Lie groups&algebras, Coxeter/reflection groups and root systems are closely related, and feature prominently throughout mathematics and physics, in particular the exceptional ones. We argue that the root system concept is the most useful for our purposes, and that since an inner product is implicit when considering reflections, one can always construct the Clifford algebra over the underlying vector space. Clifford algebra has a very simple reflection formula and via the Cartan-Dieudonne theorem provides a double cover of the orthogonal transformations. In particular, in 3D the Clifford algebra is 8-dimensional and its even subalgebra is 4-dimensional. Starting from a 3D root system one can therefore construct groups of 4D or 8D objects under Clifford multiplication. The 4D ones can in general be shown to be root systems with interesting automorphism groups - in particular D4, F4, H4 are induced from A3, B3, H3 - and for the 8D case one can show (via R. Wilson's reduced inner product) that the Clifford double cover of the 120 reflections in H3 yields the 240 roots of E8.
A new construction of E8 and the other exceptional root systems
Pierre Dechant (York)
Mon, 18/01/2016 - 16:30