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Lie group-Lie algebra correspondence

Every Lie algebra corresponds to a single simply connected Lie group.

The Baker-Campbell-Hausdorff formula basically defines how to map an algebra to the group.

Bibliography:

Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation"

Table of contents 103 7
- Lie algebra exponential covering problem Lie group-Lie algebra correspondence 40 2
  - A single exponential map is not enough to recover a simple Lie group from its algebra Lie algebra exponential covering problem 5
  - The product of a exponential of the compact algebra with that of the non-compact algebra recovers a simple Lie from its algebra Lie algebra exponential covering problem 22
- Two different Lie groups can have the same Lie algebra Lie group-Lie algebra correspondence 42 3
  - Every Lie algebra has a unique single corresponding simply connected Lie group Two different Lie groups can have the same Lie algebra 23 1
    - Universal covering group Every Lie algebra has a unique single corresponding simply connected Lie group 6
  - Every Lie group that has a given Lie algebra is the image of an homomorphism from the universal cover group Two different Lie groups can have the same Lie algebra

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