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# Baker-Campbell-Hausdorff formula (BCH formula)

| 🗖 nosplit | ↑ parent "Lie algebra" | words: 231
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Solution for given and of: $$eZ=eXeY (128)$$ where is the exponential map.
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If we consider just real number, , but when X and Y are non-commutative, things are not so simple.
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Furthermore, TODO confirm it is possible that a solution does not exist at all if and aren't sufficiently small.
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This formula is likely the basis for the Lie group-Lie algebra correspondence. With it, we express the actual group operation in terms of the Lie algebra operations.
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Notably, remember that a algebra over a field is just a vector space with one extra product operation defined.
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Vector spaces are simple because all vector spaces of the same dimension on a given field are isomorphic, so besides the dimension, once we define a Lie bracket, we also define the corresponding Lie group.
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Since a group is basically defined by what the group operation does to two arbitrary elements, once we have that defined via the Baker-Campbell-Hausdorff formula, we are basically done defining the group in terms of the algebra.
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