Lie group

A Lie group is a group that is also a smooth manifold, with smooth multiplication and inversion. Its tangent space at the identity carries a natural Lie algebra structure.

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Maurer-Cartan form

For a matrix Lie group, the left-invariant Maurer-Cartan form is

ρ = g^{- 1} d g

. It identifies each tangent space with the Lie algebra and obeys

d ρ + ρ \land ρ = 0

.

Maurer-Cartan equation

The Maurer-Cartan equation is

d ρ + ρ \land ρ = 0

for the left-invariant form

ρ = g^{- 1} d g

.

Maurer-Cartan equation in a Lie-algebra basis

If

ρ = σ^{α} T_{α}

and

[T_{α}, T_{β}] = c^{γ}_{α β} T_{γ}

, then

d σ^{γ} = - \frac{1}{2} c^{γ}_{α β} σ^{α} \land σ^{β} .

Real affine group

The orientation-preserving real affine group acts by

x \mapsto e^{a} x + b

and has multiplication

(a, b) (a^{'}, b^{'}) = (a + a^{'}, b + e^{a} b^{'})

.

Left-invariant metric

A left-invariant Riemannian metric is preserved by every left translation. Its Killing fields generated by left translations are right-invariant vector fields.

Exponential map of a Lie group ( $exp : g \to G$ )

For a matrix Lie group, the exponential map is the matrix exponential

exp X = \sum_{n \geq 0} X^{n} / n!

and sends one-parameter additive subgroups of the Lie algebra to one-parameter subgroups of the group.

Cayley transform (Lie theory, $C (X) = (I + X) (I - X)^{- 1}$ )

The Cayley transform is a rational local parametrization of a matrix Lie group from its Lie algebra wherever

I - X

is invertible. Unlike the exponential map, its image can meet nonidentity components.

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Wikipedia Bot  1

A **Lie group** is a mathematical structure that combines concepts from algebra and geometry. It is defined as a group that is also a smooth manifold, which means it has a structure that allows for differentiation and smoothness.

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Ciro Santilli  40 Updated 2025-07-16

The key and central motivation for studying Lie groups and their Lie algebras appears to be to characterize symmetry in Lagrangian mechanics through Noether's theorem, just start from there.

Notably local symmetries appear to map to forces, and local means "around the identity", notably: local symmetries of the Lagrangian imply conserved currents.

More precisely: local symmetries of the Lagrangian imply conserved currents.

TODO Ciro Santilli really wants to understand what all the fuss is about:

Oh, there is a low dimensional classification! Ciro is a sucker for classification theorems! en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras

The fact that there are elements arbitrarily close to the identity, which is only possible due to the group being continuous, is the key factor that simplifies the treatment of Lie groups, and follows the philosophy of continuous problems are simpler than discrete ones.

Bibliography:

youtu.be/kpeP3ioiHcw?t=2655 "Particle Physics Topic 6: Lie Groups and Lie Algebras" by Alex Flournoy (2016). Good SO(3) explicit exponential expansion example. Then next lecture shows why SU(2) is the representation of SO(3). Next ones appear to eventually get to the physical usefulness of the thing, but I lost patience. Not too far out though.
www.youtube.com/playlist?list=PLRlVmXqzHjURZO0fviJuyikvKlGS6rXrb "Lie Groups and Lie Algebras" playlist by XylyXylyX (2018). Tutorial with infinitely many hours
www.staff.science.uu.nl/~hooft101/lectures/lieg07.pdf
www.physics.drexel.edu/~bob/LieGroups.html

What is Lie theory? by Mathemaniac 2023

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