Symbolic Computations in Higher Dimensional Clifford Algebras

TL;DR

This paper introduces methods for symbolic computations in high-dimensional Clifford algebras using Maple packages, leveraging tensor decompositions, periodicity theorems, and matrix representations to facilitate algebraic manipulations.

Contribution

It presents new computational techniques and implementations for symbolic algebra in Clifford algebras of dimension nine or higher, including coding algebra isomorphisms and involutions.

Findings

Implemented methods in Maple packages exttt{Clifford} and exttt{Bigebra}

Demonstrated algebraic computations using tensor decompositions and periodicity theorems

Provided benchmarks showing efficiency of the methods

Abstract

We present different methods for symbolic computer algebra computations in higher dimensional (\ge9) Clifford algebras using the \Clifford\ and \Bigebra\ packages for \Maple(R). This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.