# 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?

• Before hopping into Lie Groups, make sure you are solid on understanding group theory and how the axioms of a group encode exactly what we think of when we think of "symmetry" Dec 9, 2020 at 6:41
• The easy - albeit somewhat arm waving answer - is a fluid flow. The generalization of flows and the shapes and structures that occur within them, which is what Sophus Lie was originally investigating. As for symmetry - a whirlpool for example translated along the direction of its flow - stream lines - one parameter subgroups - is identical to the "whirlpool" you started out with. Dec 9, 2020 at 21:44
• It's all a Lie.
– Mark
Dec 9, 2020 at 21:49
• @Mark Actually, its pronounced "Lee" Dec 10, 2020 at 22:32

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 ^_^

• Oh - and as for why physicists care -- the world we live in is full of continuously varying parameters. So the symmetries of space(time) very naturally have lie group structure. Iirc there's also reasons that quantum physicists care, but that's quite far from my wheelhouse so I'll let someone else discuss it Dec 9, 2020 at 5:24
• You could also define it as a topological group with a smooth structure. Dec 9, 2020 at 17:22
• @HallaSurvivor: it's far outside my wheelhouse as well, but there is a certain Lie group that reifies all known standard model interactions (in fact, the group predicted some interactions). Dec 11, 2020 at 16:18

I think symmetry was well addressed in the other answers. But since there was no mention of Lie algebras or infinitesimal generators, I'd like to add this answer, since these notions appear often in physics.

Many of the "ordinary" groups that we encounter in a first class on group theory belong to a class of groups called "finitely presentable groups". For such groups you can choose certain elements of that group as generators, and describe certain relations in terms of the group operation on those elements. It turns out that these generators and relations describe the group completely.

So for example $$\mathbb Z$$ is generated by $$1$$ (with no relations): $$\mathbb Z = <1>$$, and $$\mathbb Z_5$$ is generated by $$1$$ subject to the relation $$5*1 = 0$$, or: $$\mathbb Z_5 = <1 | 5*1 = 0>$$. In a way these generators and relations capture everything there is about the group. The generators act as "small building blocks" for the group. But there are groups for which this type of analysis doesn't work.

Lie groups are another class of groups that are also "manifolds" - i.e. they look like curves or surfaces in n-dimensional space. In such groups the "generators" are not elements of the group - instead they are infinitesimal objects (vectors!) that belong to a different space (a vector space! The "tangent space at the identity"), and the relations between them are defined not in terms of the group operation, but in terms of a new operation (the Lie bracket).

In more detail:

You need these vectors as generators because in a smooth group (think $$\mathbb R$$ as opposed to $$\mathbb Z$$, or the circle $$S^1$$ as opposed to $$\mathbb Z_5$$) you cannot choose a "smallest" element as a building block (there is no "smallest" real number in $$\mathbb R$$). So instead you need to get fancy and take limits of sequences of operations as they approach the group's identity. Taking these limits requires doing calculus on the surface, which is what you do in differential geometry. That's why this type of analysis is done on smooth groups (i.e. they are differentiable manifolds), and why Lie groups are required to be smooth. Because these generating vectors are obtained from a calculus-y limiting process, they are called "infinitesimal generators".

The way these vectors "generate" the group is that they sit with their base at the group's identity element and "point" in the direction of the operations whose limits they came from. You can then "start at the identity and follow a vector" in order to generate those operations. You can also "mix" vectors together using a special operation, by saying something like "let's follow $$v$$ and then $$w$$ and see what happens". This operation on two vectors gives you another vector (which is not obvious) - it's called the Lie bracket, written $$[v, w]$$. The Lie bracket of two "infinitesimal generators" (i.e. vectors) captures information that is similar to the relations between group elements in finitely presentable groups.

So now we have:

• finitely presentable groups: some "smallest" elements of the group can be used to build up the entire group, subject to some relations between those elements
• Lie groups: can't choose smallest elements, but we use limits and differential geometry to squeeze infinitesimal generators (i.e. vectors) out of our group. These vectors can "generate" the entire group, subject to relations between the infinitesimal generators via the Lie bracket.

To summarize: a Lie group is a smooth group that looks like a curve or a surface (possibly in n dimensions). It has "infinitesimal generators" that are tangent vectors sitting at the group's identity. We can "mix" these tangent vectors to get more tangent vectors using the Lie bracket. This vector space plus the Lie bracket is a Lie algebra. All together we get this calculus-y version of generators and relations.

Some final notes on notation that comes from Lie theory / differential geometry and that often looks like black magic:

• tangent vectors in differential geometry can be presented as "derivations", so you might write a tangent vector as $$\frac{d}{dy}$$. This can be mind-blowing when you first see it. When this happens in the context of Lie groups, that tangent vector $$\frac{d}{dy}$$ is often called an "infinitesimal generator of Y", for whatever Y represents. For example $$\frac{d}{dx}$$ might be an "infinitesimal generator of translations" (in the X direction), because the group this comes from represents translational symmetry.

• Since the Lie bracket is written $$[v, w]$$ for tangent-vectors / infinitesimal-generators $$v, w$$, and such vectors in differential geometry are often written as derivations e.g. $$\frac{d}{dx}$$, you'll often see expressions like $$[\frac{d}{dx}, \frac{d}{dy}]$$ (I'm looking at you quantum physics). Such notation is often an indication that there are Lie groups lurking in the background.

• the function that lets you "start in the direction of a tangent vector and follow the line of operations it points to" is often called... $$e$$. So for example you might write $$e^{t\frac{d}{dx}}$$. This is the "one parameter subgroup" generated by $$\frac{d}{dx}$$. If you see weird things being raised in an exponential: matrices, differential operators, etc... another clue that Lie groups are lurking about.

 subject to certain conditions of course

 if $$e^{i\theta} = cos(\theta) + isin(\theta)$$ ever bothered you, it has a beautiful interpretation in terms of Lie groups

• (+1) Suspect there's a typo in "$\mathbb Z_5$ is generated by $1$ subject to the relation $5*1 = 0$, or: $\mathbb Z_5 = <1 | 5*1 = 0>$" ... if 5 is the last element before you "wrap around" to 0, doesn't that mean you've generated $\mathbb Z_6$? Sep 2 at 20:15
• @Silverfish in this case 5 is not an element of the group, but an integer. It's shorthand for $5*1 = 1+1+1+1+1$, and setting that equal to $0$ effectively means you can never generate "$5$" in that group; the "highest" you'll ever get is $1+1+1+1=4$ Sep 3 at 17:12
• Apologies, I misinterpreted $*$ as being used to represent the group operation (i.e. addition). In case you are wondering, I was literally taught in school that (in $\mathbb Z_5$) we have $1*1=2$ and so on, up to $4*1=0$ and so, as you say, $5$ never gets generated. I was taken by surprise to see $5*1=0$ given that "$5$" in some sense "doesn't exist" in this context (except as an integer belonging in the same equivalence class as $0$), which is why it looked to me like $\mathbb Z_6$ could have been intended. Obviously I use $*$ for multiplication every day yet somehow it didn't cross my mind! Sep 13 at 19:53

There is already an answer, so let me try to give you a few simple, naive and hand-wavy ideas.