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Comment by aswanson | Hacker News Reader

I wish someone would better explain Lie algebra.

Lie algebra is a group whose underlying set is smooth. For example, angles (rotations) on a circle (or units in the complex numbers) form a Lie algebra. You can add and subtract angles on a circle, and if you perturb one angle you perturb the result slightly: (theta1 + delta) + theta2 = theta1 + theta2 + delta.

That's a trivial example, of course.

Contrast with a symmetry group of a hexagon. There's no (non-degenerate) notion of a continuous transition from identity to rotate-by-pi.

Ah! So its just a group that has a defined group of continuous transformations that can be applied to it?

Actually, a Lie Group is a group which is simultaneously a manifold, such that the group multiplication corresponds to diffeomorphisms of the manifold onto itself. The Lie Algebra corresponding to that Lie Group is the tangent space of the identity. The elements of the Lie Algebra generate the given Lie Group via the exponential map.

What do you mean by "generate" here? The exponential map is not necessarily surjective.

Not really.

Very roughly: It's a group with a an infinite number of elements which is smooth, in the sense that there is a notion that elements can be close or far from each other, and locally around each element the group looks like R^n (for some n). Of course the group operation must play nice with this notion of "close". I.e. if elements A and B are close, after multiplying both of them by X, AX and BX can't be too far: the group operation does not "tear" the structure apart.

So in the example above of rotations on the plane, you can visualize the group as a circle where a point represents the rotation by that angle. Locally it looks like R, but it has a different global structure.

This is incorrect.

A Lie _group_ is a group which is also a manifold, such that the group operations are smooth. (I don't know what it means for a set to be smooth.)

A Lie _algebra_ is a vector space together with a bilinear operation (the Lie bracket, written [a, b]) which also satisfies the Jacobi identity:

  [a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0.

How are they related? If G is a Lie group, the tangent space at the identity is a Lie algebra. What does the Lie algebra represent? Well, unlike a plain manifold, in a Lie group, given a tangent vector a, there is a path starting from a the identity in the Lie group always in the direction a. Given two tangent vectors a, b, you can flow a little in the direction a, and then a little in the direction b, or vice versa. The Lie bracket can be thought of as a measure of how the flow fails to commute.

http://www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathe... is quite readable

I'll would request the same for p-adic numbers, while they're at it!

Obligatory Schemes and Syzygies

p-adic numbers are measure of necesary- resolution to describe. A pair of p-adics is very close when you can use a very coarse ruler (of tick size p^n, large n) to measure the exact difference (1 vs 10000000001). p-adic numbers are far away when you need a very fine grid (1.0 vs 10.000000000001) to measure the exact difference.

I'd request the same for whatever mathematical concepts Quants use for stock market trading. Or at least a list of concepts useful for such an endeavor.

p-adic numbers are actually quite simple; so simple, I've actually seen children invent it on their own!! They're just like ordinary decimal notation numbers, except instead of allowing these to extend infinitely to the right (as in 3.14159...), you allow them to extend infinitely to the left. So, in addition to familiar numbers like 45 or 67.8, you also have numbers like ...95141.3 or ...999 or what have you.

You add and multiply these according to the same rules you're already familiar with. So, for example, if you add 1 to ...999, the last digit of the output is 0, and you get a carry. Making the second to last digit 0, with another carry. And so on and so on, making the result ...0000 over all. Thus, ...999 acts like -1.

(If we were working in base two, this last example would be just like the "two's complement" you are perhaps familiar with from computer arithmetic!)

In fact, most discussion of p-adic numbers isn't done in base ten. Instead, people typically focus on p-adics in a prime base (hence the p). Why? Because in a prime base, you will find that every nonzero p-adic number has a multiplicative inverse, which is very convenient (while in a composite base, you will find that sometimes nonzero numbers multiply to zero). But there's nothing actually stopping you from making use of the notion for non-prime base, should you be interested in doing so; the notion is still perfectly coherent. (That having been said, another reason mathematicians focus on p-adics in prime bases is that the Chinese Remainder Theorem essentially allows one to reduce the study of all other bases to this case.)

There's a lot of beautiful further theory to explore here, but again, the basic idea is hopefully quite simple. Let me know if you have any questions and I'll be happy to try explaining further or more clearly.

Interesting. Very similar to the way ints are represented in Python (since they have arbitrary length, negative numbers are represented with infinitely many 1's in the most significant bits).