The $E_8$ geometry from a Clifford perspective

The discussion of how to apply geometric algebra to euclidean $n$-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from $19^{th}$ century mathematics. We then introduce the dual projectivized Clifford algebra $\mathbf{P}(\mathbb{R}^*_{n,0,1})$ (euclidean PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry. We compare euclidean PG...

We give an inductive construction for irreducible Clifford systems on Euclidean vector spaces. We then discuss how this notion can be adapted to Riemannian manifolds, and outline some developments in octonionic geometry.

This is a survey on the construction of a canonical or "octonionic K\"ahler" 8-form, representing one of the generators of the cohomology of the four Cayley-Rosenfeld projective planes. The construction, in terms of the associated even Clifford structures, draws a parallel with that of the quaternion K\"ahler 4-form. We point out how these notions allow to describe the primitive Betti numbers with respect to different even Clifford structures, on most of the exceptional symme...

This paper is meant to be an informative introduction to spinor representations of Clifford algebras. In this paper we will have a look at Clifford algebras and the octonion algebra. We begin the paper looking at the quaternion algebra $\mathbb{H}$ and basic properties that relate Clifford algebras and the well know Pin and Spin groups. We then will look at generalized spinor representations of Clifford algebras, along with many examples. We conclude the paper looking at the ...

I apply the algebraic framework developed in [1] to study geometry of hyperbolic spaces in 1, 2, and 3 dimensions. The background material on projectivised Clifford algebras and their application to Cayley-Klein geometries is described in [2].

One of the main goals of these notes is to explain how rotations in reals^n are induced by the action of a certain group, Spin(n), on reals^n, in a way that generalizes the action of the unit complex numbers, U(1), on reals^2, and the action of the unit quaternions, SU(2), on reals^3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and reals^n). The group Spin(n), called a spinor group, is defined as a certain subgr...

Affine subgroups having the same Coxeter number with the affine Coxeter groups W(An), W(Dn), and W(En) are constructed by graph folding technique. The affine groups W(Cn) and W(Bn) are obtained from the Coxeter groups W(A2n-1) and W(D2n-1) respectively. The affine groups W(E6), W(D6) and W(E8) lead to the affine groups W(F4), W(H3), and W(H4) respectively by graph folding. The latter two are the non-crystallographic groups where W(H3) plays a special role in the quasicrystall...

The well-known classification of the Clifford algebras $Cl(r,s)$ leads to canonical forms of complex and real representations which are essentially unique by virtue of the Wedderburn theorem. For $s\ge 1$ representations of $Cl(r,s)$ on $R^{2N}$ are obtained from representations on $R^N$ by adding two new generators while in passing from a representation of $Cl(p,0)$ on $R^N$ to a representation of $Cl(r,0)$ on $R^{2N}$ the number of generators that can be added is either 1, ...

It is shown that classical Clifford algebras are group algebras of cyclic subgroups of arrowy rermutations. It is established that Euclidean 3-space, Pauli and Dirac algebras and groups of global guage transformations are corollary from the geometry of 8-dimensional vacuum and 9-dimensional cosmos.

This paper shows how beginning with Justus Grassmann's work, Hermann Grassmann was influenced in his mathematical thinking by crystallography. H. Grassmann's Ausdehnungslehre in turn had a decisive influence on W.K. Clifford in the genesis of geometric algebras. Geometric algebras have been expanded to conformal geometric algebras, which provide an ideal framework for modern computer graphics. Within this framework a new visualization of three-dimensional crystallographic spa...