When a Lie group
This is related to the Lie algebra action and the Maurer-Cartan form.
When two Lie groups share the same Lie algebra they are locally the same. See relation SO(3) and SU(2) for an important example.
Definition (abstract)
A Lie algebra is a vector space
obeying the following identities
Bilinearity makes equivalent alternativity and anticommutativity:
On the other hand, Jacobi identity is better understood as saying that
On an associative algebra
With this bracket,
In abstract, a Lie algebra is a vector space (finite or infinite dimensional). See about solvable algebras and solvable structures to understand things about the finite dimensional and infinite dimensional cases.
Examples:
Important cases: solvable Lie algebras, simple Lie algebras,...
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Author of the notes: Antonio J. Pan-Collantes
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