lie groups

In Some Basics of Fourier Analysis we introduced some of the basic ideas in Fourier analysis, which is ubiquitous in many parts of both pure and applied math. In this post we look at these same ideas from a different point of view, that of representation theory.

Representation theory is a way of studying group theory by turning it into linear algebra, which in many cases is more familiar to us and easier to study.

A (linear) representation is just a group homomorphism from some group $G$ we’re interested in, to the group of linear transformations of some vector space. If the vector space has some finite dimension $n$ , the group of its linear transformations can be expressed as the group of $n \times n$ matrices with nonzero determinant, also known as $\mathrm{GL}_n(k)$ ( $k$ here is the field of scalars of our vector space).

In this post, we will focus on infinite-dimensional representation theory. In other words, we will be looking at homomorphisms of a group $G$ to the group of linear transformations of an infinite-dimensional vector space.

“Infinite-dimensional vector spaces” shouldn’t scare us – in fact many of us encounter them in basic math. Functions are examples of such. After all, vectors are merely things we can scale and add to form linear combinations. Functions satisfy that too. That being said, if we are dealing with infinity we will often need to make use of the tools of analysis. Hence functional analysis is often referred to as “infinite-dimensional linear algebra” (see also Metric, Norm, and Inner Product).

Just as a vector $v$ has components $v_i$ indexed by $i$ , a function $f$ has values $f(x)$ indexed by $x$ . If we are working over uncountable things, instead of summation we may use integration.

We will also focus on unitary representations in this post. This means that the linear transformations are further required to preserve a complex inner product (which takes the form of an integral) on the vector space. To facilitate this, our functions must be square-integrable.

Consider the group of real numbers $\mathbb{R}$ (under addition). We want to use representation theory to study this group. For our purposes we want the square-integrable functions on some quotient of $\mathbb{R}$ as our vector space. It comes with an action of $\mathbb{R}$ , by translation. In other words, an element $a$ of $\mathbb{R}$ acts on our function $f(x)$ by sending it to the new function $f(x+a)$ .

So what is this quotient of $\mathbb{R}$ that our functions will live on? For now let us choose the integers $\mathbb{Z}$ . The quotient $\mathbb{R}/\mathbb{Z}$ is the circle, and functions on it are periodic functions.

To recap: We have a representation of the group $\mathbb{R}$ (the real line under addition) as linear transformations (also called linear operators) of the vector space of square-integrable functions on the circle.

In representation theory, we will often decompose a representation into a direct sum of irreducible representations. Irreducible means it contains no “subrepresentation” on a smaller vector space. The irreducible representations are the “building blocks” of other representations, so it is quite helpful to study them.

How do we decompose our representation into irreducible representations? Consider the representation of $\mathbb{R}$ on the vector space $\mathbb{C}$ (the complex numbers) where a real number $a$ acts by multiplying a complex number $z$ by $e^{2\pi i k a}$ , for $k$ an integer. This representation is irreducible.

If this looks familiar, this is just the Fourier series expansion for a periodic function. So a Fourier series expansion is just an expression of the decomposition of the representation of R into irreducible representations!

What if we chose a different vector space instead? It might have been the more straightforward choice to represent $\mathbb{R}$ via functions on $\mathbb{R}$ itself instead of on the circle $\mathbb{R}/\mathbb{Z}$ . That may be true, but in this case our decomposition into irreducibles is not countable! The irreducible representations into which this other representation decomposes is the one where a real number $a$ acts on $\mathbb{C}$ by multiplication by $e^{2 \pi i k a}$ where $k$ is now a real number, not necessarily an integer. So it’s not indexed by a countable set.

This should also look familiar to those who know Fourier analysis: This is the Fourier transform of a square-integrable function on $\mathbb{R}$ .

So now we can see that concepts in Fourier analysis can also be phrased in terms of representations. Important theorems like the Plancherel theorem, for example, also may be understood as an isomorphism between the representations we gave and other representations on functions of the indices. We also have the Poisson summation in Fourier analysis. In representation theory this is an equality obtained from calculating the trace in two ways, as a sum over representations and as a sum over conjugacy classes.

Now we see how Fourier analysis is related to the infinite-dimensional representation theory of the group $\mathbb{R}$ (one can also see this as the infinite-dimensional representation theory of the circle, i.e. the group $\mathbb{R}/\mathbb{Z}$ – the article “Harmonic Analysis and Group Representations” by James Arthur discusses this point of view). What if we consider other groups instead, like, say, $\mathrm{GL}_n(\mathbb{R})$ or $\mathrm{SL}_n(\mathbb{R})$ (or $\mathbb{R}$ can be replaced by other rings even)?

Things get more complicated, for example the group may not be abelian. Since we used integration so much, we also need an analogue for it. So we need to know much about group theory and analysis and everything in between for this.

These questions have been much explored for the kinds of groups called “reductive”, which are closely related to Lie groups. They include the examples of $\mathrm{GL}_n(\mathbb{R})$ and $\mathrm{SL}_n(\mathbb{R})$ earlier, as well as certain other groups we have discussed in previous posts such as the orthogonal and unitary (see also Rotations in Three Dimensions). There is a theory for these groups analogous to what I have discussed in this post, and hopefully this will be discussed more in future blog posts here.

