2007 will be remembered as the year in which the exceptional Lie algebra E₈ went mainstream. On 19 March the BBC published a story that a 248-dimension maths puzzle had been solved. The subject of the story was the herculean effort to work out the Kazhdan–Lusztig polynomials for the maximally split real form of E₈. (This is the real form which appears as the classical duality group in the compactification of eleven-dimensional supergravity down to three dimensions.) On 14 November, the Telegraph published the story that a surfer dude stuns physicists with theory of everything, echoing the noises coming from elsewhere in the blogosphere about this paper.
Unrelated – at least, I would like to believe — to these developments, E₈ made a surprise visit earlier this year as I was preparing some lectures on Killing superalgebras to an audience of mathematicians in Florence. Supergravity is not easy to motivate to mathematicians and the beauty and coherence of the theory usually hard to appreciate. While preparing the lectures, I felt compelled to seek an analogue of the notion of Killing superalgebra in riemannian geometry which might make the supergravity construction more palatable.
At its most elementary, the Killing superalgebra owes its existence to very simple geometric data: a spin manifold with some privileged notion of spinor, and such that this class of spinors is then preserved by the vector fields constructed out of them. Luckily this situation is not uncommon in riemannian geometry. For example, we could consider manifolds with Killing spinors, such as the round spheres.
One particular triple of spheres is the one appearing in the octonionic Hopf fibration: S⁷ → S⁹ → S⁸, where we may think of S⁸ as the octonionic projective space and S⁷ as the unit octonions. It turns out that applying the Killing superalgebra construction to these sphere we obtain (the compact real forms of) the following ℤ₂-graded Lie algebras: B₄, F₄ and E₈, respectively! One can also obtain the maximally split real forms by a simple modification. I still harbour the hope that these algebras relate to each other just like the spheres from which they are constructed relate to each other via the Hopf fibration.