Euclidean Clifford Algebra

TL;DR

This paper introduces the Euclidean Clifford algebra for real vector spaces with a Euclidean metric, providing a computational framework with multivectors, scalar products, and contraction operators.

Contribution

It develops a detailed construction of Euclidean Clifford algebra using multivectors, scalar products, and contraction operators, enhancing computational tools in geometric algebra.

Findings

Defined Euclidean Clifford algebra for real vector spaces.

Introduced multivectors and scalar products in this context.

Derived key identities and formulas for the algebra.

Abstract

Let $V$ be a $n$-dimensional real vector space. In this paper we introduce the concept of \emph{euclidean} Clifford algebra $\mathcal{C\ell}(V,G_{E})$ for a given euclidean structure on $V,$ i.e., a pair $(V,G_{E})$ where $G_{E}$ is a euclidean metric for $V$ (also called an euclidean scalar product). Our construction of $\mathcal{C\ell}(V,G_{E})$ has been designed to produce a powerful computational tool. We start introducing the concept of \emph{multivectors} over $V.$ These objects are elements of a linear space over the real field, denoted by $\bigwedge V.$ We introduce moreover, the concepts of exterior and euclidean scalar product of multivectors. This permits the introduction of two \emph{contraction operators} on $\bigwedge V,$ and the concept of euclidean \emph{interior} algebras. Equipped with these notions an euclidean Clifford product is easily introduced. We worked out with considerable details several important identities and useful formulas, to help the reader to develope a skill on the subject, preparing himself for the reading of the following papers in this series.