## Clifford Algebras

Clifford algebra is a deformation of the exterior algebra by a quadratic form. More intuitively, this means that the usual scalar and exterior products are combined into a single product xy = x

**·**y + x∧y. Due to the symmetry properties of the product, parallel vectors commute whilst orthogonal vectors anticommute. Clifford algebra and the exterior algebra are isomorphic as vector spaces, but as algebras Clifford algebra is strictly richer: unlike its constituent parts, the combined product is invertible, which leads to many simplifications, including applications to differential operators, with novel Green's function methods resulting from the invertibility of the vector derivative.

In particular, Clifford algebra has a uniquely simple way of handling reflections, which is therefore imminently useful in the Coxeter framework. Because of the commutation properties of the Clifford product of parallel and perpendicular vectors, the usual reflection formula a vector a=a⊥+a∥ in a hypersurface with unit normal n is a'=a⊥-a∥=a-2a∥ (where the perpendicular and parallel components are measured with respect to n: a∥ = (n

**·**a)a actually combines into a single term by `sandwiching': a'=-nan.

This basic result is very powerful, since by the Cartan-Dieudonné theorem every rotation is a product of an even number of reflections (spinors/versors in the algebra), and more generally due to a theorem by Artin, every orthogonal transformation is the product of reflections. This therefore provides a convenient way of dealing with orthogonal transformations by `sandwiching'. In particular, this provides a simple construction of the Spin and Pin groups, which are double covers of the special orthogonal group and the orthogonal group, respectively. Since furthermore the conformal group is homeomorphic to the special orthogonal group in two dimensions higher C(p,q) ~ SO(p+1, q+1), one can use these same techniques for conformal transformations, including reflections, rotations and translations - this is therefore of interest to affine extensions and (quasi)lattices. The conformal group contains interesting stabiliser subgroups - for timelike, spacelike and null vectors one obtains the (Anti) de Sitter groups and the Poincaré group. Spherical, hyperbolic, euclidean and conformal geometries are therefore easily handled in this framework. The usual matrix representations of the transformations are easily recovered from the spinor formulation, though the converse is more complicated, and spinorial techniques have various advantages over the conventional matrix or quaternionic formulations.