# Lie Theory Back to Basics

This blog post introduces Lie Groups and defines important concepts that are needed for state estimation or SLAM. Largely adapted from "A Micro lie-theory for state estimation in Robotics"

## Preliminaries

### Manifold

A smooth manifold is a curved and differentiable hyper-surface – one with no spikes / edges – that locally resembles a linear space $$\mathbb{R}^n$$ embedded in a higher dimensional space.

#### Examples:

• 3D Vectors $$\in \mathbb{R}^3$$ with unit-norm constraint form the spherical manifold $$\mathbb{S}^3$$
• Rotations in the 2D plane $$\in \mathbb{R}^2$$ form a circular manifold
• 3D Rotations form a hypersphere (3-sphere) in $$\mathbb{R}^4$$
Figure 1: A manifold $$\mathcal{M}$$ and its associated tangent space $$\mathcal{T}_\times \mathcal{M} (\mathbb{R}^2)$$ tangent at the point $$\mathcal{X}$$ and a side view. The velocity element $$\dot{\mathcal{X}} = \frac{\partial{X}}{\partial{t}}$$ does not belong to the manifold but to the tangent space.

### Group

A Group $$\mathcal{G}$$ is a set with a composition operator $$\circ$$, that for elements $$X, Y, Z \in G$$ satisfy the following axioms:

1. Closure under $$\circ$$: $$X \circ Y \in \mathcal{G}$$
2. Identity $$E$$: $$E \circ X = X \circ E = X \in \mathcal{G}$$
3. Inverse $$X^{-1}$$: $$X^{-1} \circ X = X \circ X^{-1} = E \in \mathcal{G}$$
4. Associativity: $$(X \circ Y) \circ Z = X \circ (Y \circ Z)$$

Notice the omission of commutativity

### Lie Group

A smooth manifold that looks identical (example: A sphere, a circle) at every point and satisfies all group properties:

• Composition of Lie group elements remain in the lie group
• Each lie group has an identity element
• Every element has an inverse.

#### Examples

1. The unit complex number group $$\mathbb{S}^1: z = \cos(\theta) + i \sin(\theta) = e^{i \theta}$$ under complex multiplication as the composition operator is a Lie group. A Lie group element $$z \in \mathbb{S}^1$$ rotates a regular complex vector $$\mathbf{x} = x + iy$$ via complex multiplication to $$\mathbf{x}^\prime = \mathbf{z} \circ \mathbf{x}$$. The unit norm constraint of the unit complex group defines a 1-sphere in 2D space.
Figure 1: A manifold $$\mathcal{M}$$ and its associated tangent space $$\mathcal{T}_\times \mathcal{M} (\mathbb{R}^2)$$ tangent at the point $$\mathcal{X}$$ and a side view. The velocity element $$\dot{\mathcal{X}} = \frac{\partial{X}}{\partial{t}}$$ does not belong to the manifold but to the tangent space.