References:

Representation theory on Wikipedia

Representation of a Lie group on Wikipedia

Fourier analysis on Wikipedia

Harmonic analysis on Wikipedia

Plancherel theorem on Wikipedia

Poisson summation formula on Wikipedia

An Introduction to the Trace Formula by James Arthur

Harmonic Analysis and Group Representations by James Arthur

In Rotating and Reflecting Vectors Using Matrices we learned how to express rotations in $2$ -dimensional space using certain special $2\times 2$ matrices which form a group (see Groups) we call the special orthogonal group in dimension $2$ , or $\text{SO}(2)$ (together with other matrices which express reflections, they form a bigger group that we call the orthogonal group in $2$ dimensions, or $\text{O}(2)$ ).

In this post, we will discuss rotations in $3$ -dimensional space. As we will soon see, notations in $3$ -dimensional space have certain interesting features not present in the $2$ -dimensional case, and despite being seemingly simple and mundane, play very important roles in some of the deepest aspects of fundamental physics.

We will first discuss rotations in $3$ -dimensional space as represented by the special orthogonal group in dimension $3$ , written as $\text{SO}(3)$ .

We recall some relevant terminology from Rotating and Reflecting Vectors Using Matrices. A matrix is called orthogonal if it preserves the magnitude of (real) vectors. The magnitude of the vector $v$ must be equal to the magnitude of the vector $Av$ , for a matrix $A$ , to be orthogonal. Alternatively, we may require, for the matrix $A$ to be orthogonal, that it satisfy the condition

$\displaystyle AA^{T}=A^{T}A=I$

where $A^{T}$ is the transpose of $A$ and $I$ is the identity matrix. The word “special” denotes that our matrices must have determinant equal to $1$ . Therefore, the group $\text{SO}(3)$ consists of the $3\times3$ orthogonal matrices whose determinant is equal to $1$ .

The idea of using the group $\text{SO}(3)$ to express rotations in $3$ -dimensional space may be made more concrete using several different formalisms.

One popular formalism is given by the so-called Euler angles. In this formalism, we break down any arbitrary rotation in $3$ -dimensional space into three separate rotations. The first, which we write here by $\varphi$ , is expressed as a counterclockwise rotation about the $z$ -axis. The second, $\theta$ , is a counterclockwise rotation about an $x$ -axis that rotates along with the object. Finally, the third, $\psi$ , is expressed as a counterclockwise rotation about a $z$ -axis that, once again, has rotated along with the object. For readers who may be confused, animations of these steps can be found among the references listed at the end of this post.

The matrix which expresses the rotation which is the product of these three rotations can then be written as

$\displaystyle g(\varphi,\theta,\psi) = \left(\begin{array}{ccc} \text{cos}(\varphi)\text{cos}(\psi)-\text{cos}(\theta)\text{sin}(\varphi)\text{sin}(\psi) & -\text{cos}(\varphi)\text{sin}(\psi)-\text{cos}(\theta)\text{sin}(\varphi)\text{cos}(\psi) & \text{sin}(\varphi)\text{sin}(\theta) \\ \text{sin}(\varphi)\text{cos}(\psi)+\text{cos}(\theta)\text{cos}(\varphi)\text{sin}(\psi) & -\text{sin}(\varphi)\text{sin}(\psi)+\text{cos}(\theta)\text{cos}(\varphi)\text{cos}(\psi) & -\text{cos}(\varphi)\text{sin}(\theta) \\ \text{sin}(\psi)\text{sin}(\theta) & \text{cos}(\psi)\text{sin}(\theta) & \text{cos}(\theta) \end{array}\right)$ .

The reader may check that, in the case that the rotation is strictly in the $x$ – $y$ plane, i.e. $\theta$ and $\psi$ are zero, we will obtain

$\displaystyle g(\varphi,\theta,\psi) = \left(\begin{array}{ccc} \text{cos}(\varphi) & -\text{sin}(\varphi) & 0 \\ \text{sin}(\varphi) & \text{cos}(\varphi) & 0 \\ 0 & 0 & 1 \end{array}\right)$ .

Note how the upper left part is an element of $\text{SO}(2)$ , expressing a counterclockwise rotation by an angle $\varphi$ , as we might expect.

Contrary to the case of $\text{SO}(2)$ , which is an abelian group, the group $\text{SO}(3)$ is not an abelian group. This means that for two elements $a$ and $b$ of $\text{SO}(3)$ , the product $ab$ may not always be equal to the product $ba$ . One can check this explicitly, or simply consider rotating an object along different axes; for example, rotating an object first counterclockwise by 90 degrees along the $z$ -axis, and then counterclockwise again by 90 degrees along the $x$ -axis, will not end with the same result as performing the same operations in the opposite order.

We now know how to express rotations in $3$ -dimensional space using $3\times 3$ orthogonal matrices. Now we discuss another way of expressing the same concept, but using “unitary”, instead of orthogonal, matrices. However, first we must revisit rotations in $2$ dimensions.

