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 $X^{♯}$ associated to $X \in g$ .

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 $g$ together with a bilinear operation

[\cdot, \cdot] : g \times g \to g

obeying the following identities

Alternativity: $[x, x] = 0$
Jacobi identity: $[x, [y, z]] = [[x, y], z] + [y, [x, z ∥ x, y], z] + [y, [x, z]]$
$◼$

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 $[x, y] = x y - y x$ .
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 $Diff (M)$ 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.

Given a basis $e_{i}$ of the vector space $g$ , the Lie bracket of any two basis elements can be expressed as a linear combination of basis elements:

[e_{i}, e_{j}] = \sum_{k} c_{i j}^{k} e_{k}

where the scalars $c_{i j}^{k}$ are called the structure coefficients (or structure constants) of the Lie algebra with respect to the chosen basis. These coefficients completely determine the Lie bracket operation once the basis is fixed, and they satisfy relations imposed by bilinearity, antisymmetry (i.e., $c_{i j}^{k} = - c_{j i}^{k}$ ), and the Jacobi identity:

\sum_{m} (c_{i j}^{m} c_{m ℓ}^{n} + c_{j ℓ}^{m} c_{m i}^{n} + c_{ℓ i}^{m} c_{m j}^{n}) = 0.

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