Global and Local Nilpotent Bases of Matrices

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.