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.
Articles by others on the same topic
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.
Bibliography: