Lie group

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The article on this topic in the Group Properties Wiki can be found at: Lie group

Definition

Symbol-free definition

A Lie group is a set equipped with two structures:

in such a way that the group multiplication operation is a differentiable map from the product manifold to the base manifold, and the inversion map is a differentialbe map from the manifold to itself.

Additional structure

Riemannian metric

Any Lie group can be equipped with a Riemannian metric. In fact, for a compact Lie group, we can find a unique bi-invariant metric.

To find a left-invariant metric, simply take the tangent space at the identity, and put any metric on it. Now, by using the group action on tangent spaces by left multiplication, obtain aa metric on each of the tangent spaces. This metric is left-invariant by consturction.

When the Lie group is compact, we can use this to obtain a bi-invariant metric by averaging out over the right translates of the metric.

Measure

Any Lie group can be equipped with a compatible Haar measure. In particular, when the Lie grop is compact, we can normalize the measure so that the total measure of the group is 1.

Futher properties satisfied by Lie groups

Lie groups are very special kinds of differential manifolds, and the invariant Riemannian metrics must also satisfy some nice properties. First, any left-invariant Riemannian metric on a Lie group is a homogeneous metric, where the map taking any point to another is simply the corresponding group multiplication. Further, if the group's adjoint action on its Lie algebra is transitive on the set of orthonormal bases for the Lie algebra, then the metric is a constant-curvature metric.

Properties that any left-invariant Riemannian metric on a Lie group must satisfy: