When a Lie group $G$ acts on a manifold, it is interesting to study its infinitesimal generators, that is, the tangent vectors at the identity $e\in G.$ They correspond to vector fields at the manifold and they are called the fundamental vector field $X^{\sharp}$ associated to $X\in \mathfrak{g}$.
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 $\mathfrak{g}$ together with a bilinear operation
$$ [\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g} $$obeying the following identities
$\blacksquare$
Bilinearity makes equivalent alternativity and anticommutativity:
$$ [x, y]=-[y, x] $$On the other hand, Jacobi identity is better understood as saying that $[-,z]$ are derivations of the Lie algebra.
On an associative algebra $A$ over a field $F$ with multiplication $( x , y ) \mapsto x y$, a Lie bracket may be defined by the commutator $[x,y]=xy-yx$.
With this bracket, $A$ is a Lie algebra. The associative algebra $A$ is called an enveloping algebra of the Lie algebra $(A,[\,\cdot \,,\cdot \,])$. Every Lie algebra can be embedded into one in such a way that it arises from an associative algebra in this fashion; see universal enveloping algebra.
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,...
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: