The exceptional symmetry of E8

Here's a little movie I made about the Lie group E₈:

Out of all the known Lie groups, E₈ stands out as the largest and most complex exceptional group. It has 248 generating elements, which by themselves have an astounding degree of symmetry. This symmetry can only be fully grasped in 8-dimensional space. But luckily it is also possible to project E₈ onto a two-dimensional plane, chosen such that the resulting image preserves a small fraction of its total symmetry. There are different choices for these two-dimensional planes, some preserving more symmetry than others. The movie rotates through a selection of these planes in succession.

LieLink: a Mathematica interface for LiE

Ever wished that you could easily transfer you computations done in LiE to Mathematica? You now can with LieLink! LieLink is an small Mathematica package that interfaces with LiE, allowing you to directly execute LiE commands in Mathematica and get the result back. Here's an small example:

<<LieLink`

SetDefaultAlgebra["A2"]

LieTensor[{1, 0}, {1, 0}]
(* => {0,1} + {2,0} *)

You can download LieLink from github.com/teake/LieLink.

The Weyl group of C4

Time for a pretty picture! It's the Weyl group of the finite Lie algebra C₄.

Although C₄ is 'only' 36 dimensional, its Weyl group is a bit bigger and has 384 elements. The pictures doesn't show the elements of the Weyl group though, but rather all possible connections between them. If you looks closely the black dots aren't dots -- they're points where are lots of lines meet.

Coxeter planes

Here are a few nice Coxeter projections of representations of finite Lie algebras. If you don't know what a Coxeter projection is, or a finite Lie algebra for that matter, have a look at my PhD thesis (from which I took these pictures). Or, visit John Stembridge's website for more information and pictures.

Update: some of these pictures do not show the full representation, but only a subset.

The orbits of the two highest dominant weights of the 1764 representation of A₅.

The orbits of the two highest dominant weights of the 5985 representation of A₁₇.

The orbits of the two highest dominant weights of the 442 representation of B₆.

The adjoint representation of B₁₉.

The adjoint representation of D₁₀.

xTras for xAct

I'm a big fan of xAct, a tensor algebra package for Mathematica. While there are other tensor packages for Mathematica on the market, xAct is by far the best. It canonicalizes tensorial expressions blazingly fast, and its perturbation capabilities are state-of-the-art. If you've always wanted to do second-order perturbations of four-derivative curvature tensors but were afraid the actual calculation might take you some weeks, xAct is your man. It does it in a few seconds, and more importantly, doesn't mess up minus signs.

In fact, I like xAct so much, I wrote an additional package for. It's aptly called xTras, and its available over on www.xact.es/xtras. It brings some functionality that I found missing in xAct, like computing equations of motion for the metric, perturbations around AdS spaces, and Young projectors (yes, this also includes Bianchi identities).

I'll demonstrate some of the new functionality by computing the linearization of the Einstein tensor around AdS spaces. Here goes!
First, open up Mathematica, and enter the following line:

   In:    <<xAct`xTras`

This loads the xTras package (assuming you're managed to download and install it). Next we'll define a manifold and a metric:

   In:    DefConstantSymbol[dimension,PrintAs->"D"]
   In:    DefManifold[M,dimension,IndexRange[a,f]]
   In:    DefMetric[-1,g[-a,-b],CD]

We'll be doing stuff on AdS spaces, which has a constant curvature. Hence we need to define a constant symbol to indicate that curvature:

   In:    DefConstantSymbol[L]

Now we're ready to define the standard Lagrangian for gravity with a cosmological constant:

   In:    lagrangian = RicciScalarCD[] -(dimension-2)(dimension-1)L

We'd like to compute the equations of motion that follow from this Lagrangian. To do so, we first write the command

   In:    DefMetricVariation[g,h,eps]

This command makes it possible to do covariant metric variations. It also registers the command VarL, which varies Lagrangians:

   In:    eom = VarL[g[a,b]][lagrangian] //TensorCollect
Out:    \( \frac{1}{2} (2 - 3 D + D^2) L g_{ab} + R_{ab} - \frac{1}{2} g_{ab} R\)

The equations of motion should allow for AdS spaces. To check this, we first generate a list of replacement rules for curvature tensors of the covariant derivative CD on symmetric spaces:

   In:    AdSrules = SymmetricSpaceRules[CD,L]

 And indeed, the equations of motion are zero for this background:

   In:    eom /. AdSrules // ToCanonical
Out:    \( 0 \)

This means we can perturb around this solution. So without any further ado, here's the linear perturbation of the Einstein tensor:

   In:    ExpandBackground[eom, BackgroundSolution -> AdSrules] // TensorCollect
Out:    \( (l -  D l) h^{1}{}_{ab} + \frac{1}{2} (-1 + D) l g_{ab} h^{1c}{}_{c} - \frac{1}{2} \triangledown_{a}\triangledown_{b}h^{1c}{}_{c} + \frac{1}{2} \triangledown_{c}\triangledown_{a}h^{1}{}_{b}{}^{c} + \frac{1}{2} \triangledown_{c}\triangledown_{b}h^{1}{}_{a}{}^{c} \)
                                      \(- \frac{1}{2} \triangledown_{c}\triangledown^{c}h^{1}{}_{ab} - \frac{1}{2} g_{ab} \triangledown_{d}\triangledown_{c}h^{1cd} + \frac{1}{2} g_{ab} \triangledown_{d}\triangledown^{d}h^{1c}{}_{c} \)

 And that's it! Granted, we could also have done this by hand. But the power of xAct is that it can do much more complicated calculations without breaking a sweat. If we would like to know the second order pertubation of the Einstein tensor, we can simply replace the above input by ExpandBackground[eom,2,BackgroundSolution->AdSrules]. Pretty cool, right?

Affine root systems

Currently I'm working on my PhD thesis. It isn't finished yet, but I decided the following images were worth a sneak peak:

Besides looking nice, the pictures actually convey some information. They're so-called Hasse diagrams of the root systems of a few affine Lie algebras. From left to right we have the following affine algebras: (a) A1+, (b) C2+, (c) D4+, (d) A8+, (e) D7+, and (f) E7+. But luckily you don't need to fully understand the mathematical background, which is admittedly quite complicated, to enjoy their beauty.

More symmetry: continuous groups

In my previous post on "What on earth has Teake been doing for the last four years?" I tried to explain the concepts of symmetry and groups. Today we're going on step beyond, and see how they are related to algebras. Hold tight! It's going to be a bumpy ride ...

Let's first start of with the triangle. As we saw in the previous post it only had six distinct symmetries, and these symmetries formed a group. It's a discrete group because there are only a finite number of symmetries, and thus a finite number of elements in that group.
If you want objects with bigger symmetry, all you have to is increase the number of sides of your polygon. Here's for example the pentagon:

From now on we ignore the reflection symmetries and focus only on the rotational ones. It's easy to see the pentagon has six symmetries: rotations over 0˚, 72˚, 144˚, 216˚, and 288˚ all leave it invariant.

When we move up to the hendecagon (the 11-sided regular polygon), it will come as no surprise that the thing has 12 distinct rotational symmetries. But what happens if we crank the number of sides up to infinity? Then our polygon become a circle:

You can rotate it over any angle, and it remains the same. This means it has an infinite amount of symmetry! The mathematical object that describes these symmetries is still a group, but no longer a discrete (finite) one. The symmetry group of the circle is continuous. The reason why we call it continuous is because you can smoothly get from one rotation to another one by continuously applying infinitessimal (i.e. very small) rotations. Another way of phrasing this is to say that every angle between e.g. 72˚ and 144˚ corresponds to a symmetry. This was not so for the pentagon: in that case there are 'gaps' between the rotations. That's why that kind of symmetry is called discrete.

Continuous groups are known as Lie groups (pronounced as "lee"; they're named after Sophus Lie). They contain an infinite amount of elements. But because they're continuous we can parametrize the elements in one or more parameters. For the circle we can write any rotation over an angle θ as R(θ) as

This is just the rotation matrix in two dimensions. If R(θ) is still a group element, it should satisfy the group multiplication rule: R(θ₁) • R(θ₂) = R(θ₁ + θ₂). Or in plain English: the result of two succesive rotations over angles θ₁ and θ₂ should give a new rotation over an angle θ₁ + θ₂. Sure enough, if we brush up on our linear algebra and trigonometry, we find that

