Atlas of Lie Groups and Representations

For anyone curious, a Lie group is a continuous collection of symmetries. While a square, for example, has 8 symmetries (identity, three rotations, and 4 reflections), a circle has infinitely many: you can reflect it about any axis and rotate it by any angle and get the same circle you started with.

Since Lie groups are indexed by continuous parameters, they can be studied as spaces in their own right. For example, the Lie group of rotations of the circle (called SO(2)) is topologically the same as the circle: you can put the rotations of the circle in 1-1 correspondence with the points on the circle. For the (2-)sphere, the Lie group of rotations actually looks more like a higher dimensional (3-)sphere.

Lie algebras consist of vectors (rather than symmetries), and encode most of the structure of corresponding lie groups. For example, the Lie algebra for the circle rotation group is the vector space of the real line: each rotation is indexed by a real number (the angle, with redundancy) and composition of rotations is obtained by adding angles. In higher dimensions, especially where symmetries don't commute, the Lie algebra is indispensable and often the starting point for study.

What's the best resource on the machine learning/Lie group relationship?

Edit: NVM I found something https://people.cs.uchicago.edu/~risi/papers/KondorThesis.pdf

It has nothing to do with social groups dedicated to eliminating truth, as I've now learned.

Wikipedia article [1] states:

> In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups.

[1] https://en.wikipedia.org/wiki/Lie_group