---
title: "Learning some Lie theory for fun and profit "
date: 2020-11-10
toc: false
---

[fn::{-} The phrase "for fun and profit" seems to be a pretty old
expression: according to the answers to [[https://english.stackexchange.com/q/25205][this StackExchange question]],
it might date back to Horace's [[https://en.wikipedia.org/wiki/Ars_Poetica_(Horace)][/Ars Poetica/]] ("prodesse et
delectare"). I like the idea that books (and ideas!) should be both
instructive and enjoyable...]

While exploring [[./quaternions.html][quaternions]] and the theory behind them, I noticed an
interesting pattern: in the exposition of
cite:sola2017_quater_kinem_error_state_kalman_filter, quaternions and
rotations matrices had exactly the same properties, and the derivation
of these properties was rigorously identical (bar some minor notation
changes).

This is expected because in this specific case, these are just two
representations of the same underlying object: rotations. However,
from a purely mathematical and abstract point of view, it cannot be a
coincidence that you can imbue two different types of objects with
exactly the same properties.

Indeed, this is not a coincidence: the important structure that is
common to the set of rotation matrices and to the set of quaternions
is that of a /Lie group/.

* Why would that be interesting?

From a mathematical point of view, seeing a common structure like this
should raise alarm signals in our heads. Is there a deeper concept at
play here? If we can find that two objects are two examples of the
same abstract structure, maybe we'll also be able to identify that
structure elsewhere, maybe where it's less obvious. And then, if we
prove interesting theorems on the abstract structure, we'll
essentially get the same theorems on every example of this structure,
and /for free!/ (i.e. without any additional work!)[fn:structure]

[fn:structure]{-} When you push that idea to its extremes, you get
[[https://en.wikipedia.org/wiki/Category_theory][category theory]], which is just the study of (abstract) structure. This
in a fun rabbit hole to get into, and if you're interested, I
recommend the amazing [[https://www.math3ma.com/][math3ma]] blog, or
cite:riehlCategoryTheoryContext2017 for a complete and approachable
treatment. cite:fongSevenSketchesCompositionality2018 gives an
interesting perspective on why category theory is interesting in the
real world.


We can think of it as a kind of factorization: instead of doing the
same thing over and over, we can basically do it /once/ and recall the
general result whenever it is needed, as one would define a function
and call it later in a piece of software.

* Important structure

Continuing on the example of rotations, what common properties can we
identify?

1. Quaternions and rotation matrices can be multiplied together (to
   compose rotations), have an identity element, along with nice
   properties.
2. Quaternions and rotation matrices can be differentiated, and we can
   map them with usual vectors.

These two group of properties actually correspond to common
mathematical structures: a /group/ and a /differentiable manifold/.

You're probably already familiar with [[https://en.wikipedia.org/wiki/Group_(mathematics)][groups]], but let's recall the
basic properties:
- It's a set $G$ equipped with a binary operation $\cdot$.
- The group is closed under the operation: for any element $x,y$ in G,
  $x \cdot y$ is always in $G$.
- The operation is associative: $x \cdot (y \cdot z) = (x \cdot y)
  \cdot z$.
- There is a special element $e$ of $G$ (called the /identity
  element/), such that $x \cdot e = e \cdot x$ for all $x \in G$.
- For every element $x$ of $G$, there is a unique element of $G$
  denoted $x^{-1}$ such that $x \cdot x^{-1} = x^{-1} \cdot x = e$.

A [[https://en.wikipedia.org/wiki/Differentiable_manifold][differentiable manifold]] is a more complex beast. Although the
definition is more complex, we can loosely imagine it as a surface (in
higher dimension) on which we can compute derivatives at every
point. This means that there is a tangent hyperplane at each point,
which is a nice vector space where our derivatives will live.

* References