The group $\text{SO}(2)$ is not the only way we have of expressing rotations in $2$ -dimensions. For example, we can also make use of the unitary (we will explain the meaning of this word shortly) group in $1$ -dimension, also written $\text{U}(1)$ . It is the group formed by the complex numbers with magnitude equal to $1$ . The elements of this group can always be written in the form $e^{i\theta}$ , where $\theta$ is the angle of our rotation. As we have seen in Connection and Curvature in Riemannian Geometry, this group is related to quantum electrodynamics, as it expresses the gauge symmetry of the theory.

The groups $\text{SO}(2)$ and $\text{U}(1)$ are actually isomorphic. There is a one-to-one correspondence between the elements of $\text{SO}(2)$ and the elements of $\text{U}(1)$ which respects the group operation. In other words, there is a bijective function $f:\text{SO}(2)\rightarrow\text{U}(1)$ , which satisfies $ab=f(a)f(b)$ for $a$ , $b$ elements of $\text{SO}(2)$ . When two groups are isomorphic, we may consider them as being essentially the same group. For this reason, both $\text{SO}(2)$ and $U(1)$ are often referred to as the circle group.

We can now go back to rotations in $3$ dimensions and discuss the group $\text{SU}(2)$ , the special unitary group in dimension $2$ . The word “unitary” is in some way analogous to “orthogonal”, but applies to vectors with complex number entries.

Consider an arbitrary vector

$\displaystyle v=\left(\begin{array}{c}v_{1}\\v_{2}\\v_{3}\end{array}\right)$ .

An orthogonal matrix, as we have discussed above, preserves the quantity (which is the square of what we have referred to earlier as the “magnitude” for vectors with real number entries)

$\displaystyle v_{1}^{2}+v_{2}^{2}+v_{3}^{2}$

while a unitary matrix preserves

$\displaystyle v_{1}^{*}v_{1}+v_{2}^{*}v_{2}+v_{3}^{*}v_{3}$

where $v_{i}^{*}$ denotes the complex conjugate of the complex number $v_{i}$ . This is the square of the analogous notion of “magnitude” for vectors with complex number entries.

Just as orthogonal matrices must satisfy the condition

$\displaystyle AA^{T}=A^{T}A=I$ ,

unitary matrices are required to satisfy the condition

$\displaystyle AA^{\dagger}=A^{\dagger}A=I$

where $A^{\dagger}$ is the Hermitian conjugate of $A$ , a matrix whose entries are the complex conjugates of the entries of the transpose $A^{T}$ of $A$ .

An element of the group $\text{SU}(2)$ is therefore a $2\times 2$ unitary matrix whose determinant is equal to $1$ . Like the group $\text{SO}(3)$ , the group $\text{SU}(2)$ is also a group which is not abelian.

Unlike the analogous case in $2$ dimensions, the groups $\text{SO}(3)$ and $\text{SU}(2)$ are not isomorphic. There is no one-to-one correspondence between them. However, there is a homomorphism from $\text{SU}(2)$ to $\text{SO}(3)$ that is “two-to-one”, i.e. there are always two elements of $\text{SU}(2)$ that get mapped to the same element of $\text{SO}(3)$ under this homomorphism. Hence, $\text{SU}(2)$ is often referred to as a “double cover” of $\text{SO}(3)$ .

In physics, this concept underlies the weird behavior of quantum-mechanical objects called spinors (such as electrons), which require a rotation of 720, not 360, degrees to return to its original state!

The groups we have so far discussed are not “merely” groups. They also possesses another kind of mathematical structure. They describe certain shapes which happen to have no sharp corners or edges. Technically, such a shape is called a manifold, and it is the object of study of the branch of mathematics called differential geometry, which we have discussed certain basic aspects of in Geometry on Curved Spaces and Connection and Curvature in Riemannian Geometry.

For the circle group, the manifold that it describes is itself a circle. The elements of the circle group correspond to the points of the circle. The group $\text{SU}(2)$ is the surface of the $4$ – dimensional sphere, or what we call a $3$ -sphere (for those who might be confused by the terminology, recall that we are only considering the surface of the sphere, not the entire volume, and this surface is a $3$ -dimensional, not a $4$ -dimensional, object). The group $\text{SO}(3)$ is $3$ -dimensional real projective space, written $\mathbb{RP}^{3}$ . It is a manifold which can be described using the concepts of projective geometry (see Projective Geometry).

A group that is also a manifold is called a Lie group (pronounced like “lee”) in honor of the mathematician Marius Sophus Lie who pioneered much of their study. Lie groups are very interesting objects of study in mathematics because they bring together the techniques of group theory and differential geometry, which teaches us about Lie groups on one hand, and on the other hand also teaches us more about both group theory and differential geometry themselves.

References:

Orthogonal Group on Wikipedia

Rotation Group SO(3) on Wikipedia

Euler Angles on Wikipedia

Unitary Group on Wikipedia

Spinor on Wikipedia

Lie Group on Wikipedia

Real Projective Space on Wikipedia

Algebra by Michael Artin