A Scholar’s Review of Lie Groups and Algebras

Abstract

These lecture notes, prepared for the Perimeter Scholars International master’s program at Perimeter Institute, provide a detailed and self-contained introduction to Lie groups, Lie algebras, and their representations. The notes first review fundamental concepts in differential geometry and abstract algebra, define Lie groups and algebras, and discuss the relation between a Lie group and its associated Lie algebra in full mathematical detail, along with some subtleties. An exhaustive list of common matrix Lie groups, their associated Lie algebras, and their topological properties is provided, along with a detailed discussion of the Lorentz group. Representation theory is then introduced, along with some relevant concepts, followed by a thorough derivation of some of the irreducible representations of several popular Lie groups.

1 Introduction

These lecture notes are intended to provide students with some essential knowledge of Lie groups, algebras, and their representations, which are ubiquitous in theoretical physics, especially in the context of quantum mechanics, gauge theories, and quantum gravity.

The notes are structured as follows:

• Chapter 2 provides the mathematical concepts and definitions necessary to understand Lie groups and algebras at the abstract level, and in particular the precise relation between Lie groups and their corresponding Lie algebras.

• Chapter 3 presents an organized and comprehensive discussion of matrix Lie groups, which are the most common Lie groups encountered in theoretical physics, along with their associated algebras and topological properties.

• Chapter 4 discusses representations of Lie groups and algebras at the abstract level, along with some relevant concepts.

• Finally, Chapter 5 derives some of the representations of several popular matrix Lie groups: $\mathrm{U}\left(1\right)$ , $\mathrm{SO}\left(2\right)$ , $\mathrm{SU}\left(2\right)$ , and $\mathrm{SO}\left(3\right)$ .

Should the reader encounter any typos or mistakes, please report them to the author at the email address provided above.

2 Abstract Lie Groups and Algebras

2.1 Manifolds

2.1.1 Smooth Manifolds

The Earth is shaped roughly like a spheroid. Therefore it has curvature, and its topology is compact. However, a person standing on the surface of the planet and looking around will not see this curvature; their immediate surroundings will look flat. Moreover, they will not notice the compact topology; the Earth seems to keep going as far as the eye can see. They might then conclude that the entire Earth is, in fact, a flat Euclidean plane, and not a spheroid! However, this is only true locally; satellite photos, for example, reveal the global structure of the Earth.

An $n$ -dimensional manifold, or $n$ -manifold, generalizes this idea: it is a space which locally looks like ${ℝ}^n$ , although globally it may be more complicated. More precisely, an $n$ -manifold is a topological space equipped with an atlas, which is a collection of charts. These charts, also known as coordinate charts or neighborhoods, are open sets which cover the manifold – that is, their union is the whole manifold. This means that any point on the manifold is in at least one chart. Each chart is homeomorphic to ${ℝ}^n$ , or in other words, there is a continuous invertible function from each chart to ${ℝ}^n$ .

The charts are open sets, so to cover the whole manifold they must overlap. When two charts overlap, there is a transition function which allows us to move from one chart to the other in a well-defined way. More precisely, this transition function provides a change of coordinates from one coordinate chart to another.

Generally, it is impossible to cover a whole manifold with just one chart. As a simple example, consider the circle $S^1$ . Points on the circle may be given by an angle $\theta$ ; however, the angles are in the interval $[0,2\pi )$ , which is not open and thus cannot be a chart. Furthermore, we cannot take our chart to be a larger interval, such as $(-\pi ,2\pi )$ , since then some points will be described by two different values of the coordinate, and the map cannot be invertible. Instead, we must cover the circle with two or more overlapping charts, related by transition functions.

In a differentiable manifold, the transition functions are all continuously differentiable a finite number of times, that is, they belong to $C^k$ for some $k\in \mathbb{N}$ . In a smooth manifold, the transition functions are smooth, that is, infinitely differentiable or $C^{\mathrm{\infty }}$ . We will assume all manifolds are smooth in these notes.

2.1.2 Tangent Spaces

A tangent space to a manifold at a point is simply the collection of vectors that are tangent to the manifold at that point. The tangent space to the manifold $M$ at a point $p\in M$ is denoted $T_{p}M$ . For example, the tangent space $T_p{S}^2$ to the sphere $S^2$ at the point $p\in S^2$ is the plane orthogonal to the radial vector pointing from the origin to $p$ . Of course, this definition relies on the fact that we can embed the sphere in a 3-dimensional ambient space. We should instead define a tangent space in an abstract way, without using an embedding.

A nice way to define the tangent space $T_{p}M$ is as follows. Let $f,g\in C^{\mathrm{\infty }}\left(M\right)$ be smooth functions on the manifold, $f,g:M\to ℝ$ . We define a tangent vector $𝐱$ at the point $p\in M$ as a map which takes a smooth function $f\in C^{\mathrm{\infty }}\left(M\right)$ to a number $𝐱\left[f\right]$ ,

$$𝐱:C^{\mathrm{\infty }}\left(M\right)\to ℝ,f\mapsto 𝐱\left[f\right],$$

and satisfies the following axioms:

1. $𝐱\left[f+g\right]=𝐱\left[f\right]+𝐱\left[g\right]$ ,

2. $𝐱\left[\alpha f\right]=\alpha 𝐱\left[f\right]$ where $\alpha \in ℝ$ ,

3. $𝐱\left[fg\right]=𝐱\left[f\right]g+f𝐱\left[g\right]$ .

A map satisfying these axioms is also called a derivation. The first two axioms ensure that it is a linear map, while the third is a generalization of the familiar Leibniz rule. Furthermore, given a number $\alpha \in ℝ$ and another tangent vector $𝐲$ , we define

$$\left(\alpha 𝐱\right)\left[f\right]\equiv \alpha 𝐱\left[f\right],$$

$$\left(𝐱+𝐲\right)\left[f\right]\equiv 𝐱\left[f\right]+𝐲\left[f\right].$$

Then it is easy to see that the tangent vectors indeed form a vector space. We call that space the tangent space to $M$ at $p$ , or $T_{p}M$ .

Now, a curve $\gamma$ is a function $\gamma :ℝ\to M$ mapping real numbers to points on the manifolds in a smooth way, that is, such that for any smooth function $f\in C^{\mathrm{\infty }}\left(M\right)$ , the composition $\left(f\circ \gamma \right)\left(t\right)\equiv f\left(\gamma \left(t\right)\right)$ depends smoothly on the real parameter $t$ . The curve has a tangent vector ${\gamma }^{\prime }\left(t\right)$ at each point $\gamma \left(t\right)\in M$ . We may imagine the curve as indicating the position of a particle on the manifold at time $t$ , in which case ${\gamma }^{\prime }\left(t\right)$ is the velocity vector, indicating the particle’s instantaneous velocity at time $t$ .

