Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups

This is a short review of the algebraic properties of Clifford algebras and spinors. Their use in the description of fundamental physics (elementary particles) is also summarized. Lecture given at the ICCA7 conference, Toulouse (23/05/2005)

Introducing a quaternionic structure on Euclidean space, the fundaments for quaternionic and symplectic Clifford analysis are studied in detail from the viewpoint of invariance for the symplectic group action.

Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ...

In a recent paper, algebraic descriptions for all non-relativistic spins were derived by elementary means directly from the Lie algebra $\specialorthogonalliealgebra{3}$, and a connection between spin and the geometry of Euclidean three-space was drawn. However, the details of this relationship and the extent to which it can be developed by elementary means were not expounded. In this paper, we will reveal the geometric content of the spin algebras by realising them within a ...

Quaternions were appeared through Lagrangian formulation of mechanics in Symplectic vector space. Its general form was obtained from the Clifford algebra, and Frobenius' theorem, which says that "the only finite-dimensional real division algebra are the real field ${\bf R}$, the complex field ${\bf C}$ and the algebra ${\bf H}$ of quaternions" was derived. They appear also through Hamilton formulation of mechanics, as elements of rotation groups in the symplectic vector space...

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...

The line geometric model of 3-D projective geometry has the nice property that the Lie algebra sl(4) of 3-D projective transformations is isomorphic to the bivector algebra of CL(3,3), and line geometry is closely related to the classical screw theory for 3-D rigid-body motions. The canonical homomorphism from SL(4) to Spin(3,3) is not satisfying because it is not surjective, and the projective transformations of negative determinant do induce orthogonal transformations in th...

4-dimensional $A_{4}$ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group $W(A_{4})$ where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary $W(A_{4})$ orbit into three dimensions is made using the subgroup $W(A_{3})$. A generalization of the Catalan solids for 3D polyhedra has been developed and dual polytopes of the uniform $A_{4}$ polytopes have been constructed.

In this paper we address the problem of constructing a class of representations of Clifford algebras that can be named "alphabetic (re)presentations". The Clifford algebras generators are expressed as m-letter words written with a 3-character or a 4-character alphabet. We formulate the problem of the alphabetic presentations, deriving the main properties and some general results. At the end we briefly discuss the motivations of this work and outline some possible applications...

In this paper we classify reflection subgroups of Euclidean Coxeter groups.