In Spring 2016 I was taking 18.757 Representations of Lie Algebras. Since I knew next to nothing about either Lie groups or algebras, I was forced to quickly learn about their basic facts and properties. These are the notes that I wrote up accordingly. Proofs of most of these facts can be found in standard textbooks, for example Kirillov.
1. Lie groups
Let or , depending on taste.
Throughout, we will let denote the identity, or if we need further emphasis.
Note that in particular, every group can be made into a Lie group by endowing it with the discrete topology. This is silly, so we usually require only focus on connected groups:
In fact, we can also reduce this to the study of simply connected Lie groups as follows.
Here are some examples of Lie groups.
As geometric objects, Lie groups enjoy a huge amount of symmetry. For example, any neighborhood of can be “copied over” to any other point by the natural map . There is another theorem worth noting, which is that:
2. Haar measure
Recall the following result and its proof from representation theory:
Proof: Take a representation and equip it with an arbitrary inner form . Then we can average it to obtain a new inner form
which is -invariant. Thus given a subrepresentation we can just take its orthogonal complement to decompose .
We would like to repeat this type of proof with Lie groups. In this case the notion doesn’t make sense, so we want to replace it with an integral instead. In order to do this we use the following:
Note that we have:
This follows by just noting that if is Radon measure on , then . This now lets us deduce that
Indeed, we can now consider
as we described at the beginning.
3. The tangent space at the identity
In light of the previous comment about neighborhoods of generating , we see that to get some information about the entire Lie group it actually suffices to just get “local” information of at the point (this is one formalization of the fact that Lie groups are super symmetric).
To do this one idea is to look at the tangent space. Let be an -dimensional Lie group (over ) and consider the tangent space to at the identity . Naturally, this is a -vector space of dimension . We call it the Lie algebra associated to .
4. The exponential map
Right now, is just a vector space. However, by using the group structure we can get a map from back into . The trick is “differential equations”:
We will write to emphasize the argument being thought of as “time”. Thus this proposition should be intuitively clear: the theory of differential equations guarantees that is defined and unique in a small neighborhood of . Then, the group structure allows us to extend uniquely to the rest of , giving a trajectory across all of . This is sometimes called a one-parameter subgroup of , but we won’t use this terminology anywhere in what follows.
This lets us define:
The exponential map gets its name from the fact that for all the examples I discussed before, it is actually just the map . Note that below, for a matrix ; this is called the matrix exponential.
Actually, taking the tangent space at the identity is a functor. Consider a map of Lie groups, with lie algebras and . Because is a group homomorphism, . Now, by manifold theory we know that maps between manifolds gives a linear map between the corresponding tangent spaces, say . For us we obtain a linear map
In fact, this fits into a diagram
Here are a few more properties of :
- , which is immediate by looking at the constant trajectory .
- , i.e. the total derivative is the identity. This is again by construction.
- In particular, by the inverse function theorem this implies that is a diffeomorphism in a neighborhood of , onto a neighborhood of .
- commutes with the commutator. (By the above diagram.)
5. The commutator
Right now is still just a vector space, the tangent space. But now that there is map , we can use it to put a new operation on , the so-called commutator.
The idea is follows: we want to “multiply” two elements of . But is just a vector space, so we can’t do that. However, itself has a group multiplication, so we should pass to using , use the multiplication in and then come back.
Here are the details. As we just mentioned, is a diffeomorphism near . So for , close to the origin of , we can look at and , which are two elements of close to . Multiplying them gives an element still close to , so its equal to for some unique , call it .
One can show in fact that can be written as a Taylor series in two variables as
where is a skew-symmetric bilinear map, meaning . It will be more convenient to work with than itself, so we give it a name:
Now we know multiplication in is associative, so this should give us some nontrivial relation on the bracket . Specifically, since
we should have that , and this should tell us something. In fact, the claim is:
Proof: Although I won’t prove it, the third-order terms (and all the rest) in our definition of can be written out explicitly as well: for example, for example, we actually have
The general formula is called the Baker-Campbell-Hausdorff formula.
Then we can force ourselves to expand this using the first three terms of the BCS formula and then equate the degree three terms. The left-hand side expands initially as , and the next step would be something ugly.
This computation is horrifying and painful, so I’ll pretend I did it and tell you the end result is as claimed.
There is a more natural way to see why this identity is the “right one”; see Qiaochu. However, with this proof I want to make the point that this Jacobi identity is not our decision: instead, the Jacobi identity is forced upon us by associativity in .
In any case, with the Jacobi identity we can define an general Lie algebra as an intrinsic object with a Jacobi-satisfying bracket:
Note that a Lie algebra may even be infinite-dimensional (even though we are assuming is finite-dimensional, so that they will never come up as a tangent space).
6. The fundamental theorems
We finish this list of facts by stating the three “fundamental theorems” of Lie theory. They are based upon the functor
we have described earlier, which is a functor
- from the category of Lie groups
- into the category of finite-dimensional Lie algebras.
The first theorem requires the following definition:
If we drop the “simply connected” condition, we obtain a functor which is faithful and exact, but not full: non-isomorphic Lie groups can have isomorphic Lie algebras (one example is and ).