The meaning of the derivative ${\gamma }^{\prime }\left(t\right)$ is intuitively clear, but to define it rigorously, we must use its action on a function $f\in C^{\mathrm{\infty }}\left(M\right)$ :

$${\gamma }^{\prime }\left(t\right):C^{\mathrm{\infty }}\left(M\right)\to ℝ,$$

$${\gamma }^{\prime }\left(t\right)\left[f\right]\equiv {\left(f\circ \gamma \right)}^{\prime }\left(t\right).$$

One can easily check that ${\gamma }^{\prime }\left(t\right)$ satisfies the three axioms above. Therefore, we may define the tangent space $T_{p}M$ as the collection of tangent vectors ${\gamma }^{\prime }\left(0\right)$ to all the curves passing through $p$ at time $t=0$ , that is, curves satisfying $\gamma \left(0\right)=p$ .

2.2 Lie Groups

2.2.1 Groups

Let us recall that a group $G$ is a set of elements, along with a product, satisfying the following axioms:

1. Closure: For any two elements $g,h\in G$ , the product $gh$ is also in $G$.

2. Associativity: For any three elements $f,g,h\in G$ , we have $\left(fg\right)h=f\left(gh\right)$ .

3. Identity element : There exists a (unique) element $I\in G$ such that, for any element $g\in G$ , we have $Ig=gI=g$ .

4. Inverse element: For any element $g\in G$ there exists a unique element $g^{-1}$ such that $g{g}^{-1}=g^{-1}g=I$ .

If the product is commutative, $gh=hg$ for all $g,h\in G$ , we say that the group is Abelian. However, the product is usually not commutative.

Note that the product can be any binary operation, not just multiplication; for example, the integers with the addition operation, $m+n$ for $m,n\in \mathbb{Z}$ , form an (Abelian) group. However, in theoretical physics, and especially in the context of representation theory, we mostly use the notation $gh$ for the group product, since the group elements are represented as matrices, and the product is then matrix multiplication.

2.2.2 Lie Groups and Their Actions

We are now ready to define a Lie group: it is a group which is also a manifold. In other words, it is a set of elements which satisfy the group axioms, with the additional requirement that a neighborhood of each element is homeomorphic to ${ℝ}^n$ . Furthermore, the group product and the inversion $g\mapsto g^{-1}$ are both required to be smooth maps. The dimension of the Lie group is $n$, the dimension of the manifold.

One of the most powerful concepts in physics is symmetries: transformations under which a particular object is invariant. Some symmetries are discrete; for example, a square is invariant under four rotations: $0$, $\pi /2$ , $\pi$, and $3\pi /2$ radians, and four flips: vertical, horizontal, and the two diagonals. This forms a discrete symmetry group with 8 elements, known as the dihedral group of order 8 and denoted ${\mathrm{D}}_4$ . However, symmetries can also be continuous; for example, a circle is invariant under rotations by any angle. This forms a continuous symmetry group with a continuum of elements, known as the circle group, or the unitary group of degree 1, and denoted $\mathrm{U}\left(1\right)$ .

Lie group are generally used in physics as continuous symmetry groups, and thus their elements represent symmetries of a certain object. The group action defines how elements of the group act on the object; for example, the action of an element $g\in \mathrm{U}\left(1\right)$ on the circle rotates the circle by the angle corresponding to this element.

Given a group $G$ and a space $X$, we define the action of $G$ on $X$ as follows. The left group action is a function $G\times X\to X$ , denoted for $g\in G$ and $x\in X$ as $g▷x$ , or simply $gx$ , such that

1. $I▷x=x$ for all $x\in X$ , where $I\in G$ is the identity element, and

2. $\left(gh\right)▷x=g▷\left(h▷x\right)$ for all $g,h\in G$ and $x\in X$ , that is, the action of the product $gh$ is the same as first applying $h$ and then applying $g$ to the result.

Note that if $X=G$ , that is, the group is acting on itself (via the usual group product), then these two axioms are automatically satisfied. The axioms mean that, for every $g\in G$ , the left action is an invertible map from $X$ to itself, which is exactly what we would expect from a group of transformations.

We can similarly define the right group action, a function $X\times G\to X$ , denoted for $g\in G$ and $x\in X$ as $x◁g$ , or simply $xg$ , such that $x◁I=x$ and $x◁\left(gh\right)=\left(x◁g\right)◁h$ . Whether the action is left or right ultimately amounts to which elements acts first when the product $gh$ acts on $x$: $h$ for a left action or $g$ for a right action, as can be seen from the axioms $\left(gh\right)▷x=g▷\left(h▷x\right)$ and $x◁\left(gh\right)=\left(x◁g\right)◁h$ . Thus, if we are given a right action, we can always construct a left action from it simply by mapping $x◁g\mapsto g^{-1}▷x$ . Indeed, we have that ${\left(gh\right)}^{-1}▷x=h^{-1}{g}^{-1}▷x$ , so $g$ would still end up acting first, as would be expected from a right action. Therefore, we can limit ourselves to discuss left actions exclusively without loss of generality.

2.3 Lie Algebras

2.3.1 Algebras and Lie Algebras

An algebra is a vector space $𝔤$ over a field $𝔽$ (which we will take to be either $ℝ$ or $ℂ$) equipped with a binary operation $[\cdot ,\cdot ]:𝔤\times 𝔤\to 𝔤$ satisfying the following axioms:

• Right distributivity: For all $𝐱,𝐲,𝐳\in 𝔤$ , $[𝐱+𝐲,𝐳]=[𝐱,𝐳]+[𝐲,𝐳]$ .

• Left distributivity: For all $𝐱,𝐲,𝐳\in 𝔤$ , $[𝐱,𝐲+𝐳]=[𝐱,𝐲]+[𝐱,𝐳]$ .

• Compatibility with scalars: For all $𝐱,𝐲\in 𝔤$ and $\alpha ,\beta \in 𝔽$ , $[\alpha 𝐱,\beta 𝐲]=\alpha \beta [𝐱,𝐲]$ .

Note that the operation $[\cdot ,\cdot ]$ is not necessarily commutative, or even associative!

In the case of a Lie algebra, the binary operator $[\cdot ,\cdot ]$ is called the Lie bracket or commutator, and it is required to satisfy two additional axioms:

• Alternativity: For all $𝐱\in 𝔤$ , $[𝐱,𝐱]=0$ .

• The Jacobi identity: For all $𝐱,𝐲,𝐳\in 𝔤$ , $[𝐱,[𝐲,𝐳]]+[𝐲,[𝐳,𝐱]]+[𝐳,[𝐱,𝐲]]=0$ .

The alternativity axiom implies anti-commutativity of the bracket:

$$0=[𝐱+𝐲,𝐱+𝐲]=[𝐱,𝐲]+[𝐲,𝐱]$$

$$\Rightarrow [𝐱,𝐲]=-[𝐲,𝐱].$$

A well-known example of a Lie algebra is the vector space ${ℝ}^3$ with the cross product $\times$. It’s easy to check that it indeed satisfies all of the axioms above. Another well-known example is the set $\mathrm{M}(n,𝔽)$ of real or complex $n\times n$ matrices, with the binary operation being the matrix commutator $[A,B]\equiv AB-BA$ .

2.3.2 Generators of Lie Algebras and Structure Constants

Since a Lie algebra is a vector space, it has a basis. In this context, we call the basis vectors generators. The number of generators defines the dimension of the Lie algebra. We will denote the generators ${𝝉}_i$ , where the index $i$ goes from 1 to the number of dimensions. Now, if we take the commutator of two generators, the result is by definition another vector in the algebra, and therefore it must be a linear combination of the generators. Thus we define:

$$[{𝝉}_i,{𝝉}_j]\equiv f_{ij}{}^k𝝉_k,$$ (2.1)

where $f_{ij}{}^k$ are called the structure constants. (Note that the Einstein summation convention is implied: any index which appears twice, once as an upper index and once as a lower index, is summed over.) Since the commutator is anti-symmetric and satisfies the Jacobi identity, the structure constants satisfy

$$f_{ij}{}^k=-f_{ji}{}^k,$$

$$f_{ij}{}^{l}f_{kl}{}^m+f_{jk}{}^{l}f_{il}{}^m+f_{ki}{}^{l}f_{jl}{}^m=0.$$

Sometimes in physics we instead use the convention $[{𝝉}_i,{𝝉}_j]\equiv \mathrm{i}{f}_{ij}{}^k𝝉_k$ , which differs from the math convention by a factor of $\mathrm{i}$. In these notes we will use the math convention, except in Subsection 5.2.5 where we will use the physics convention.

2.4 The Lie Algebra Associated to a Lie Group

Every Lie group has a Lie algebra associated to it. In this section, we will see that the Lie algebra is the tangent space to the Lie group at the identity element, and it inherits its Lie bracket from the action of tangent vectors. However, in order to define this relation rigorously, we will first have to introduce some fundamental concepts in differential geometry.

It is important to note that while every Lie group has a Lie algebra, the relation is not one-to-one; two groups can have the same algebra. There are some other subtleties involved, depending on the topology of the group. We will discuss some of them below.

In general, the notation for the Lie algebra will use the same letters designating the Lie group it is associated with, written in lowercase Fraktur font. For example, the Lie algebra of $\mathrm{SU}\left(2\right)$ is $𝔰𝔲\left(2\right)$ . For a general Lie group $G$, the associated Lie algebra is written $𝔤$. We will denote group elements with Roman font, $g\in G$ , and algebra elements in bold font, $𝐱\in 𝔤$ . This is to remind the reader that the Lie algebra elements are actually vectors in a vector space.

2.4.1 The Differential and Left-Invariant Vector Fields

Given a map $\varphi :M\to N$ between a manifold $M$ and another manifold $N$, the differential or pushforward of $\varphi$ at a point $p\in M$ , denoted $\mathrm{d}{\varphi }_p$ , is the linear map

$$\mathrm{d}{\varphi }_p:T_{p}M\to T_{\varphi \left(p\right)}N,$$

$$\mathrm{d}{\varphi }_p\left({\gamma }^{\prime }\left(0\right)\right)\equiv {\left(\varphi \circ \gamma \right)}^{\prime }\left(0\right),$$

where $\gamma :ℝ\to M$ is a curve such that $\gamma \left(0\right)=p$ . To understand what this means, recall that ${\gamma }^{\prime }\left(0\right)$ is a tangent vector at $p$. We can map the curve $\gamma :ℝ\to M$ to a curve $\varphi \circ \gamma :ℝ\to N$ , and the differential “pushes us forward” from the tangent vector ${\gamma }^{\prime }\left(0\right)\in T_{p}M$ at $p=\gamma \left(0\right)\in M$ to the tangent vector ${\left(\varphi \circ \gamma \right)}^{\prime }\left(0\right)\in T_{\varphi \left(p\right)}N$ at $\varphi \left(p\right)=\left(\varphi \circ \gamma \right)\left(0\right)\in N$ .

In Subsection 2.2.2 we defined the group action, and we noted that any group can also act on itself, with the group action being the usual group product. For each group element $g\in G$ , we define a corresponding left action:

$$L_g:G\to G\mathit{}⟹\mathit{}{L}_g\left(h\right)\equiv gh.$$

Now, let $𝐱\in \mathrm{\Gamma }\left(TG\right)$ be a vector field on the Lie group $G$, that is, a function assigning a tangent vector $𝐱\left(g\right)\in T_{g}G$ to each point $g\in G$ in the manifold. Given some $h\in G$ , we can employ two different actions to combine it with $g$:

1. First compose $g$ with $h$ to get $L_g\left(h\right)=gh$ , and then calculate $𝐱\left(L_g\left(h\right)\right)=𝐱\left(gh\right)$ ;

2. First calculate $𝐱\left(h\right)$ , and then push it forward from the point $h$ to the point $gh$ using the differential of the left action, $\left(\mathrm{d}{L}_g\right)\left(𝐱\left(h\right)\right)$ .

In general, these two actions will produce difference results. The vector field $𝐱$ is called left-invariant if, for all $g,h\in G$ , these two actions coincide:

$$𝐱\left(gh\right)=\left(\mathrm{d}{L}_g\right)\left(𝐱\left(h\right)\right).$$

In other words, for a left-invariant vector field, if we calculate $𝐱$ at $gh$ , or first calculate $𝐱$ at $h$ and then push the result forward to $gh$ , we will get the same result.

The reason we care about left-invariant vector fields is that, if $𝐱$ is a left-invariant vector field, and we take $h=I$ (the identity element), then

$$𝐱\left(g\right)=\left(\mathrm{d}{L}_g\right)\left(𝐱\left(I\right)\right).$$

Hence, the value of $𝐱$ at any point $g\in G$ is completely determined by its value at the identity. Instead of looking at an entire vector field – a tangent vector at each group element – we can just look at the field’s value at the identity, provided that it’s left-invariant.

Conversely, if we have a tangent vector at the identity element, $𝐯\in T_{I}G$ , we may define a vector field from it as follows:

$$𝐱\left(g\right)\equiv \left(\mathrm{d}{L}_g\right)\left(𝐯\right),\forall g\in G.$$

Obviously, $𝐱\left(I\right)=𝐯$ . Furthermore, we have

$𝐱\left(gh\right)$ $=(\mathrm{d}{L}_{gh})\left(𝐯\right)$

$=\mathrm{d}\left(L_g\circ L_h\right)\left(𝐯\right)$

$(*)$ $=\left(\mathrm{d}{L}_g\right)\left(\mathrm{d}{L}_h\left(𝐯\right)\right)$

$=\left(\mathrm{d}{L}_g\right)\left(𝐱\left(h\right)\right),$

where in $(*)$ we used the chain rule. Hence, the vector field $𝐱$ thus defined is left-invariant. We conclude that there is an isomorphism – an invertible homomorphism – between the space of left-invariant vector fields on $G$ and the space of tangent vectors at the identity, $T_{I}G$ .

2.4.2 The Lie Algebra of a Lie Group

Now we can properly define the Lie algebra $𝔤$ of a Lie group $G$: it is the set of all left-invariant vector fields on $G$. One can check that this set is indeed a vector space. However, we are still missing one crucial ingredient – the Lie bracket. In Subsection 2.1.2 we defined tangent vectors as derivations: linear maps satisfying the Leibniz rule. Given two vector fields $𝐱,𝐲$ on $G$, we define their commutator by its action on functions $f\in C^{\mathrm{\infty }}\left(G\right)$ as follows:

$$[𝐱,𝐲]\left[f\right]\equiv 𝐱\left[𝐲\left[f\right]\right]-𝐲\left[𝐱\left[f\right]\right].$$ (2.2)

Here, by $𝐱\left[𝐲\left[f\right]\right]$ we mean: first act with the derivation $𝐲$ on the function $f$, which produces another function (a real number at each point on the manifold), and then act with $𝐱$ on the resulting function. One can check that $[𝐱,𝐲]$ satisfies the axioms for a derivation, and therefore it is a vector field as well.

2.4.3 Integral Curves and Flows

Let $𝐱\in \mathrm{\Gamma }\left(TG\right)$ be a vector field on a manifold $G$, let $g\in G$ be a point on the manifold, and let $R\subseteq ℝ$ be an open interval containing $0$. The integral curve of $𝐱$ passing through $g$ is a curve $\gamma :R\to G$ such that

$${\gamma }^{\prime }\left(t\right)=𝐱\left(\gamma \left(t\right)\right),\gamma \left(0\right)=g.$$ (2.3)

In other words, the curve passes through $g$ and the tangent vector to the curve at each point is given by the value of the vector field $𝐱$ at that point. If we imagine a vector field as a collection of arrows sprinkled on the manifold, then the velocity of the integral curve follows these arrows. ( 2.3 ) is an ordinary differential equation, and by the standard existence and uniqueness theorems for such equations, there exists a unique integral curve for any vector field $𝐱$ passing through any point $g$ for some interval $R\subseteq ℝ$ with $0\in R$ .

In particular, there exists a unique maximal integral curve $\gamma :R\to G$ , where the interval $R$ is maximal. A vector field $𝐱\in \mathrm{\Gamma }\left(TG\right)$ is called complete if every maximal integral curve $\gamma$ of that vector field is defined on all of $ℝ$, that is, $\gamma :ℝ\to G$ . Given an integral curve $\gamma$ of a left-invariant vector field $𝐱$ passing through a group element $g$, the curve ${\gamma }_g\equiv L_g\circ \gamma$ is also an integral curve, due to $𝐱$ being left-invariant. Using this property, we may in fact extend any integral curve to $ℝ$, and thus $𝐱$ is complete (this is left as an exercise for the reader). Since the Lie algebra is the set of left-invariant vector fields, any vector in a Lie algebra is complete.

2.4.4 The Exponential Map

Using the fact that $𝐱$ is complete, we may define a smooth map ${\varphi }_{𝐱}:ℝ\times G\to G$ such that, for all $g\in G$ ,

1. ${\varphi }_{𝐱}(0,g)=g$ ,

2. $t\mapsto {\varphi }_{𝐱}(t,g)$ is an integral curve of $𝐱$ passing through $g$.

We may collect all of these maps into a map $\varphi :ℝ\times G\times 𝔤\to G$ defined by $\varphi (t,g,𝐱)\equiv {\varphi }_{𝐱}(t,g)$ . This map satisfies:

1. $\varphi (0,g,𝐱)=g$ ,

2. $t\mapsto \varphi (t,g,𝐱)$ is an integral curve of $𝐱$ passing through $g$,

3. $\varphi (t,g,𝐱)=g\varphi (t,I,𝐱)$ ,

4. $\varphi (t,g,s𝐱)=\varphi (st,g,𝐱)$ ,

5. $\varphi (s,I,𝐱)\varphi (t,I,𝐱)=\varphi (s+t,I,𝐱)$ .

Properties 1 and 2 follow automatically from the definition of ${\varphi }_{𝐱}$ . Property 3 follows from the fact that $\gamma \left(t\right)\equiv g\varphi (t,I,𝐱)$ is an integral curve of $𝐱$ with initial condition

$$\gamma \left(0\right)=g\varphi (0,I,𝐱)=gI=g,$$

and since $\varphi (t,g,𝐱)$ is also an integral curve of $𝐱$ with the same initial condition, by the uniqueness property they must be equivalent. Properties 4 and 5 follow similarly.

By combining properties 3 and 4, we find:

$$\varphi (t,g,𝐱)=g\varphi (t,I,𝐱)=g\varphi (1,I,t𝐱).$$

Therefore, the map $\varphi$ is in fact completely determined by $𝐱$ alone: if we know $\varphi (1,I,𝐱)$ , then for any $g$ and $t$ we can easily find $\varphi (t,g,𝐱)$ . We call this map the exponential map:

$$\exp 𝐱:𝔤\to G,\exp 𝐱\equiv \varphi (1,I,𝐱).$$

Then $g\exp \left(t𝐱\right)$ is the integral curve for $𝐱$ which passes through $g$ at $t=0$ . Furthermore, from property 5 we have

$$\exp \left(s𝐱\right)\exp \left(t𝐱\right)=\exp \left(\left(s+t\right)𝐱\right).$$

We may also write ${\mathrm{e}}^{𝐱}\equiv \exp 𝐱$ for short. However, note that the usual identity ${\mathrm{e}}^{𝐱+𝐲}={\mathrm{e}}^{𝐱}{\mathrm{e}}^{𝐲}$ does not apply here, unless $𝐱$ and $𝐲$ are linearly dependent.

In summary, using the exponential map, we may obtain a group element $\exp 𝐱\in G$ from an algebra element $𝐱\in 𝔤$ .

3 The Matrix Lie Groups and Their Algebras

So far, we’ve dealt with abstract Lie groups and algebras. Now that we have a fairly good understanding of how they are defined, we will talk about the Lie groups that are the most common in physics: the matrix Lie groups.

Let $\mathrm{M}(n,𝔽)$ be the set of $n\times n$ matrices with entries from the field $𝔽$, which we will assume is either the real numbers $ℝ$ or the complex numbers $ℂ$. This set is not a group with respect to matrix multiplication, since some matrices are not invertible. However, if we take only the invertible matrices, we obtain the Lie group $\mathrm{GL}(n,𝔽)$ , also known as the general linear group. Note that for $n\ge 2$ this group is non-Abelian, since matrix multiplication is non-commutative.

The group $\mathrm{GL}(n,𝔽)$ and its subgroups are collectively known as the matrix Lie groups. Their associated Lie algebras are matrix Lie algebras, and the Lie bracket is the usual matrix commutator. In this chapter we will give a more down-to-earth definition of the exponential map for the case of a matrix Lie group, review some subtleties, recall the definition of a normal subgroup, and then list the most important matrix Lie groups, along with their dimensions, topological properties, and associated Lie algebras.

