Facts about Lie Groups and Algebras

Power Overwhelming 2018-03-12

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 ${K = \mathbb R}$ or ${K = \mathbb C}$ , depending on taste.

Definition 1

A Lie group is a group ${G}$ which is also a ${K}$ -manifold; the multiplication maps ${G \times G \rightarrow G}$ (by ${(g_1, g_2) \mapsto g_1g_2}$ ) and the inversion map ${G \rightarrow G}$ (by ${g \mapsto g^{-1}}$ ) are required to be smooth.

A morphism of Lie groups is a map which is both a map of manifolds and a group homomorphism.

Throughout, we will let ${e \in G}$ denote the identity, or ${e_G}$ if we need further emphasis.

Note that in particular, every group ${G}$ 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:

Proposition 2 (Reduction to connected Lie groups)

Let ${G}$ be a Lie group and ${G^0}$ the connected component of ${G}$ which contains ${e}$ . Then ${G^0}$ is a normal subgroup, itself a Lie group, and the quotient ${G/G^0}$ has the discrete topology.

In fact, we can also reduce this to the study of simply connected Lie groups as follows.

Proposition 3 (Reduction to simply connected Lie groups)

If ${G}$ is connected, let ${\pi : \widetilde G \rightarrow G}$ be its universal cover. Then ${\widetilde G}$ is a Lie group, ${\pi}$ is a morphism of Lie groups, and ${\ker \pi \cong \pi_1(G)}$ .

Here are some examples of Lie groups.

Example 4 (Examples of Lie groups)

${\mathbb R}$ under addition is a real one-dimensional Lie group.
${\mathbb C}$ under addition is a complex one-dimensional Lie group (and a two-dimensional real Lie group)!
The unit circle ${S^1 \subseteq \mathbb C}$ is a real Lie group under multiplication.
${\text{GL }(n, K) \subset K^{\oplus n^2}}$ is a Lie group of dimension ${n^2}$ . This example becomes important for representation theory: a representation of a Lie group ${G}$ is a morphism of Lie groups ${G \rightarrow \text{GL }(n, K)}$ .
${\text{SL }(n, K) \subset \text{GL }(n, K)}$ is a Lie group of dimension ${n^2-1}$ .

As geometric objects, Lie groups ${G}$ enjoy a huge amount of symmetry. For example, any neighborhood ${U}$ of ${e}$ can be “copied over” to any other point ${g \in G}$ by the natural map ${gU}$ . There is another theorem worth noting, which is that:

Proposition 5

If ${G}$ is a connected Lie group and ${U}$ is a neighborhood of the identity ${e \in G}$ , then ${U}$ generates ${G}$ as a group.

2. Haar measure

Recall the following result and its proof from representation theory:

Claim 6

For any finite group ${G}$ , ${\mathbb C[G]}$ is semisimple; all finite-dimensional representations decompose into irreducibles.

Proof: Take a representation ${V}$ and equip it with an arbitrary inner form ${\left< -,-\right>_0}$ . Then we can average it to obtain a new inner form

$\displaystyle \left< v, w \right> = \frac{1}{|G|} \sum_{g \in G} \left< gv, gw \right>_0.$

which is ${G}$ -invariant. Thus given a subrepresentation ${W \subseteq V}$ we can just take its orthogonal complement to decompose ${V}$ . $\Box$ We would like to repeat this type of proof with Lie groups. In this case the notion ${\sum_{g \in G}}$ doesn’t make sense, so we want to replace it with an integral ${\int_{g \in G}}$ instead. In order to do this we use the following:

Theorem 7 (Haar measure)

Let ${G}$ be a Lie group. Then there exists a unique Radon measure ${\mu}$ (up to scaling) on ${G}$ which is left-invariant, meaning

$\displaystyle \mu(g \cdot S) = \mu(S)$

for any Borel subset ${S \subseteq G}$ and “translate” ${g \in G}$ . This measure is called the (left) Haar measure.

Example 8 (Examples of Haar measures)

The Haar measure on ${(\mathbb R, +)}$ is the standard Lebesgue measure which assigns ${1}$ to the closed interval ${[0,1]}$ . Of course for any ${S}$ , ${\mu(a+S) = \mu(S)}$ for ${a \in \mathbb R}$ .
The Haar measure on ${(\mathbb R \setminus \{0\}, \times)}$ is given by
$\displaystyle \mu(S) = \int_S \frac{1}{|t|} \; dt.$

In particular, ${\mu([a,b]) = \log(b/a)}$ . One sees the invariance under multiplication of these intervals.
Let ${G = \text{GL }(n, \mathbb R)}$ . Then a Haar measure is given by
$\displaystyle \mu(S) = \int_S |\det(X)|^{-n} \; dX.$
For the circle group ${S^1}$ , consider ${S \subseteq S^1}$ . We can define
$\displaystyle \mu(S) = \frac{1}{2\pi} \int_S d\varphi$

across complex arguments ${\varphi}$ . The normalization factor of ${2\pi}$ ensures ${\mu(S^1) = 1}$ .

