What are the practical applications and uses of Lie Algebras?

[mnb96]

Hello,
Can anyone explain in extremely simple words what Lie Algebras deal with, and are useful for? Could you also point out a very simple, toy example, in which the use of Lie Algebras is vital?

Thanks!

 

[Pythagorean]

Lie Groups

:P

I know a few things about Lie Groups, but I'm a n00b to it. I don't understand it overall because so much of the terminology requires a good foundation in abstract algebra (which I do not have). I'd be interested to hear if anyone can give you a satisfactory answer, maybe I'd learn something about the concept of it myself.

But, simple 3d space that you are familiar with is considered a Lie group.

 

[malawi_glenn]

The Lie Algebra is the commutator relations for the infinitesimal elements around the identity element in a certain transformation group.

e.g. the SU(2) group, the group of unitary 2x2 matrices with determinant = +1.
We can parametrize all these matrices in terms of three matrices and the identity.
These three matrices are the Pauli matrices, and we can write the element infinitelsimal close to the identiy as:
1 + epsilon_a * T^a

where a is an index a = 1,2,3, and we have employed the einstein summation convention.
The T^1, T^2, T^3 are the pauli matrices, and the commutation relation between these specify the Lie Algebra.

We use this since this is related to symmetries, e.g. spin and quantum numbers (in physics, I am a physicist)

Read more:
http://arxiv.org/abs/0810.3328
http://en.wikipedia.org/wiki/Lie_algebra