3.1 The Exponential Map for Matrix Lie Groups

If our Lie group $G$ is a matrix group with a matrix Lie algebra $𝔤$, then the exponential map has a concrete definition:

$${\mathrm{e}}^{𝐱}\equiv \sum _{k=0}^{\mathrm{\infty }}\frac{{𝐱}^k}{k!},$$

where $𝐱\in 𝔤$ , and ${𝐱}^k$ is simply the product of the matrix $𝐱$ with itself $k$ times. The identity element $I\in G$ , which is the $n\times n$ identity matrix, is the exponential of the zero element $\mathrm{𝟎}\in 𝔤$ , which is the $n\times n$ zero matrix:

$$I={\mathrm{e}}^{\mathrm{𝟎}}.$$

The exponential of a general algebra element $𝐱\in 𝔤$ will always result in some group element $g\in G$ :

$${\mathrm{e}}^{𝐱}=g\in G.$$

If the matrices $𝐱\in 𝔤$ and $𝐲\in 𝔤$ commute, then

$${\mathrm{e}}^{𝐱+𝐲}={\mathrm{e}}^{𝐱}{\mathrm{e}}^{𝐲}.$$

In particular, since $𝐱$ commutes with $-𝐱$ , we have ${\mathrm{e}}^{𝐱}{\mathrm{e}}^{-𝐱}=I$ . Therefore, the inverse $g^{-1}\in G$ will be the exponential of $-𝐱\in 𝔤$ :

$$g^{-1}={\mathrm{e}}^{-𝐱}.$$

Furthermore, for two scalars $s,t\in 𝔽$ , the matrices $s𝐱$ and $t𝐱$ commute, so

$${\mathrm{e}}^{s𝐱}{\mathrm{e}}^{t𝐱}={\mathrm{e}}^{\left(s+t\right)𝐱}.$$

We also have, for a scalar $t\in ℝ$ and a matrix $𝐱\in 𝔤$ ,

$$\frac{\mathrm{d}}{\mathrm{d}t}{\mathrm{e}}^{t𝐱}=𝐱{\mathrm{e}}^{t𝐱}={\mathrm{e}}^{t𝐱}𝐱$$

$$\Rightarrow \frac{\mathrm{d}}{\mathrm{d}t}{\mathrm{e}}^{t𝐱}{|}_{t=0}=𝐱.$$

Finally, we have the important relation

$$det{\mathrm{e}}^{𝐱}={\mathrm{e}}^{\mathrm{tr}𝐱}.$$

Now, if $G$ is connected and compact , then the map $𝐱\mapsto {\mathrm{e}}^{𝐱}$ is surjective (onto), meaning that each group element $g\in G$ has at least one element $𝐱\in 𝔤$ such that $g={\mathrm{e}}^{𝐱}$ . This is a sufficient condition, but not a necessary one; for example, even though $\mathrm{GL}(n,ℂ)$ is non-compact, the map from the Lie algebra $𝔤𝔩(n,ℂ)$ to the Lie group $\mathrm{GL}(n,ℂ)$ is surjective.

Moreover, if the map $𝐱\mapsto {\mathrm{e}}^{𝐱}$ is injective (one-to-one), meaning that each group element $g\in G$ has at most one element $𝐱\in 𝔤$ such that $g={\mathrm{e}}^{𝐱}$ , then $G$ is simply connected . This is a necessary condition, but not a sufficient one; for example, even though $\mathrm{SU}\left(2\right)$ is simply connected, the matrix $𝐱\equiv \mathrm{diag}(\mathrm{i}\pi ,-\mathrm{i}\pi )\in 𝔰𝔲\left(2\right)$ and its negative both map to the same matrix ${\mathrm{e}}^{𝐱}={\mathrm{e}}^{-𝐱}\in \mathrm{SU}\left(2\right)$ .

3.2 Normal Subgroups

3.2.1 Subgroups and Cosets

A subgroup is a subset of elements in the group which themselves form a group. For example, consider the general linear group $\mathrm{GL}(n,𝔽)$ , which consists of all $n\times n$ matrices over the field $𝔽$. If we restrict to the subset of matrices with determinant 1, we get the special linear group $\mathrm{SL}(n,𝔽)$ . It is easy to see that the properties of closure, associativity, identity element and inverse element are all satisfied for $\mathrm{SL}(n,𝔽)$ , so it is indeed a group, and hence a subgroup of $\mathrm{GL}(n,𝔽)$ .

Now, given a group $G$ and a subgroup $H$, the left coset $gH$ and right coset $Hg$ are the sets of elements formed by acting on every single element of $H$ with a particular element $g\in G$ , either from the left or from the right:

$$gH\equiv \left\{gh\right|h\in H\},$$

$$Hg\equiv \left\{hg\right|h\in H\}.$$

Note that if $G$ is Abelian, then the left and right cosets are the same. Also, since ${\left(gH\right)}^{-1}=H{g}^{-1}$ , the number of left cosets is equal to the number of right cosets.

As a simple example, consider the group $(\mathbb{Z},+)$ of the integers with the addition operation. The even integers $2\mathbb{Z}$ are clearly a subgroup of $(\mathbb{Z},+)$ . There are two possible cosets with respect to this subgroup: $2\mathbb{Z}$ itself (acting on every element of $2\mathbb{Z}$ with the identity element $0$) and $2\mathbb{Z}+1$ , which is the subgroup of odd integers. However, note that, in general, cosets are not necessarily subgroups!

3.2.2 Definition of a Normal Subgroup

A normal subgroup $N$ of $G$ is a subgroup for which the left and right cosets are the same:

$$gN=Ng,\forall g\in G.$$

In other words, the normal subgroup $N$ “commutes” with every element of the group $G$. (Obviously, if $G$ is Abelian, every subgroup is automatically normal.) An equivalent definition is that a normal subgroup $N$ of $G$ is invariant under conjugation with any element of $G$:

$$gh{g}^{-1}\in N,\forall h\in N,\forall g\in G.$$

As an example, consider the group $\mathrm{GL}(n,𝔽)$ and its subgroup $\mathrm{SL}(n,𝔽)$ . Given an element $g\in \mathrm{GL}(n,𝔽)$ and an element $h\in \mathrm{SL}(n,𝔽)$ , we have:

$$det\left(gh{g}^{-1}\right)=det(g)det(h)det(g^{-1})=1,$$

since $det(g^{-1})=1/det(g)$ and $det(h)=1$ . We see that $gh{g}^{-1}$ has determinant 1, and since it is also invertible, we conclude that it is in $\mathrm{SL}(n,𝔽)$ . Hence $\mathrm{SL}(n,𝔽)$ is a normal subgroup of $\mathrm{GL}(n,𝔽)$ .

3.2.3 Quotient Groups

The quotient group $G/N$ , where $N$ is a normal subgroup, is defined as the set of left cosets:

$$G/N\equiv \left\{gN\right|g\in G\},$$

where

$$gN\equiv \left\{gh\right|h\in N\}.$$

Every element $gN$ in $G/N$ is actually the set of all of the possible ways elements from the normal subgroup $N$ can be applied to $g$ from the right. In order to make this set into a group, we must also define a group product. We take the product to be:

$$\left(g_{1}N\right)\left(g_{2}N\right)\equiv \left(g_1{g}_2\right)N,$$

$$g_{1}N,g_{2}N\in G/N.$$

Here, the fact that $N$ is normal becomes crucial. If $N$ was not normal, then we could have had a situation where $g_{1}N=g_1^{\prime }N$ and $g_{2}N=g_2^{\prime }N$ for $g_1,g_2,g_1^{\prime },g_2^{\prime }\in G$ , and yet $\left(g_1{g}_2\right)N\ne \left(g_1^{\prime }{g}_2^{\prime }\right)N$ , which means that the product does not have a well-defined outcome! However, if $N$ is normal, then by definition we know that $N$ “commutes” with any element of $G$. Thus

$(g_1{g}_2)N$ $=g_1{g}_2^{\prime }N$

$=g_{1}N{g}_2^{\prime }$

$=g_1^{\prime }N{g}_2^{\prime }$

$=N{g}_1^{\prime }{g}_2^{\prime }$

$=(g_1^{\prime }{g}_2^{\prime })N,$

and we have a well-defined product.