Note that we have:

Corollary 9

If the Lie group ${G}$ is compact, there is a unique Haar measure with ${\mu(G) = 1}$ .

This follows by just noting that if ${\mu}$ is Radon measure on ${X}$ , then ${\mu(X) < \infty}$ . This now lets us deduce that

Corollary 10 (Compact Lie groups are semisimple)

${\mathbb C[G]}$ is semisimple for any compact Lie group ${G}$ .

Indeed, we can now consider

$\displaystyle \left< v,w\right> = \int_G \left< g \cdot v, g \cdot w\right>_0 \; dg$

as we described at the beginning.

3. The tangent space at the identity

In light of the previous comment about neighborhoods of ${e}$ generating ${G}$ , we see that to get some information about the entire Lie group it actually suffices to just get “local” information of ${G}$ at the point ${e}$ (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 ${G}$ be an ${n}$ -dimensional Lie group (over ${K}$ ) and consider ${\mathfrak g = T_eG}$ the tangent space to ${G}$ at the identity ${e \in G}$ . Naturally, this is a ${K}$ -vector space of dimension ${n}$ . We call it the Lie algebra associated to ${G}$ .

Example 11 (Lie algebras corresponding to Lie groups)

${(\mathbb R, +)}$ has a real Lie algebra isomorphic to ${\mathbb R}$ .
${(\mathbb C, +)}$ has a complex Lie algebra isomorphic to ${\mathbb C}$ .
The unit circle ${S^1 \subseteq \mathbb C}$ has a real Lie algebra isomorphic to ${\mathbb R}$ , which we think of as the “tangent line” at the point ${1 \in S^1}$ .

Example 12 ( ${\mathfrak{gl}(n, K)}$ )

Let’s consider ${\text{GL }(n, K) \subset K^{\oplus n^2}}$ , an open subset of ${K^{\oplus n^2}}$ . Its tangent space should just be an ${n^2}$ -dimensional ${K}$ -vector space. By identifying the components in the obvious way, we can think of this Lie algebra as just the set of all ${n \times n}$ matrices.

This Lie algebra goes by the notation ${\mathfrak{gl}(n, K)}$ .

Example 13 ( ${\mathfrak{sl}(n, K)}$ )

Recall ${\text{SL }(n, K) \subset \text{GL }(n, K)}$ is a Lie group of dimension ${n^2-1}$ , hence its Lie algebra should have dimension ${n^2-1}$ . To see what it is, let’s look at the special case ${n=2}$ first: then

$\displaystyle \text{SL }(2, K) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid ad - bc = 1 \right\}.$

Viewing this as a polynomial surface ${f(a,b,c,d) = ad-bc}$ in ${K^{\oplus 4}}$ , we compute

$\displaystyle \nabla f = \left< d, -c, -b, a \right>$

and in particular the tangent space to the identity matrix ${\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}}$ is given by the orthogonal complement of the gradient

$\displaystyle \nabla f (1,0,0,1) = \left< 1, 0, 0, 1 \right>.$

Hence the tangent plane can be identified with matrices satisfying ${a+d=0}$ . In other words, we see

$\displaystyle \mathfrak{sl}(2, K) = \left\{ T \in \mathfrak{gl}(2, K) \mid \text{Tr } T = 0. \right\}.$

By repeating this example in greater generality, we discover

$\displaystyle \mathfrak{sl}(n, K) = \left\{ T \in \mathfrak{gl}(n, K) \mid \text{Tr } T = 0. \right\}.$

4. The exponential map

Right now, ${\mathfrak g}$ is just a vector space. However, by using the group structure we can get a map from ${\mathfrak g}$ back into ${G}$ . The trick is “differential equations”:

Proposition 14 (Differential equations for Lie theorists)

Let ${G}$ be a Lie group over ${K}$ and ${\mathfrak g}$ its Lie algebra. Then for every ${x \in \mathfrak g}$ there is a unique homomorphism

$\displaystyle \gamma_x : K \rightarrow G$

which is a morphism of Lie groups, such that

$\displaystyle \gamma_x'(0) = x \in T_eG = \mathfrak g.$

We will write ${\gamma_x(t)}$ to emphasize the argument ${t \in K}$ being thought of as “time”. Thus this proposition should be intuitively clear: the theory of differential equations guarantees that ${\gamma_x}$ is defined and unique in a small neighborhood of ${0 \in K}$ . Then, the group structure allows us to extend ${\gamma_x}$ uniquely to the rest of ${K}$ , giving a trajectory across all of ${G}$ . This is sometimes called a one-parameter subgroup of ${G}$ , but we won’t use this terminology anywhere in what follows.

