Method of averaging in Clifford algebras

TL;DR

This paper explores averaging operators in Clifford algebras, analyzing their properties and relations to projections, and applies these methods to solve systems of commutator equations.

Contribution

It introduces and studies averaging operators like the Reynolds operator in Clifford algebras, providing new solutions to commutator systems.

Findings

Reynolds operator averages Salingaros' vee group actions

Relations between averaging operators and projection operators

Solutions to systems of commutator equations using averaging methods

Abstract

In this paper we consider different operators acting on Clifford algebras. We consider Reynolds operator of Salingaros' vee group. This operator average" an action of Salingaros' vee group on Clifford algebra. We consider conjugate action on Clifford algebra. We present a relation between these operators and projection operators onto fixed subspaces of Clifford algebras. Using method of averaging we present solutions of system of commutator equations.