Revisiting Clifford Algebras

TL;DR

This paper explores the structure and properties of Clifford algebras in various contexts, including those associated with Hessians of functionals, infinite-dimensional Banach spaces, and explicit kernel formulas, providing new insights and definitions.

Contribution

It introduces a tensorial topology on Clifford algebras, studies Clifford algebras of infinite-dimensional Banach spaces, and derives explicit formulas for Hilbertian kernels.

Findings

Defined tensorial topology on Clifford algebras

Analyzed properties of Clifford algebras in Banach spaces

Provided explicit formulas for Hilbertian kernels

Abstract

This paper is divided in three parts. In the first part, I study the Clifford algebra associated to the hessian of a functional $f$ defined on an open subset of $\mathbb{R}^n$ \ and the Clifford algebra associated to the hessian of the Legendre transform of $f$. I give also the definition of a tensorial topology on a Clifford algebra. In the second part, I study the Clifford algebra $C\left( E,q\right) $ of an infinite dimensional Banach space $E$ and its main properties. Finally, in the third part, I give the explicit formula of the hilbertian kernel of a Clifford algebra with examples.