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).

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.

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

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


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

are smooth maps.

Examples to follow…

III Meeting on Lie systems

The III meeting on Lie systems is going to be held next week (21.09.2015 – 26.09.2015) here in Warsaw. It should be a great chance to catch up with some friends in the ‘Spanish Group’.

Of course you are all wondering what a Lie system is. Well, basically a Lie system is a systems of first-order ordinary differential equations whose general solution can be written in terms of a finite family of particular solutions and a superposition rule. There is a rich geometric theory here and many motivating examples that arise from physics.

From Poisson Geometry to Quantum Fields on Noncommutative Spaces

I will be attending the autumn school “From Poisson Geometry to Quantum Fields on Noncommutative Spaces” Oct 05–10, in Würzburg, Germany.

There will be a series of lectures:

  • Francesco D’Andrea (University of Naples)
    Topics in Noncommutative Differential Geometry
  • Martin Bordemann (Univ. Haute Alsace, Mulhouse)
    Algebraic Aspects of Deformation Quantization
  • Henrique Bursztyn (IMPA, Rio de Janeiro)
    Poisson Geometry and Beyond
  • Simone Gutt (ULB, Brussels)
    Symmetries in Deformation Quantization
  • Gandalf Lechner (University of Cardiff)
    Strict Deformation Quantization and Noncommutative Quantum Field Theories
  • Eva Miranda (University of Barcelona)
    Poisson Geometry and Normal Forms: A Guided Tour through Examples

It should be very interesting and I hope to learn a lot about subjects that are aligned with my general research area, but alas I have not yet looked into properly.

Also I will be presenting a poster on ‘Graded bundle in the category of Lie groupoids’ which is based on recent work with K. Grabowska and J. Grabowski (arXiv preprint)

The website for the school states that places may still be available.

On a variant of rhodonea curves

Rhodonea curves or rose curves are plots of a polar equation of the form
\(r = \cos(k \theta)\).

If we specialise to equations with

\(k= \frac{n}{d}\)

for n and d integers (>0), then we have plots of the form below. In the table n runs across and d down

Now, just for fun I considered a slight variant of this given by

\(r = \cos( k \theta) – k\)

The plots are as follows

For another variant I considered

\(r = \cos( k \theta) – k^{-1}\)

I am not sure there is anything mathematically deep here, I just like the images and classify this as some basic mathematical art.

On the physics of chocolate

Researchers at Technische Universität München, Germany, have reported that molecular dynamics can be used to gain new insights into the chocolate conching [1].

Chocolate conching is the stage of manufacturing where aromatic sensation, texture and mouthfeel are developed.

This work seems to be the first to attempt to properly understand the role of lecithins in chocolate production.

Physics, helping to build a tasty more palatable world.

[1] M Kindlein, M Greiner, E Elts and H Briesen, Interactions between phospholipid head groups and a sucrose crystal surface at the cocoa butter interface, 2015 J. Phys. D: Appl. Phys. 48 384002.

Chocolate physics: how modelling could improve mouthfeel, IOP website.

An interview with Prof. Christopher Lintott

Prof. Christopher Lintott of Oxford University, winner of this years Kelvin Medal and Prize from the Institute of Physics and regular on the BBC’s Sky at Night agreed to answer a few questions.

Science and Popularisation

1. What first got you involved in science, and in particular astronomy?

I was a small kid who loved looking through telescopes – first of all a neighbour’s small reflector, then a larger telescope at school. I loved the idea that we could understand what’s happening in space despite being stuck on the surface of a small insignificant planet – and that there was lots left for us still to find out.

2. What was your first telescope?

The same one I have now, a 6” reflector. It’s nothing fancy – it doesn’t even have a motor – but it allows me to explore the sky. I’m a great fan of astrophotography, but I spend too much time looking at my computer as it is. When I’m observing I want the photons to be hitting my eyeballs!

3. How did you get involved in the BBC’s Sky at Night?

I’d been doing some science writing and got invited to be a guest on the show. From there, I was lucky enough to be part of the team and I gradually did more and more. I think Sky at Night’s a wonderful show, with the chance to explore so many fascinating aspects of our relationship with the Universe.

4. What is ‘Citizen Science’?

It’s a modern term for an old idea, which is that anyone can participate in the scientific process. These days, we use the term to cover the kind of projects we build on Zooniverse.org – projects which allows professional astronomers and volunteers together to comb through the vast stores of data which modern surveys produce.

5. Which medium do you think is the most effective at popularising science?

It depends what you’re trying to do, but one of the things that I think we need to remember is that we can’t rely on people choosing to seek out scientific content. A large proportion of the public have been put off science through experiences at school, or through a lack of confidence, and we need to find ways to reach them. In the old days, that meant big budget TV shows, but now that audience is fragmenting we need to find new ways for people to stumble across science. As an example, the Adler Planetarium in Chicago runs a Telescopes in the City program in which they take scopes (and astronomers) to random locations, surprising people with the sky. I think that kind of experience can be life-changing.

6. What, in your opinion, should be the ultimate goal of science popularisation?

I’m not sure it’s an ultimate goal, but there are lots of people who I believe would enjoy following science as it happens, and maybe even participating. I want that crowd to feel like they’re part of the journey, rather than just consumers of pre-packaged scientific results. We just reported on the New Horizons encounter with Pluto, which threw up all sorts of wonderful surprises. Someone said to me that they hadn’t realised that scientists smile when they say they don’t know something – I’d like more people to participate in the joy of not knowing.


1. Can you say a few words about your research?

These days I’m interested in galaxies – in particular, we’re trying to find out why some galaxies form stars and why some appear to have shut down. Most of this work is done with the wonderful data provided by Galaxy Zoo volunteers.