So group multiplication is indeed satisfied.
The above parametrization makes for easier bookkeeping of the infinite amount of group elements. But things can be simplified even further! Because the parametrization is continuous, we can take the derivative of R(θ) with respect to θ:

The magic happens when you consider the value of dR(θ)/dθ at zero angle, θ = 0 :

which we call T, for short. This thing is independent of the angle θ. What's more, you can recover all the rotations by simply exponentiating T:

We say that T generates the symmetry group of the circle. In proper mathematical lingo, it is called a generator. This single object captures all the important properties of the infinite symmetry group (well, almost all, but we'll not go into that right now). The bookkeeping now becomes very simply: we can just focus on the generator T, instead of the infinite amount of group elements.

You can show that for more complicated groups (e.g. the symmetry group of the sphere) the above simplification also holds. All the group elements can be led back to a finite number of generators (in the generic case there is more than one generator). These generators no longer are elements of a group. Instead, they form what is known as Lie algebra. But more on that in one of the upcoming episodes of "What on earth has Teake been doing for the last four years?"!

Symmetry

This is the first post of what eventually should become the "What on earth has Teake been doing for the last four years?"-series. Brace yourself: it's about maths and physics. Run while you still can!

I'll try to keep things simple by starting of with a concept that is as mundane as it is fascinating: symmetry. Not only makes it our world round, but it’s also what makes it go round. From the perfect circular wheels on our bikes and cars that deliver an enjoyable ride, to the error-correction protocols that keep e-mails from turning into junk; it’s literally all around us.

So what is symmetry exactly? A symmetry is an action on an object that, once you’re done performing it, does not change that object. It's a somewhat abstract definition, but take for example the triangle, which has 6 symmetries.  There are two different rotations (over 120˚ and 240˚), three reflections, and finally the action of doing nothing at all (yes, that's also a symmetry). You can try them out in the following applet. Clicking on the arrows causes the triangle to rotate and reflect.

This is all pretty straightforward, right? But things start to get interesting we you keep track of the effect of the different rotations and reflections. Let's paint the corners so we can see where they end up:

One thing you'll notice is that doing twice a clockwise rotation is equal to doing one counter-clockwise rotation. The same is true for any other combination of actions -- it will always yield the net effect of one single reflection or rotation. It might also happen that the triangle ends up in the original configuration, but remember that doing nothing is also a symmetry.

The combined actions describe what mathematicians call a group. A group is a set of elements plus a rule of multiplying those elements. Let's call the set G and the multiplication rule "•". Then the precise definition of a group (in which a, b are elements of the set G) is the validity of the following four statements:

• Closure.
  The result of the operation a • b is also in G.
• Identity element.
  There exists an element 1 in G, such that for all elements a in G, the equation 1 • a = a • 1 = a holds.
• Inverse element.
  There exists an element a^-1 in G such that a • a^-1 = a^-1 • a = 1.
• Associativity.
  The equation (a • b) • c = a • (b • c) holds.
  

These four set of rules are called the group axioms. They might sound a bit abstract, but in fact, they're not. Let's have a look at our triangle again.

The symmetry actions are labeled as follows:

• 1: identity element ("doing nothing at all").
• y: counter-clockwise rotation by 120˚.
• p: clockwise rotation by 120˚.
• R: reflection in the top vertex.
• G: reflection in the lower-right vertex.
• B: reflection in the lower-left vertex.

We already noticed that the closure axiom holds. The existence of the identity element is also pretty obvious. What about the identity axiom? This indeed also holds: every action has an inverse. The reflections are their own inverse, whereas the rotations are each other's. The last thing to check is associativity. It's a bit harder to verify, but believe me, it holds too.

Just for completeness sake, here's one last version of the triangle applet. This one includes the group multiplication table, which keeps track of what happens when you first do the action in the first row, followed by the action in the first column. (If you didn't believe me on the validity of associativity you can use this table to check it.)

What I've shown you so far is that the symmetries of the triangle can be described in terms of the mathematical concept of a group. The importance of group theory lies in the fact that any symmetry you can think of can be described as a group, and that conversely all groups describe a symmetry.

By now the answer to the question "What on earth has Teake been doing for the last four years?" will hardly come as a surprise: it's group theory. More on that in part two of this series!