3.2.4 Semidirect Products and Simple Groups

Finally, let $Q$ be a group isomorphic to a quotient group, $Q\cong G/N$ . Then we say that $G$ is the semidirect product of $Q$ and the normal subgroup $N$:

$$G=Q⋉N.$$

Any group $G$ possesses two trivial normal subgroups: the group $\left\{I\right\}$ , consisting of only the identity element, and the group $G$ itself. If $G$ does not possess any non-trivial normal subgroups, we say that it is a simple group. If $G$ is not simple, then we can break it into two smaller groups by taking the quotient with a non-trivial normal subgroup. In the example of $\mathrm{GL}(n,𝔽)$ and its (non-trivial) normal subgroup $\mathrm{SL}(n,𝔽)$ , we find

$$\mathrm{GL}(n,𝔽)={𝔽}^{\times }⋉\mathrm{SL}(n,𝔽),$$

$${𝔽}^{\times }\cong \mathrm{GL}(n,𝔽)/\mathrm{SL}(n,𝔽),$$

where ${𝔽}^{\times }\equiv 𝔽\backslash \left\{0\right\}$ is the multiplicative group of the field $𝔽$.

3.3 The General and Special Linear Groups

The general linear group $\mathrm{GL}(n,𝔽)$ consists of all invertible $n\times n$ matrices:

$\mathrm{GL}(n,𝔽)$ $\equiv \{A\in \mathrm{M}(n,𝔽)$

$|detA\ne 0\}.$

• Action: $\mathrm{GL}(n,𝔽)$ is the group of all invertible linear transformations from the vector space ${𝔽}^n$ to itself (also known as automorphisms).

• Dimension: The dimension of $\mathrm{GL}(n,ℝ)$ is $n^2$ , since real $n^2$ parameters are needed to specify an $n\times n$ matrix. The requirement that it is invertible does not reduce the number of parameters. To see that, note that removing the matrices with $detA=0$ is analogous to removing one point (the origin) from $ℝ$; you still need 1 parameter to specify where on the real line you are. Similarly, $\mathrm{GL}(n,ℂ)$ has dimension $2n^2$ , since its $n^2$ complex entries require $2n^2$ real parameters.

• Compactness: $\mathrm{GL}(n,𝔽)$ is non-compact.

• Connectedness: $\mathrm{GL}(n,ℝ)$ is disconnected. It consists of two connected components, corresponding to positive and negative determinants respectively. To see this, note that the image of $detA$ is ${ℝ}^{\times }\equiv ℝ\backslash \left\{0\right\}$ , which is disconnected, since it can be divided into the two disjoint open sets $(-\mathrm{\infty },0)$ and $(0,+\mathrm{\infty })$ . Since $detA$ is a continuous function, $\mathrm{GL}(n,ℝ)$ must also be disconnected . In contrast, $\mathrm{GL}(n,ℂ)$ is connected, since the image of $detA$ is ${ℂ}^{\times }\equiv ℂ\backslash \left\{0\right\}$ , which is connected (but not simply connected – why?)

• Lie algebra: The associated Lie algebra $𝔤𝔩(n,𝔽)$ is simply the set $\mathrm{M}(n,𝔽)$ of all $n\times n$ matrices over the field $𝔽$. Indeed, for any matrix $𝐱\in \mathrm{M}(n,𝔽)$ , the corresponding group element ${\mathrm{e}}^{𝐱}$ is automatically invertible, with its inverse being ${\mathrm{e}}^{-𝐱}$ .

The special linear group $\mathrm{SL}(n,𝔽)$ consists of all $n\times n$ matrices with determinant 1:

$\mathrm{SL}(n,𝔽)$ $\equiv \{A\in \mathrm{M}(n,𝔽)$

$|detA=1\}.$

• Action: $\mathrm{SL}(n,𝔽)$ is the group of all automorphisms of ${𝔽}^n$ that preserve volume and orientation.

• Dimension: The dimension of $\mathrm{SL}(n,ℝ)$ is $n^2-1$ , and the dimension of $\mathrm{SL}(n,ℂ)$ is $2\left(n^2-1\right)$ .

• Compactness: $\mathrm{SL}(n,𝔽)$ is still non-compact.

• Connectedness: $\mathrm{SL}(n,𝔽)$ is connected, even for $𝔽=ℝ$ .

• Lie algebra: The associated Lie algebra $𝔰𝔩(n,𝔽)$ is the set of all $n\times n$ matrices over $𝔽$ with vanishing trace, since $det{\mathrm{e}}^{𝐱}={\mathrm{e}}^{\mathrm{tr}𝐱}$ .

Note that $\mathrm{SL}(n,𝔽)$ is a normal subgroup of $\mathrm{GL}(n,𝔽)$ , and we have

$$\mathrm{GL}(n,𝔽)={𝔽}^{\times }⋉\mathrm{SL}(n,𝔽),$$

where ${𝔽}^{\times }\equiv 𝔽\backslash \left\{0\right\}$ is the multiplicative group of $𝔽$. This means that we can decompose any general linear transformation in $\mathrm{GL}(n,𝔽)$ into a volume- and orientation-preserving transformation in $\mathrm{SL}(n,𝔽)$ , times an element of ${𝔽}^{\times }$ which may change the volume and/or orientation. Or, more simply, any matrix in $\mathrm{GL}(n,𝔽)$ can be written as its determinant, which is a number in ${𝔽}^{\times }$ , times a matrix in $\mathrm{SL}(n,𝔽)$ .

3.4 The Orthogonal and Special Orthogonal Groups

The orthogonal group $\mathrm{O}\left(n\right)$ consists of all real orthogonal $n\times n$ matrices:

$\mathrm{O}\left(n\right)$ $\equiv \{A\in \mathrm{M}(n,ℝ)$

$|A^{\mathrm{T}}A=I\},$

where $A^{\mathrm{T}}$ is the transpose of $A$ such that ${\left(A^{\mathrm{T}}\right)}_{ij}\equiv A_{ji}$ .

• Action: $\mathrm{O}\left(n\right)$ is the group of distance-preserving automorphisms of ${ℝ}^n$ (also known as isometries) which leave the origin invariant. Indeed, the inner product of two vectors $𝐱,𝐲\in {ℝ}^n$ is given by $𝐱\cdot 𝐲\equiv {𝐱}^{\mathrm{T}}𝐲$ , and if $A$ is orthogonal, the inner product is invariant under the orthogonal transformation $𝐱\mapsto A𝐱$ , $𝐲\mapsto A𝐲$ . Hence the norm $\parallel 𝐱\parallel \equiv \sqrt{𝐱\cdot 𝐱}$ , and in particular the Euclidean distance $\parallel 𝐱-𝐲\parallel$ , is invariant under the action of $\mathrm{O}\left(n\right)$ .

