articles by others on the same topic
Differentiation sends Lie groups to Lie algebras and smooth group homomorphisms to bracket-preserving linear maps. On connected simply connected Lie groups, this is an equivalence with finite-dimensional real Lie algebras: the Lie third theorem gives existence, and integration of a Lie algebra homomorphism gives unique maps. General connected groups require discrete central quotient data, and disconnected groups additionally require component and conjugation data.
The Lie group–Lie algebra correspondence is a fundamental concept in mathematics that relates Lie groups and Lie algebras, which are both central in the study of continuous symmetries and their structures. Here’s a breakdown of the concepts and their relationship: ### Lie Groups - A **Lie group** is a smooth manifold that also has a group structure such that the group operations (multiplication and inversion) are smooth maps. Lie groups are used to describe continuous symmetries (e.g.