# A first look at Lie theory

A friend of mine made a request…

Any chance you could make an expository post on Lie Theory for those of us who only known some abstract algebra and calculus? The topic seems very inaccessible otherwise, but I hear Lie Groups and Lie Algebras mentioned regularly.

As your friendly neighbourhood mathematician I will try to oblige.

Disclaimer What I do is give an informal overview and not worry too much about details and proper proofs. Proofs you can find in textbooks. Rather I want to present the ideas and sketch some constructions.

I will build this account up over the period of a few weeks.

Rough Plan
The things I would like to cover are the following.

1. Abstract Lie algebras
2. Lie groups
3. The Lie algebra of a Lie group
4. Lie’s theorems
5. Some odds and ends (Maybe a few words about Lie groupoids etc)

There maybe some changes here as the work develops.

I will also include some simple exercises for those that are interested. I will post solutions at the end.

Part 0: Introduction
Anybody who reads anything about modern physics will encounter the terms ‘Lie group’ and ‘Lie algebra’. Lie theory is all about the relation between these two structures.

A Lie group is a group that also has a smooth manifold structure, importantly the group operations are compatible with this smooth structure. Groups represent transformations and symmetries of mathematical objects. Lie groups are the mathematical framework for studying continuous symmetries of mathematical objects. Thus, Lie groups are fundamental in geometry and theoretical physics.

Now, every Lie group has associated with it a Lie algebra, whose vector space structure is the tangent space of the Lie group at the identity element. The Lie algebra describes the local structure of the group. Informally one can think of the Lie algebra as describing the elements of the Lie group that are ‘very close to the identity element’.

The theory of Lie groups and Lie algebras was initiated by Sophus Lie, and hence the nomenclature. Lie’s motivation was to extend Galois theory, which proved useful in the study of algebraic equations, to cope with continuous symmetries of differential equations. Lie laid down much of the basic theory of continuous symmetry groups.

The plan is with these notes is to sketch the relation between Lie groups and Lie algebras. I will stick to the finite dimensional case for this first look.

Part I: Abstract Lie algebras
Let us start with a completely algebraic set-up. Informally, a Lie algebra is a vector space with a non-associative product, known as a ‘bracket’ that satisfies some nice properties. We will only consider algebras over the reals or complex here, though everything will generalise to more arbitrary fields (with some minor modifications if necessary).

Definition
A Lie algebra is a vector space $$\mathfrak{g}$$ together with a bilinear operation $$[\bullet,\bullet]: \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}$$, that satisfies the following conditions

1. Skewsymmetry
$$[x,y] = -[y,x]$$
2. Jacobi identity

$$[x,[y,z]] + [z,[x,y]] +[y,[z,x]]=0$$

for all $$x,y, z \in \mathfrak{g}$$.

Note that Lie algebras are non-associative. Thinking of the bracket as a form of multiplication we see that the Jacobi identity is related to the ‘associator’ which is non-zero in general

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

The Jacobi identity can also be written in ‘Loday form’

$$[x,[y,z]] = [[x,y],z] + [y,[x,z]]$$,

which means that the operator $$Ad_{x}:= [x, \bullet]$$ satisfies the Leibniz rule, the so called adjoint operator is a derivation. Note that this form of the Jacobi identity has this interpretation even if the bracket is not skewsymmetric. In fact such bracket algebras are well studied and are usually called “Loday” or “Leibniz-Loday” algebras.

The dimension of a Lie algebra is defined to be the dimension of the underlying vector space. Elements of a Lie algebra are said to generate that Lie algebra if they form the smallest subalgebra that contains these elements is the Lie algebra itself.

Example Any vector space equipped with a vanishing bracket $$[x,y]=0$$, is a Lie algebra. We call any Lie algebra with a vanishing bracket an abelian Lie algebra.

Example The (real) vector space of all n×n skew-hermitian matrices together with the standard commutator is Lie algebra. This Lie algebra is denoted $$\mathfrak{u}(n)$$.

Example The Heisenberg algebra is the Lie algebra generated by three elements x,y,z and the Lie brackets are defined as
$$[x,y] =z$$, $$[x,z] =0$$ and $$[y,z] =0$$.

Given a set of generators $$\{T_{a}\}$$ we can define the Lie algebra in terms of its structure constants. As the Lie bracket of any pair of generators must be a linear combination of the generators we have

$$[T_{a}, T_{b}] = C^{c}_{ab}\: T_{c}$$,

and so the Lie algebra is determined by the structure constants $$C^{c}_{ab}$$.

Exercise How many one dimensional Lie algebras are there up to isomorphisms?

Exercise There are exactly two Lie algebras of dimension two over the real numbers, up to isomorphism. Can you write these down in terms of generators?

Exercise What conditions do the structure constants need to satisfy in order to have a Lie algebra? (Hint: think about the two defining conditions of a Lie algebra)

People study Lie algebras in their own right, but historically they arose from the study of Lie groups. From my own perspective, it is the fact that Lie algebras are ‘infinitesimal Lie groups’ that makes them interesting and useful. In the next section I will move on to groups and in particular Lie groups.

Part II: Lie groups
Before we move on to Lie groups, let us recall the notion of a group. Generically, one thinks of groups as encoding transformations and symmetries of mathematical objects, so they arise all across mathematics.

Definition
A group is a set $$G$$ together with a binary operation $$\circ: G \times G \rightarrow G$$ that satisfies the following axioms

1. Associativity
For every $$a,b,c \in G$$ we have $$(a\circ b) \circ c = a\circ (b \circ c)$$.
2. Existence of the identity
There exists an element $$e \in G$$ such that $$e\circ a = a \circ e$$ for all $$a \in G$$.
3. Existence of inverse elements
For every $$a \in G$$ there exists an element $$b := a^{-1}$$ such that $$a\circ b = b \circ a =e$$.

It can be shown that the identity element $$e$$ is unique. There is only one identity element. Note we have said noting about commutativity. Generally $$a\circ b$$ is not the same as $$b\circ a$$. Groups for which these two expression are always equal are called abelian groups.

Example The set of integers $$\mathbb{Z}$$ together with standard addition form an abelian group. The identity element is zero and the inverse of any element is $$a^{-1} = {-}a$$.

Exercise Does the set of real numbers $$\mathbb{R}$$ equipped with standard addition form a group? Does the set of real numbers with standard multiplication form a group?

Example A symmetric group a set consists of permutations on the given set; ie. bijective maps from the set to itself. The product is just composition of the permutations as functions. The identity element is just the identity function from the set to itself. The inverse of an element is just the inverse as a function.

Example Probably the simplest non-abelian group is the rotation group $$SO(3)$$. This group consists of all rotations about the origin of three-dimensional Euclidean space and the composition is just standard composition of linear maps. Because all linear transformations can be represented by matrices (once a basis has been chosen) the group $$SO(3)$$ can be represented by the set of orthogonal 3×3 matrices and standard matrix multiplication. This group is non-abelian as the order of which rotations are composed matters.

Now, Lie groups are both groups and smooth manifolds at the same time. Before we make this statement a bit more precise I should say a few words about manifolds…

For an informal overview of the idea of manifolds you can consult an earlier post I made here. I will assume everyone had read this, or is at least familiar with the basic idea. I will review the minimum needed to define a Lie group.

A manifold is a ‘space’ that is locally similar to $$\mathbb{R}^{n}$$ for some n. A smooth manifold is a refinement of that notion to allow us to do calculus. Any manifold can be described by a collection of charts, also known as an atlas.

An atlas on a topological space $$X$$ (say) is a collection of pairs $$\{(U_{\alpha},\phi_{\alpha})\}$$ called charts, where the $$U_{\alpha}$$ are open sets that cover the topological space, such that

$$\phi_{\alpha}: U_{\alpha} \rightarrow \mathbb{R}^{n},$$

is a homomorphism of $$U_{\alpha}$$ onto an open subset of $$\mathbb{R}^{n}$$. Loosley this means that locally we can ways think about cutting our topological space up into small pieces of the real linear space.

The transition maps are defined as

$$\phi_{\alpha \beta}:= \phi_{\beta} \circ \phi^{-1}_{\alpha}|_{\phi_{\alpha}(U_{\alpha} \cap U_{\beta})}: \phi_{\alpha}(U_{\alpha} \cap U_{\beta}) \rightarrow \phi_{\beta}(U_{\alpha} \cap U_{\beta}).$$

Any topological space with an atlas is a topological manifold. Loosley, the transition maps allow you to sew together the local patches by telling you what happens on the overlap of such patches.

We will be interested smooth manifolds, that is we insist that the transition maps be infinitely differentiable in the standard sense. Because we can describe everything locally on a smooth manifold in terms of smooth transition functions and local patches of $$\mathbb{R}$$ we can extend all our knowledge of standard multi-variable calculus to smooth manifolds.

In particular we know what a smooth map between two smooth manifolds is. As topological spaces a map between smooth manifolds is a continuous map. To define it as ‘smooth’ we compose the function with a chart on our source and target manifolds and as we know what smoothness means for map from $$\mathbb{R}^{n}$$ to say $$\mathbb{R}^{m}$$ we can accordingly define smoothness for maps between smooth manifolds.

Exercise Fill in details for the above paragraph.

We can now state what a Lie group is…

Definition A Lie group $$G$$ is a smooth manifold that also carries a group structure whose product and inversion operations are smooth maps.

That is both

$$\mu : G \times G \rightarrow G$$
$$(x,y) \mapsto \mu(x,y) = x\cdot y$$

and

$$inv : G \rightarrow G$$
$$x \mapsto x^{-1}$$

are smooth maps.

Examples to follow…

# What is a topological space?

Introduction
A topological space is a rather general notion of a space in terms of a set of points endowed with some extra structure that gives us some notion of “nearness” of points. Topological spaces are quite general objects and appear all over the place in modern mathematics.

I my early post on manifolds, I used the notion of a topological space in defining the domains that are patched together to build a manifold, though I did not use the word “topology” at all. So manifolds are very nice examples of topological spaces.

Intuitively topology is the study of the properties of topological spaces that do not change under deformations, stretching and bending, but not cutting and gluing. That is we keep the notion of “nearness” of points.

There are several equivalent ways of defining a topological space, but here we will take the most intuitive route in terms of neighborhoods of points.

Neighborhoods
Our topological space is a set of points, which we will denote $$X$$. By a neighborhood of a point $$x \in X$$, we mean a subset of $$X$$ that consists of all points “sufficiently close” to $$x$$. What we mean by “sufficiently close” depends on the situation and can depend on the different neighborhoods. Anyway, neighborhoods satisfy some natural axioms (properties):

1.Each point $$x$$ belongs to every one of its neighborhoods.
2.Every subset of $$X$$ that contains a neighborhood of a point $$x$$ is also a neighborhood of $$x$$.
3.When two neighborhoods of a point $$x$$ overlap, this overlap is also a neighborhood of $$x$$.

The above three axioms are very natural and clear. The fourth is less so, but very important in patching neighborhoods together.

4.Any neighborhood $$N_{1}$$ of $$x$$ contains another neighborhood $$N_{2}$$ of $$x$$ such that $$N_{1}$$ is a neighborhood of each of the points in $$N_{2}$$.

The feeling one should get from this these axioms is that we have a notion of “closeness” given by points sharing neighborhoods. Moreover, given a point and one of its neighborhoods, one can “move the point a little” and still remain in the original neighborhood.

Continuous maps
As hinted to earlier, topology studies spaces up to changes that do not “cut and paste”. The notion of topological spaces and continuous maps (“small changes in input give small changes in output”) between them formalises this. So, as i want this to be a rather informal post lets not get bogged down with details and look at an example.

Courtesy of Wikipedia

Above is a graphical representation of a continuous deformation between a mug and a torus. Heuristically, you see that points that are near each other on the mug remain near each other on the torus. Also note that such a map does not change the overall shape, there is one hole and always one hole here.

In conclusion
Intuitively topological spaces are just sets in which we have a good notion of “nearness” of points. Topology is interested not in the details of the geometric shape of the spaces but only on how the space is put together.

# What is a Manifold?

Preamble
So I have decided to create a few posts about some of the important mathematical ideas that I regularly encounter. The idea is to be rather informal and try to give a general feeling about these objects rather than a proper mathematical definition. I am hoping these posts will be readable to anyone with a reasonable high school education in mathematics. I will assume basic algebra and elementary geometry in the plane for example.

Please email me if you have something you would like me to cover.

So on to manifolds…

Introduction
Manifolds are an important notion in geometry and topology. Basically they are spaces that locally look like the much more familiar Euclidean spaces. This allows the well-understood notions on Euclidean spaces to be generalised to manifolds rather directly.