• Dimension: The dimension of $\mathrm{O}\left(n\right)$ is $n\left(n-1\right)/2$ , since that is the number of real parameters needed to specify an orthogonal matrix .

• Compactness: $\mathrm{O}\left(n\right)$ is our first example of a compact Lie group. Indeed, first recall that, by the Heine-Borel theorem, a subset of ${ℝ}^n$ is compact if and only if it is closed and bounded. We can identify $\mathrm{M}(n,ℝ)$ with ${ℝ}^{n^2}$ by taking the $n^2$ entries of the vector in ${ℝ}^{n^2}$ to be the matrix entries of a matrix in $\mathrm{M}(n,ℝ)$ ; for example . Then $\mathrm{O}\left(n\right)$ may be similarly identified with a subset of ${ℝ}^{n^2}$ . To see that it is closed, consider the continuous function $f\left(A\right)\equiv A^{\mathrm{T}}A$ . Then we have $\mathrm{O}\left(n\right)=f^{-1}\left(\left\{I\right\}\right)$ , that is, $\mathrm{O}\left(n\right)$ is the preimage of the closed set $\left\{I\right\}$ ; therefore, since $f$ is continuous, $\mathrm{O}\left(n\right)$ must itself be closed . Furthermore, $\mathrm{O}\left(n\right)$ is bounded, since each of the rows (or columns) of an orthogonal matrix is a vector of norm 1, and thus no entry in the matrix may be larger than 1. Thus $\mathrm{O}\left(n\right)$ is compact.

• Connectedness: $\mathrm{O}\left(n\right)$ is disconnected, since an orthogonal matrix has $det\left(A^{\mathrm{T}}A\right)={\left(detA\right)}^2=1$ and thus $detA=\pm 1$ ; the two connected components correspond to the two possible values of the determinant.

• Lie algebra: The associated Lie algebra $𝔬\left(n\right)$ is the set of real anti-symmetric $n\times n$ matrices. Indeed, if $𝐱$ is anti-symmetric then ${𝐱}^{\mathrm{T}}=-𝐱$ and thus, if $A={\mathrm{e}}^{𝐱}$ , we have $A^{\mathrm{T}}={\left({\mathrm{e}}^{𝐱}\right)}^{\mathrm{T}}={\mathrm{e}}^{-𝐱}=A^{-1}$ , so $A$ is orthogonal.

The special orthogonal group $\mathrm{SO}\left(n\right)$ consists of all real orthogonal $n\times n$ matrices with determinant 1:

$\mathrm{SO}\left(n\right)$ $\equiv \{A\in \mathrm{M}(n,ℝ)$

$|A^{\mathrm{T}}A=I\text{and}detA=1\}.$

• Action: $\mathrm{SO}\left(n\right)$ is the group of isometries of ${ℝ}^n$ which leave the origin invariant and preserve orientation. These transformations are also known as rotations.

• Dimension: The dimension of $\mathrm{SO}\left(n\right)$ , like that of $\mathrm{O}\left(n\right)$ , is $n\left(n-1\right)/2$ . The condition $detA=1$ doesn’t reduce the number of parameters. To see why, note that $\mathrm{O}\left(n\right)$ is a space of dimension $n\left(n-1\right)/2$ consisting of two disjoint open sets, one with determinant $+1$ and another with determinant $-1$ . Just like dividing the real numbers into positive and negative numbers does not reduce the dimension of the real line, so does selecting only the component with determinant $+1$ not reduce the dimension of the overall space.

• Compactness: Like $\mathrm{O}\left(n\right)$ , $\mathrm{SO}\left(n\right)$ is compact.

• Connectedness: $\mathrm{SO}\left(n\right)$ is connected, and it is in fact one of the two connected components of $\mathrm{O}\left(n\right)$ : the one with determinant 1.

• Lie algebra: The associated Lie algebra $𝔰𝔬\left(n\right)$ is the set of real anti-symmetric $n\times n$ matrices, and it is identical to $𝔬\left(n\right)$ . Note that since an anti-symmetric matrix always has zero trace, its exponential always has determinant 1. Thus, even though both groups have the same Lie algebra, exponentiating that algebra only allows access to the component of $\mathrm{O}\left(n\right)$ connected to the identity matrix, namely $\mathrm{SO}\left(n\right)$ . This is to be expected, since the Lie algebra is the tangent space at the identity, and the identity has determinant 1 .

$\mathrm{SO}\left(n\right)$ is a normal subgroup of $\mathrm{O}\left(n\right)$ , and we have

$$\mathrm{O}\left(n\right)={\mathbb{Z}}_2⋉\mathrm{SO}\left(n\right),$$

where ${\mathbb{Z}}_2$ is the group $\{+1,-1\}$ with the group product being multiplication. This results from the fact that any matrix in $\mathrm{O}\left(n\right)$ can be written as its determinant, which is a number in ${\mathbb{Z}}_2$ , times a matrix in $\mathrm{SO}\left(n\right)$ .

3.5 The Indefinite Orthogonal and Special Orthogonal Groups

3.5.1 General $p$ and $q$

Let $p$ and $q$ be positive integers. The indefinite orthogonal group $\mathrm{O}(p,q)$ consists of all real $\left(p+q\right)\times \left(p+q\right)$ matrices which leave a bilinear form (or metric) of signature $(p,q)$ invariant:

$\mathrm{O}(p,q)$ $\equiv \{A\in \mathrm{M}(p+q,ℝ)$

$|A^{\mathrm{T}}\eta A=\eta \},$

$$\eta \equiv \mathrm{diag}(\underset{p}{\underbrace{-1,\mathrm{\dots },-1}},\underset{q}{\underbrace{+1,\mathrm{\dots },+1}}).$$

• Action: $\mathrm{O}(p,q)$ is the group of isometries of ${ℝ}^{p+q}$ which leave the origin invariant. ${ℝ}^{p+q}$ is the $\left(p+q\right)$ -dimensional real vector space with the dot product $𝐱\cdot 𝐲\equiv {𝐱}^{\mathrm{T}}\eta 𝐲$ , and it is invariant under the transformation $𝐱\mapsto A𝐱$ , $𝐲\mapsto A𝐲$ . Note that for $p=0$ and $q=n$ , the metric $\eta$ is the $n\times n$ identity matrix, and we recover the condition for $\mathrm{O}\left(n\right)$ , $A^{\mathrm{T}}A=I$ .

• Dimension: The dimension of $\mathrm{O}(p,q)$ is $n\left(n-1\right)/2$ where $n=p+q$ .

• Compactness: $\mathrm{O}(p,q)$ is non-compact for $p,q>0$ , since the elements of the matrices are not bounded.

• Connectedness: $\mathrm{O}(p,q)$ is disconnected and consists of four connected components, described below.

