
Atlas of Lie Groups and Representations - the-mitr
http://www.liegroups.org/software/documentation/atlasofliegroups-docs/index.html
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orlandpm
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.

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winkv
Wow! thanks for explaining this, i was aware of the discrete group but never
had the intution of lie groups,never understood continuous symmetries. It is
sad that math books are full of proofs but never care to explain the intution
behind it.

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nils-m-holm
Same here! I find math fascinating, but everything I know about it is
painfully gained knowledge. Usually I read three to four books about the same
subject and then learn from the intersection. It's a sad state of affairs!

I'd be interested in working on better math books (I have lots of experience
in writing CS textbooks), but I'd need someone who can explain math topics to
me in the first place. Math books in general are completely useless.

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adamnemecek
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](https://people.cs.uchicago.edu/~risi/papers/KondorThesis.pdf)

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disabledalpha
That thesis discusses bispectral invariants on the group of rigid motions. I
worked on a recent monograph on the same subject, which could interest you :
[https://arxiv.org/pdf/1704.03069.pdf](https://arxiv.org/pdf/1704.03069.pdf)

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wyldfire
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](https://en.wikipedia.org/wiki/Lie_group)

