[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.

[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.

[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.