What is Lie Theory/ a Lie Group, simply?

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?

This was going to be a comment, but it got too long.

I don't know how "simple" this is, but the 5 word summary is "a group with manifold structure". Or perhaps if you're a topologist, "a manifold with group structure". Now that the snarky answer is out of the way, I can try to be a bit more helpful.

Remember that groups measure the symmetries of other objects. The first examples that you see are often the symmetries of discrete objects. I.e. the symmetries of a pentagon correspond to $D_{10}$, and more generally, you get a dihedral group from looking symmetries of regular polygons. If you have a polygon with $n$ sides, then you can rotate by an angle of $\frac{2\pi}{n}$ or reflect through any of a number of axes.

What happens when the object you're studying is smooth in some sense, though? For instance, instead of looking at the symmetries of a polygon, let's look at the symmetries of a circle. Now there's no "smallest angle" to rotate through. You have a continuous parameter of group elements. For each $\theta \in [0,2\pi)$ you can rotate through that angle $\theta$. This (to me) is the defining feature of a lie group. Let's forget the reflections going forward and focus on the rotations.

How do we make the idea of a "continuous parameter" of group elements precise? It turns out the "right approach" is to give your group the structure of a smooth manifold. Remember a manifold is (roughly) a thing that locally looks like $\mathbb{R}^n$. So in the case of the symmetries of a circle (for instance), every rotation $\theta$ has a neighborhood of "nearby" rotations $(\theta - \epsilon, \theta + \epsilon)$, and this neighborhood looks like a neighborhood of $\mathbb{R}$. This is what formalizes the idea that the group elements "vary continuously". You also want to be smart about how the group structure and the manifold structure interact: The multiplication/inversion operations $m : G \times G \to G$ and $i : G \to G$ should both be differentiable. There's a lot more to say, but in the interest of keeping the answer short and relatively elementary I'll leave it there.

If you're looking for a good first reference on lie groups, and you haven't at least skimmed Stillwell's "Naive Lie Theory", you're in for a treat. Like all of his books, it's a very polite read, and it covers a lot of ground with almost no prerequisites at all. He doesn't go into the nitty gritty of manifold theory, which can bog down a lot of the discussion. Instead, he focuses on groups of matrices (whose manifold structure is obvious: after all, you can see the smooth parameters in the entries of the matrix!). This brings the entire text down to a very concrete level, and makes the subject very approachable.

I hope this helps ^_^

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.