This lets us define:

Definition 15

Retain the setting of the previous proposition. Then the exponential map is defined by

$\displaystyle \exp : \mathfrak g \rightarrow G \qquad\text{by}\qquad x \mapsto \gamma_x(1).$

The exponential map gets its name from the fact that for all the examples I discussed before, it is actually just the map ${e^\bullet}$ . Note that below, ${e^T = \sum_{k \ge 0} \frac{T^k}{k!}}$ for a matrix ${T}$ ; this is called the matrix exponential.

Example 16 (Exponential Maps of Lie algebras)

If ${G = \mathbb R}$ , then ${\mathfrak g = \mathbb R}$ too. We observe ${\gamma_x(t) = e^{tx} \in \mathbb R}$ (where ${t \in \mathbb R}$ ) is a morphism of Lie groups ${\gamma_x : \mathbb R \rightarrow G}$ . Hence
$\displaystyle \exp : \mathbb R \rightarrow \underbrace{\mathbb R}_{=G} \qquad \exp(x) = \gamma_x(1) = e^t \in \mathbb R = G.$
Ditto for ${\mathbb C}$ .
For ${S^1}$ and ${x \in \mathbb R}$ , the map ${\gamma_x : \mathbb R \rightarrow S^1}$ given by ${t \mapsto e^{itx}}$ works. Hence
$\displaystyle \exp : \mathbb R \rightarrow S^1 \qquad \exp(x) = \gamma_x(1) = e^{it} \in S^1.$
For ${\text{GL }(n, K)}$ , the map ${\gamma_X : K \rightarrow \text{GL }(n, K)}$ given by ${t \mapsto e^{tX}}$ works nicely (now ${X}$ is a matrix). (Note that we have to check ${e^{tX}}$ is actually invertible for this map to be well-defined.) Hence the exponential map is given by
$\displaystyle \exp : \mathfrak{gl}(n,K) \rightarrow \text{GL }(n,K) \qquad \exp(X) = \gamma_X(1) = e^X \in \text{GL }(n, K).$
Similarly,
$\displaystyle \exp : \mathfrak{sl}(n,K) \rightarrow \text{SL }(n,K) \qquad \exp(X) = \gamma_X(1) = e^X \in \text{SL }(n, K).$

Here we had to check that if ${X \in \mathfrak{sl}(n,K)}$ , meaning ${\text{Tr } X = 0}$ , then ${\det(e^X) = 1}$ . This can be seen by writing ${X}$ in an upper triangular basis.

Actually, taking the tangent space at the identity is a functor. Consider a map ${\varphi : G_1 \rightarrow G_2}$ of Lie groups, with lie algebras ${\mathfrak g_1}$ and ${\mathfrak g_2}$ . Because ${\varphi}$ is a group homomorphism, ${G_1 \ni e_1 \mapsto e_2 \in G_2}$ . Now, by manifold theory we know that maps ${f : M \rightarrow N}$ between manifolds gives a linear map between the corresponding tangent spaces, say ${Tf : T_pM \rightarrow T_{fp}N}$ . For us we obtain a linear map

$\displaystyle \varphi_\ast = T \varphi : \mathfrak g_1 \rightarrow \mathfrak g_2.$

In fact, this ${\varphi_\ast}$ fits into a diagram

exp-commute

Here are a few more properties of ${\exp}$ :

${\exp(0) = e \in G}$ , which is immediate by looking at the constant trajectory ${\phi_0(t) \equiv e}$ .
${\exp'(x) = x \in \mathfrak g}$ , i.e. the total derivative ${D\exp : \mathfrak g \rightarrow \mathfrak g}$ is the identity. This is again by construction.
In particular, by the inverse function theorem this implies that ${\exp}$ is a diffeomorphism in a neighborhood of ${0 \in \mathfrak g}$ , onto a neighborhood of ${e \in G}$ .
${\exp}$ commutes with the commutator. (By the above diagram.)

5. The commutator

Right now ${\mathfrak g}$ is still just a vector space, the tangent space. But now that there is map ${\exp : \mathfrak g \rightarrow G}$ , we can use it to put a new operation on ${\mathfrak g}$ , the so-called commutator.

The idea is follows: we want to “multiply” two elements of ${\mathfrak g}$ . But ${\mathfrak g}$ is just a vector space, so we can’t do that. However, ${G}$ itself has a group multiplication, so we should pass to ${G}$ using ${\exp}$ , use the multiplication in ${G}$ and then come back.

