Lie algebras

Idea

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 $X^{\mathrm{♯}}$ 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 and remarks

Definition (abstract)
A Lie algebra is a vector space $\mathfrak{g}$ together with a bilinear operation

obeying the following identities

• Alternativity: $[x,x]=0$
• Jacobi identity: $[x,[y,z]]=[[x,y],z]+[y,[x,z\parallel x,y],z]+[y,[x,z]]$
  $◼$

Bilinearity makes equivalent alternativity and anticommutativity:

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 $(x,y)\mapsto xy$, 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:

• Given a Lie group, its Lie algebra is the set of all the left invariant vector fields.
• Vector fields on a manifold: infinite dimensional Lie algebra. We can think of it, loosely speaking, as the Lie algebra of the group $\text{Diff}(M)$ of all the diffeomorphisms of $M$.
• Killing vector fields on a pseudo-Riemannian manifold. It is a subalgebra of the previous one. Analogously, it can be thought as the Lie algebra of the group of isometries of $M$.
• Symmetries of a given distribution on a manifold: infinite dimensional Lie algebra.

Important cases: solvable Lie algebras, simple Lie algebras,...