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I have been reading lately about Lie groups, and although all books keep listing the groups, and talk about Lie algebras and all that, one thing I still don't know how is it made, and I guess it's the most important question as much as my understanding goes, but most textbooks keep ignoring.

My question is, Now if I have a symmetry, and I could put it into equation (symmetry of sea waves - simplified for example, or symmetry of planetary motion), how to use Lie groups to express this symmetry? How to identify which Lie group responsible for that?

I am still at the very beginning but for some reason whatever I research I cannot find an answer to this question. I hope I would find help here!

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up vote 6 down vote accepted

A group is a set of transformations that don't change the internal relationship in "something", whether it's another mathematical structure or a set of equations describing a physical object or the physical object itself.

So putting a symmetry into equations means to find a set of functions or maps $f$ such that $$ x \text{ is OK} \Leftrightarrow f(x) \text{ is OK}$$ where "is OK" means that some conditions are satisfied, for example it may mean that the equations of motion are solved. To identify the group of symmetries of a physical system means to find all the maps $f$ for which the equivalence above is obeyed – and to compare the multiplicative table with possible tables for different groups, and find which one is the right one.

Lie group is just a special subset of groups in which the functions or maps $f$ make up a differentiable manifold, i.e. in which one needs continuous parameters such as real numbers to uniquely pinpoint an element of the group. A special mathematical toolkit, one that involves Lie algebras and their classification and representations etc., is useful to analyze Lie groups and systems with Lie group symmetries.

The very defining property of the groups is that their inner structure – especially the composition rule $(g_1,g_2)\mapsto g_1\cdot g_2$ – is independent from the system on which the group acts. This allows us to treat many aspects of these systems with the same symmetries by universal tools.

I am afraid my answer won't satisfy what you want to hear and I am also afraid that the reason is that you want to hear something that isn't true. ;-) A group itself isn't an equation.

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If I understood you correctly, and please correct me if I am wrong, Lie groups are just an abstract representation tool like matrices, they make solving problems easier, but problems could be solved somehow else if we try hard enough. – mitstudent Dec 20 '12 at 18:54
Lie groups are literally sets of matrices – and the matrices may be written down in many ways. So in this sense, Lie groups are even more abstract than matrices. Matrices are very specific tables of numbers, aren't they? Otherwise yes. They're abstract tools that allow us new ways to solve problems but they could be solved without groups, too. Well, in some cases, we would be just describing the methods of group theory in an awkward language that avoids the groups. They're really supernatural. Symmetries are the most natural properties or attributes of any system. – Luboš Motl Dec 21 '12 at 20:24
@Luboš - although it is certainly true that the vast majority of Lie groups of interest to physicists are finite dimensional, compact, meet a few other requirements, and therefore are isomorphic to matrix groups, not every Lie group is: there are infinite dimensional Lie groups and even uncountable infinite dimensional Lie groups as well. Then there are non-compact Lie groups like the Lorentz group. Not all of these are linear groups, though again, the most interesting ones are, thanks largely to Ado's theorem. – Matt J. Jul 16 '13 at 2:56

The key to answering part of your question is knowledge of generators of the Lie Group under consideration. You want to know what a group is made out of. You can use the Lie Algebra of the generators of a Lie Group to determine what simpler Groups that Group is made out of. For example, the Lorentz Group is made out of two SU(2) Groups because the algebra of its generators separates into two complexified SU(2) algebras. Exponentiating a 4X4 matrix- formed from two 2X2 matrices that are complexified generators of SU(2) on the diagonal- will result in a Lorentz Transformation. This is covered well in QFT Demystified, QFT in a Nutshell by Zee, Lie Groups for Pedestrians, and excellent video Lectures by Robert De Mello Koch that are available online.

You must learn to 1.construct the generators and 2. construct higher or lower dimensional representations using "Ladder Operators" formed from the algebra. Both these tasks are mathematically straightforward and can be found in the Koch lectures.

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