Here are the details. As we just mentioned, ${\exp}$ is a diffeomorphism near ${e \in G}$ . So for ${x}$ , ${y}$ close to the origin of ${\mathfrak g}$ , we can look at ${\exp(x)}$ and ${\exp(y)}$ , which are two elements of ${G}$ close to ${e}$ . Multiplying them gives an element still close to ${e}$ , so its equal to ${\exp(z)}$ for some unique ${z}$ , call it ${\mu(x,y)}$ .

One can show in fact that ${\mu}$ can be written as a Taylor series in two variables as

$\displaystyle \mu(x,y) = x + y + \frac{1}{2} [x,y] + \text{third order terms} + \dots$

where ${[x,y]}$ is a skew-symmetric bilinear map, meaning ${[x,y] = -[y,x]}$ . It will be more convenient to work with ${[x,y]}$ than ${\mu(x,y)}$ itself, so we give it a name:

Definition 17

This ${[x,y]}$ is called the commutator of ${G}$ .

Now we know multiplication in ${G}$ is associative, so this should give us some nontrivial relation on the bracket ${[,]}$ . Specifically, since

$\displaystyle \exp(x) \left( \exp(y) \exp(z) \right) = \left( \exp(x) \exp(y) \right) \exp(z).$

we should have that ${\mu(x, \mu(y,z)) = \mu(\mu(x,y), z)}$ , and this should tell us something. In fact, the claim is:

Theorem 18

The bracket ${[,]}$ satisfies the Jacobi identity

$\displaystyle [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0.$

Proof: Although I won’t prove it, the third-order terms (and all the rest) in our definition of ${[x,y]}$ can be written out explicitly as well: for example, for example, we actually have

$\displaystyle \mu(x,y) = x + y + \frac{1}{2} [x,y] + \frac{1}{12} \left( [x, [x,y]] + [y,[y,x]] \right) + \text{fourth order terms} + \dots.$

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 ${\mu\left( x, y + z + \frac{1}{2} [y,z] + \frac{1}{12} \left( [y,[y,z]] + [z,[z,y] \right) \right)}$ , 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. $\Box$ 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 ${G}$ .

Example 19 (Examples of commutators attached to Lie groups)

If ${G}$ is an abelian group, we have ${-[y,x] = [x,y]}$ by symmetry and ${[x,y] = [y,x]}$ from ${\mu(x,y) = \mu(y,x)}$ . Thus ${[x,y] = 0}$ in ${\mathfrak g}$ for any abelian Lie group ${G}$ .
In particular, the brackets for ${G \in \{\mathbb R, \mathbb C, S^1\}}$ are trivial.
Let ${G = \text{GL }(n, K)}$ . Then one can show that
$\displaystyle [T,S] = TS - ST \qquad \forall S, T \in \mathfrak{gl}(n, K).$
Ditto for ${\text{SL }(n, K)}$ .

In any case, with the Jacobi identity we can define an general Lie algebra as an intrinsic object with a Jacobi-satisfying bracket:

Definition 20

A Lie algebra over ${k}$ is a ${k}$ -vector space equipped with a skew-symmetric bilinear bracket ${[,]}$ satisfying the Jacobi identity.

A morphism of Lie algebras and preserves the bracket.

Note that a Lie algebra may even be infinite-dimensional (even though we are assuming ${G}$ is finite-dimensional, so that they will never come up as a tangent space).

Example 21 (Associative algebra ${\rightarrow}$ Lie algebra)

Any associative algebra ${A}$ over ${k}$ can be made into a Lie algebra by taking the same underlying vector space, and using the bracket ${[a,b] = ab - ba}$ .

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

$\displaystyle \mathscr{L} : G \mapsto T_e G$

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:

Definition 22

A Lie subgroup ${H}$ of a Lie group ${G}$ is a subgroup ${H}$ such that the inclusion map ${H \hookrightarrow G}$ is also an injective immersion.

A Lie subalgebra ${\mathfrak h}$ of a Lie algebra ${\mathfrak g}$ is a vector subspace preserved under the bracket (meaning that ${[\mathfrak h, \mathfrak h] \subseteq \mathfrak h]}$ ).

Theorem 23 (Lie I)

Let ${G}$ be a real or complex Lie group with Lie algebra ${\mathfrak g}$ . Then given a Lie subgroup ${H \subseteq G}$ , the map

$\displaystyle H \mapsto \mathscr{L}(H) \subseteq \mathfrak g$

is a bijection between Lie subgroups of ${G}$ and Lie subalgebras of ${\mathfrak g}$ .

Theorem 24 (The Lie functor is an equivalence of categories)

Restrict ${\mathscr{L}}$ to a functor

from the category of simply connected Lie groups over ${K}$
to the category of finite-dimensional Lie algebras over ${K}$ .

Then

(Lie II) ${\mathscr{L}}$ is fully faithful, and
(Lie III) ${\mathscr{L}}$ is essentially surjective on objects.

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 ${\text{SO }(3)}$ and ${\text{SU }(2)}$ ).