TITLE: What is Lie Theory/ a Lie Group, simply?
QUESTION [45 upvotes]: I'm studying physics, and I continually come across mentions of "Lie Theory" and "Lie Groups" as they relate to such topics as particle physics and String Theory, as well as vague mentions of "symmetry".
I've attempted to read some texts on the topic and, while I feel I could probably sift through it in due time, it is very terse and esoteric. I've had group theory, calculus up to university calc II, and a teensy bit of analysis. Until I'm able to study this proper, what is Lie Theory/ a Lie Group, put simply? What is with these vague mentions of "symmetry"? What can be understood in terms of what I know now?

REPLY [3 votes]: There is already an answer, so let me try to give you a few simple, naive and hand-wavy ideas.

About symmetry:

"Symmetry" refers to the fact that some things can transform into themselves without changing too much. For example, if someone shows you a cube on a table, ask you to close your eyes, and then to open them, you will not be able to tell if this person has rotated the cube or not. This is the fact that a cube possesses a lot of symmetries. Now, let us assume that this cube has coloured faces (each face has a different colour). Then, you will be able to detect any rotations: the colours will change. Thus, colouring breaks the symmetry; otherwise said, a coloured cube is less symmetric than a cube.
Now let us look at a physical theory as a mathematical object. Then one often makes a lot of symmetry assumptions: you cannot tell where in the universe or when in time you are just making experiments and using newtonian mechanics, since it is a space and time invariant theory! I mean that in an empty universe, two balls will attract themselves in the same way, no matter where they are, nor what time it is.
Now, history has shown that one can understand objects by understanding its symmetries. And group theory is a way to formulate symmetry. This is quite a big topic!

About the Lie things:

Differential calculus is powerful, so Sophus Lie thought that instead of studying symmetry in its full generality, it would be easier to study symmetry with the help of differential calculus. Hence Lie theory.