• Lie algebra: The associated Lie algebra $𝔬(p,q)$ is the set of real $n\times n$ matrices $𝐱$ satisfying ${𝐱}^{\mathrm{T}}=-\eta 𝐱{\eta }^{-1}$ . Indeed, we then have

  ${\left({\mathrm{e}}^{𝐱}\right)}^{\mathrm{T}}\eta {\mathrm{e}}^{𝐱}$ $={\mathrm{e}}^{-\eta 𝐱{\eta }^{-1}}\eta {\mathrm{e}}^{𝐱}$

  $=\left(\eta {\mathrm{e}}^{-𝐱}{\eta }^{-1}\right)\eta {\mathrm{e}}^{𝐱}$

  $=\eta ,$

  as required. Note that for $p=0$ and $q=n$ , this condition reduces to the anti-symmetry condition defining $𝔬\left(n\right)$ . As for $𝔬\left(n\right)$ , the Lie algebra only allows us access to the component connected to the identity.

The indefinite special orthogonal group $\mathrm{SO}(p,q)$ consists of all of the matrices in $\mathrm{O}(p,q)$ which have determinant 1:

$\mathrm{SO}(p,q)$ $\equiv \{A\in \mathrm{M}(p+q,ℝ)$

$|A^{\mathrm{T}}\eta A=\eta \text{and}detA=1\},$

where $\eta$ is defined as before.

• Action: It is the group of isometries of ${ℝ}^{p+q}$ which leave the origin invariant and preserve orientation.

• Dimension: $\mathrm{SO}(p,q)$ still has dimension $n\left(n-1\right)/2$ where $n=p+q$ .

• Compactness: Like $\mathrm{O}(p,q)$ , the group $\mathrm{SO}(p,q)$ is non-compact for $p,q>0$ .

• Connectedness: $\mathrm{SO}(p,q)$ is still disconnected for $p,q>0$ , but with only two connected components, the ones where $detA=1$ .

• Lie algebra: The associated Lie algebra $𝔰𝔬(p,q)$ is the set of real $n\times n$ matrices $𝐱$ satisfying ${𝐱}^{\mathrm{T}}=-\eta 𝐱{\eta }^{-1}$ . It is identical to $𝔬(p,q)$ , and only allows us access to the component of $\mathrm{SO}(p,q)$ connected to the identity.

3.5.2 $p=1$ : The Lorentz Group

The special case $p=1$ is the most important one in physics, since it defines the Lorentz group, with $\eta$ the Minkowski metric. The group $\mathrm{O}(1,n-1)$ is the Lorentz group in $n$ dimensions. To describe the four connected components of this group, let us consider the simplest non-trivial case, that of $\mathrm{O}(1,1)$ , the Lorentz group in 2 dimensions. The Minkowski metric is

and we take a general matrix $A\in \mathrm{O}(1,1)$ ,

From the defining relation $A^{\mathrm{T}}\eta A=\eta$ , we get

Thus

$$ab=cd,c^2-a^2=-1,d^2-b^2=1.$$

These are 3 equations for 4 unknowns, so we must have one free parameter. Indeed, this is consistent with the fact that the dimension of the group is 1. Let us choose the free parameter to be $a$. Then we immediately get $c^2=a^2-1$ and thus we must have $a^2\ge 1$ . Furthermore, $b=cd/a$ , and plugging that into $1=d^2-b^2$ we obtain

$1$ $=d^2-{\displaystyle \frac{c^2{d}^2}{a^2}}$

$=d^2-{\displaystyle \frac{\left(a^2-1\right)d^2}{a^2}}$

$={\displaystyle \frac{d^2}{a^2}},$

so $d=\pm a$ . Then from $b=cd/a$ we get $b=\pm c$ , where the sign corresponds to the same sign as that chosen for $d=\pm a$ . In conclusion, we have two sets of solutions, one with positive determinant:

$$b=c=\pm \sqrt{a^2-1},$$

$$d=+a,$$

$$detA=ad-bc=+1,$$

and one with negative determinant:

$$b=\pm \sqrt{a^2-1},$$

$$c=\mp \sqrt{a^2-1},$$

$$d=-a,$$

$$detA=ad-bc=-1.$$

In addition, the condition $a^2\ge 1$ is satisfied by either $a\ge +1$ or $a\le -1$ . We can now classify the solutions into four types, depending on the sign of $a$ and the sign of the determinant:

• Proper: $detA=+1$ ,

• Improper: $detA=-1$ ,

• Orthochronous: $a\ge +1$ ,

• Non-orthochronous: $a\le -1$ .

The four connected components of $\mathrm{O}(1,1)$ are then:

• Proper, orthochronous transformations: $detA=+1$ and $a\ge +1$ . This is the component connected to the identity matrix $I=\mathrm{diag}(1,1)$ , which is obtained for $a=+1$ .

• Improper, orthochronous transformations: $detA=-1$ and $a\ge +1$ . This component includes the parity inversion matrix $P\equiv \mathrm{diag}(1,-1)$ , which is obtained for $a=+1$ .

• Improper, non-orthochronous transformations: $detA=-1$ and $a\le -1$ . This component includes the time reversal matrix $T\equiv \mathrm{diag}(-1,1)$ , which is obtained for $a=-1$ .

• Proper, non-orthochronous transformations: $detA=+1$ and $a\le -1$ . This component includes the matrix $PT=\mathrm{diag}(-1,-1)=-I$ , which is obtained for $a=-1$ .

Now, by definition $\mathrm{SO}(1,1)$ contains the two proper components, both orthochronous and non-orthochronous. The subgroup of proper orthochronous transformations is denoted ${\mathrm{SO}}^{+}(1,1)$ . It can be shown that this is a normal subgroup of $\mathrm{O}(1,1)$ .

Furthermore, note that we can move between the four connected components by applying $P$, $T$, or both. One can check that the elements $\{I,P,T,PT\}$ form a group; in fact, this group is isomorphic to the quotient group $\mathrm{O}(1,1)/{\mathrm{SO}}^{+}(1,1)$ . Thus we have the semidirect product

$$\mathrm{O}(1,1)=\{I,P,T,PT\}⋉{\mathrm{SO}}^{+}(1,1).$$

This analysis readily generalizes to the 4-dimensional Lorentz group $\mathrm{O}(1,3)$ .

3.6 The Unitary and Special Unitary Groups

The unitary group $\mathrm{U}\left(n\right)$ consists of all complex unitary $n\times n$ matrices:

$\mathrm{U}\left(n\right)$ $\equiv \{A\in \mathrm{M}(n,ℂ)$

$|A^{†}A=I\},$

where $A^{†}$ is the conjugate transpose of $A$ such that ${\left(A^{†}\right)}_{ij}\equiv {\overline{A}}_{ji}$ .

• Action: $\mathrm{U}\left(n\right)$ is the group of isometries of ${ℂ}^n$ , with respect to the Hermitian inner product given by $⟨𝐱,𝐲⟩\equiv {𝐱}^{†}𝐲$ where $𝐱,𝐲\in {ℂ}^n$ , which leave the origin invariant. It is the analogue of $\mathrm{O}\left(n\right)$ for complex spaces.

• Dimension: The dimension of $\mathrm{U}\left(n\right)$ is $n^2$ . To see this, note that a general complex matrix requires $2n^2$ real parameters. The matrix $A^{†}A$ is Hermitian, since ${\left(A^{†}A\right)}^{†}=A^{†}A$ . The diagonal entries of a Hermitian matrix must be real, so they correspond to $n$ real parameters, and there are ${\sum }_{k=1}^{n-1}k=n($