On exact representations of the motions group of Galilean plane

TL;DR

This paper explores algebraic representations of the Galilean group in the plane using Pimenov and Grassmann algebras, providing exact matrix forms and geometric interpretations.

Contribution

It introduces new exact matrix representations of the Galilean group using Pimenov and Grassmann algebras, along with their geometric interpretations.

Findings

Defined properties of the Pimenov algebra with two generators

Constructed exact matrix representations of the Galilean group

Provided geometric interpretation of the algebraic representations

Abstract

The Pimenov algebra with two generators is defined and some of its properties are shown. Some exact matrix over the Pimenov algebra representations of the motions group of Galilean plane (the Galilean group) are considered. A geometric interpretation of them is giving. We consider also a exact representation of the Galilean group by elements of Grassmann algebra.