2. Which one of your papers are you most proud of, and why?

The discovery paper for the Voorwerp – I had to learn a lot to write it, and we had the most tremendous battles with the referee, but it came out well in the end. Plus it’s about a wonderful object, and consisted of what I thought astronomers did when I was a kid. Find an interesting object, and point telescopes at it until you know what it is…

Chris Lintott’s homepage at Oxford
BBC Sky at Night

On contact and Jacobi geometry

I have placed a preprint on the arXiv Remarks on contact and Jacobi geometry, which is joint work with K. Grabowska and J. Grabowski.

In the preprint we explain how the proper framework of contact and Jacobi geometry is that of \(\mathbb{R}^{\times}\)-principal bundles equipped with homogeneous Poisson structures. Note that in our approach homogeneity is with respect to a principal bundle structure and not just a vector field. This framework allows a drastic simplification of many standard results in Jacobi geometry while simultaneously generalising them to the case of non-trivial line bundles. Moreover, based on what we learned from our previous work, it became clear that this framework gives a very natural and general definition of contact and Jacobi groupoids.

The key concepts of the preprint are Kirillov manifolds and Kirillov algebroids, i.e. homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. Among other results we

  • describe the structure of Lie groupoids with a compatible principal G-bundle structure
  • present the `integrating objects’ for Kirillov algebroids
  • define contact groupoids, and show that any contact groupoid has a canonical realisation as a contact subgroupoid of the latter

Our motivation
The main motivation for this work was to put some order and further geometric understanding into the subject of contact and Jacobi geometry. We take the ‘poissonisation’ as the true starting definition of a ‘Jacobi structure’ and accept all the consequences of that choice. Importantly, once phrased in the correct way, that is in terms of \(\mathbb{R}^{\times}\)-principal bundles and their actions, the true nature of Jacobi geometry as a specialisation and not a generalisation of Poisson geometry becomes clear.

Non-trivial line bundles makes it easier?
Almost oxymoronically, passing to structures on non-trivial line bundles and then the language of \(\mathbb{R}^{\times}\)-principal bundles really does simplify the overall understanding.

This is particularly evident for contact and Jacobi groupoids where insisting on working with a trivialisation leads to unnecessary complications.

In conclusion
We hope that this work will really convince people that contact and Jacobi geometry need not be as complicated as it is often presented in the literature. Quite often the constructions become very ‘computational’ and ‘algebraic’, and in doing so the underlying geometry is obscured. In this work we really try to stick to geometry and avoid algebraic computations.

One nation science

The UK Science Minister Jo Johnson wants to reshape the funding of science and make sure that the funds are distributed wider.

Currently the ‘Golden Triangle’, Oxford, Cambridge and London’s UCL, Imperial and King’s receive 35% of the total £2.7bn annual funding.

I think that spreading the money could be a good idea, though it has to be done carefully and objectively.

Science minister signals shift towards ‘one nation science’ BBC News

The 2nd Conference of the Polish Society on Relativity

I will be attending the 2nd conference of the Polish Society of Relativity which will celebrate 100 years of general relativity.

The conference is in Warsaw and will be held over the period 23-28 November 2015.

The invited speakers include George Ellis, Roy Kerr, Roger Penrose and Kip Thorne. I am a little excited about this.

Registration is now open and you can follow the link below to find out more.

Polskie Towarzystwo Relatywistyczne

Homotopy versions of Jacobi structures

I have placed a preprint on the arXiv ‘Jacobi structures up to homotopy’ (arXiv:1507.00454 [math.DG]) which is joint work with Alfonso G. Tortorella (a PhD student from Universita degli Studi di Firenze, Italy). Our motivation for the work comes from the increasing presence of higher Poisson and Schouten structures in mathematical physics.

In the preprint we ask and answer the question of ‘how to equip sections of (even) line bundles over a supermanifold with the structure of an L-algebra’.

It turns out that the most conceptionally simple way to do this is to adopt the philosophy of [1] and study homogeneous higher Poisson geometry. In essence, we take the ‘higher Poissonisation’ of a ‘homotopy Kirillov structure’ as the starting definition. In this way we go around trying to carefully define homotopy Kirillov structure in the so-called ‘intrinsic set-up’; which would be quite complicated for non-trivial line bundles. We define a higher Kirillov manifold as a principal ℜx-bundle equipped with a homogeneous higher Poisson structure.

We also study the notion of a higher Kirillov algebroid, which is essentially a higher Kirillov manifold with an addition compatible regular action of ℜ. This additional action encodes a vector bundle structure [2].

Interestingly, from the structure of a higher Kirillov algebroid we derive a line bundle equipped with a ‘higher representation’ of an associated L-algebroid. This is very similar to the classical case of Jacobi algebroids and is a geometric realisation of Sh Lie-Rinehart representations as define by Vitagliano [4]. As a special example we see that the line bundle underlying a higher Kirillov manifold comes equipped with a higher representation of the first jet vector bundle of the said line bundle (which is naturally an L-algebroid).

Finally we present the higher BV-algebra associated with a higher Kirillov manifold following the ideas of Vaisman [3].

[1] J. Grabowski, Graded contact manifolds and contact Courant algebroids, J. Geom. Phys. 68 (2013) 27-58.

[2] J. Grabowski & M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), 1285-1305

[3] I. Vaisman, Annales Polonici Mathematici (2000) Volume: 73, Issue: 3, page 275-290.

[4] L. Vitagliano, Representations of Homotopy Lie-Rinehart Algebras, Math. Proc. Camb. Phil. Soc. 158 (2015) 155-191.

Random thoughts on mathematics, physics and more…