Coordinates
Recall that the space $latex\mathbb{R}^{n}$ for some integer $latexn$, can be understood by assigning an n-tuple $$x^{\mu}:=(x^{1}, x^{2}, \cdots , x^{n})$$ of real numbers to each point; that is we can pick some coordinate system. The choice of coordinates is far from unique and a large part of geometry is related to the freedom in picking coordinates. We will not worry about changes of coordinates here.

Above we have a Cartesian coordinate system on the plane. Every point is assigned a pair of numbers.

It was Rene Descartes (1596-1650) who pioneered this analytic approach to geometry; today we honor him the term Cartesian when referring to coordinates. By employing coordinates, algebra and analysis can come to bare on question in geometry. This is in contrast to the axiomatic or synthetic approach that goes back to Euclid (circa 300 BC). The synthetic approach to geometry uses axioms, theorems and logical arguments to study spaces.

Manifolds
We can now describe the notion of a manifold, which is a higher dimensional analogue of a curve or a surface. A manifold of dimension $$n$$ is a space, you can think of a collection of points, that locally looks like $latex\mathbb{R}^{n}$ for some integer $$n$$. All curves in the two dimensional plane that do not intersect locally look like part of a line. All smooth surfaces in $$\mathbb{R}^{3}$$, that is surfaces that do not have sharp kinks, edges, or points all locally look like the two dimensional plane, see figure below. Thus such surfaces are two dimensional manifolds. One should imagine being able to tear any small piece off the smooth surface, then being able to stretch it, push down any “hills” and push up any “valleys”” to end up with a flat piece of the 2-plane.

From Wikipedia

A little more formally, a manifold of dimension $$n$$ can be thought of as being built up out of domains, which are open subsets of $$\mathbb{R}^{n}$$. Intuitively, an open set is a set of points, in this case a subset of points belonging to $$\mathbb{R}^{n}$$, such that any individual point can be displaced and still remain in the set. This gives a notion of points being “near” without the need for the notion of a distance between points. Such domains are commonly denoted as $$U \subset \mathbb{R}^{n}$$. One can then “build” a manifold by patching these domains together, a bit like how one would make a patchwork quilt. It is how these domains are patched together that really defines the manifold.

In essence, the domains allows us to employ local coordinates when dealing with manifolds. Thus, one can build the theory of manifolds based on our understanding of the space $$\mathbb{R}^{n}$$.

A quite familiar example to us all is the relation between a globe and a map. (I mean the common notion of a map, not the mathematical one!) The globe is a representation of the Earth (or any other planet) on the surface of the two sphere. Small pieces of the globe can always be represented as a map, which you think of as a piece of the two plane. Useful local coordinates on the maps can then be employed.

Image by Christian Fischer.

The picture to have in your mind when thinking about manifolds is the relation between a globe and a map. Small pieces of the globe can always be described by maps (pieces of the 2-plane). Moreover, the entire globe can be covered by a collection of maps: an atlas.

Etymology
The etymology of the word “manifold” is old English. The word literally means many folds. Today, generally the word has come to mean any object having many different parts or features. The use of manifold in the mathematical context is adapt, manifolds are generalisations of surfaces that are built up from many domains and have many features not seen on $latex\mathbb{R}^{n}$.

Not manifolds
It is also worth highlighting some spaces that are not manifolds. A very simple example would be a self-intersecting curve in the plane. Such a curve will have regions that look like “X”. At these intersection points we no longer have a manifold structure. Such spaces are known as manifolds with a singularity.

Image courtesy of Wolfram

The double cone above looks very much like a manifold apart from the point at which the two cones meet. Everywhere not near this apex locally looks like the plane. This is also an example of a manifold with a singularity.

Manifolds with a boundary are not manifold!. For example the finite cylinder has two circles which are one dimensional manifolds as it’s boundaries.

Applications
Manifolds can have a variety of extra structures on them. Indeed I have been very loose with the class of manifolds I have discussed here. Anyway, manifolds can have a differentiable structure on them meaning that we can do calculus on them. This is great for physics and indeed smooth manifolds are important in theoretical physics.

For example, smooth manifolds that come with a metric, that is a notion of distance on them, are at the heart of Einstein’s general relativity. Also, smooth manifolds also appear as the phase spaces in classical mechanics and these carry another interesting structure, that of a Poisson bracket.

I will say more about Poisson brackets in another post.