Canonical bases for real representations of Clifford algebras

TL;DR

This paper develops algorithms and methods to transform real representations of Clifford algebras into canonical forms, leveraging algebraic structures and providing explicit computational tools with potential applications in graphics and robotics.

Contribution

It introduces explicit algorithms for converting Clifford algebra representations to canonical forms, utilizing abelian subalgebras and invariant subspaces, with implementations for lower dimensions.

Findings

Algorithms for canonical form transformation are provided.

Explicit change of basis matrices are constructed and computed.

The methods demonstrate applications in computer graphics and robotics.

Abstract

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, 2 or 4, according as the Clifford algebra represented on $R^N$ is of real, complex or quaternionic type. We have expressed canonical forms of these representations in terms of the complex and quaternionic structures in the half dimension and we obtained algorithms for transforming any given representation of $Cl(r,s)$ to a canonical form. Our algorithm for the transformation of the representations of $Cl(8d+c,0)$, $c\le 7$ to canonical forms is based on finding an abelian subalgebra of $Cl(8d+c,0)$ and its invariant subspace. Computer programs for determining explicitly the change of basis matrix for the transformation to canonical forms are given for lower dimensions. The construction of the change of basis matrices uniquely up to the commutant provides a constructive proof of the uniqueness properties of the representations and may have applications in computer graphics and robotics.