Global and Local Nilpotent Bases of Matrices

TL;DR

This paper explores the properties of Witt bases in Clifford geometric algebras, introducing local duality and linking it to Fourier and Wavelet transforms, with implications for matrix representations.

Contribution

It introduces a new concept of local duality in Witt bases and connects it to Fourier and Wavelet transforms, expanding understanding of geometric algebra structures.

Findings

Local duality enables defining real geometric algebras via complexification.

A relationship between duality concepts and matrix representations is established.

A connection between duality and Fourier/Wavelet transforms is demonstrated.

Abstract

In studying the unusual properties of a special Witt basis of a Clifford geometric algebra with a Lorentz metric, a new concept of local duality makes it possible to define any real geometric algebra by complexifying this structure. Whereas a global basis of Witt null vectors is defined in terms of a pair of correlated Grassmann algebras in a geometric algebra of neutral signature, the special Witt basis of a Lorentz geometric algebra is defined in terms of a single Grassmann algebra. The relationship between these different concepts of duality, and their matrix representations, is studied in terms of simple examples. A surprising connection is exhibited between discrete Fourier and Wavelet transforms and the concept of local duality.