lie algebra

The Theory of Everything - The E8 Lie group, a perfectly symmetrical 248-dimensional object and possibly the structure that underlies everything in our Universe.

Mathematics is the language of the Universe. Absolutely everything, from a plane crash to your skin pigment to the shape of a Sphere can all be expressed using mathematical equations. This last example is most important in pursuit of the Theory of Everything.

In the 19th century, the mathematician Sophus Lie created algebraic formulas to describe the shape of symmetrical objects. These are called Lie Fields. In the 1890s, Wilhelm Killing found a set of Lie fields that described perhaps the most complex shape in our Universe, the E8 Group. The E8 group, an interrelated 248-dimensional symmetrical object, is an extremely complex one.
This dense object is so complex, in fact, that it was plotted by computer for the first time in 2007. It took a team of 18 mathematicians four years to calculate and plot the formula for E8. So how can a symmetrical shape be the key to the Universe? First remember that a geometrical shape is merely the graphic representation of mathematical formulae. It is a pattern that is expressed in math and forms a shape when plotted. In this sense, the E8 could be the framework into which everything - all forces and particles - fits in our Universe.

Physicists generally held that gravity couldn’t be expressed mathematically in the same way as electromagnetism and strong and weak nuclear forces could.
But Garrett Lisi had heard about a mathematical way of expressing gravity, called MacDowell-Mansouri gravity. Using this expression, Lisi can use mathematical expressions to plug gravity into E8, along with electromagnetism, and weak and strong nuclear forces. All four of the forces in the Universe create a distinct effect on all of the most basic subatomic forms of matter - called elementary particles. When these particles interact with force carriers (called bosons), they become different particles. For example, when one of the most basic quantum particles - the lepton - encounters a weak-force boson, it becomes a neutrino. A lepton interacting with a photon (a boson that carries an electromagnetic charge) becomes an electron. So while there are limited numbers of the most basic particles, when they encounter the different forces, they change to become other, distinct particles. What’s more, for every particle, there is an equally distinct anti-particle, for example an anti-quark or anti-neutrino. In total, these make up the elementary particles, and there are 28 of them. Each of these distinct elementary particles has eight quantum numbers assigned to it, based on the charges each particle has. This brings the number of distinct particles to 224. These numbers helped Lisi make the particles fit into the E8 model. While the E8 is expressed as a 248-dimensional object in one way, it can also be expressed as an eight-dimensional object with 248 symmetries. Lisi used E8 within eight dimensions for his calculations. For the remaining 24 places unfilled by distinct known particles, Lisi used theoretical particles which are yet to be observed.

Take another look at E8, and notice how the lines radiate from each point:
Lisi assigned each of these 248 points to a particle, using the eight numbers based on their charges as coordinates within the eight dimensions. What he found was that, like the symmetries in the E8 group, quantum particles share the same relationship within the symmetrical object. He has hope that he has figured out a way to crack the Theory of Everything, because when he rotated the E8 filled with the force-influenced (including gravity) quantum particles, he found patterns emerging between particles and forces - photons interacting with leptons, for example, created electrons. The connections shown within points on the E8 match up to real, known connections between particles in our physical world.

Generalisation is not the point of mathematics. To be honest, it’s usually rather dry. The challenge is to generalise in a rich and revealing direction.

Terry Gannon, Moonshine Beyond the Monster §3.3

(generalisations of the affine algebras: Kac-Moody algebras, Borcherds’ algebras, toroidal algebras)


Revised the first half of topology today and remembered how much I like it. Its one of the few courses where I can still sit down and scribble pictures and stare in to space and use plenty of intuition and eventually arrive at the correct reasoning; pretty much what got me into maths in the first place. It really is a beautiful and rich subject!

Going to start the algebraic topology part tomorrow, much harder but also really interesting. The thrust of algebraic topology as I understand it is to translate topological problems into problems in group theory. These are generally much easier to understand and a great deal of group theory is very well understood.

And then the ideas of Lie Algebras help us translate harder group theory problems into linear algebra which is easier again! I love maths, why don’t I listen in lectures?