Differential Forms, Hopf Algebra and General Relativity I

TL;DR

This paper reviews differential forms and their algebraic structures, applying them to Riemannian Geometry and General Relativity, and introduces a Hopf algebra structure to Cartan-Grassmann algebra.

Contribution

It introduces a new co-multiplication product and demonstrates how Cartan-Grassmann algebra can be structured as a Hopf algebra.

Findings

Structure equations and symmetries derived using the new product

Cartan-Grassmann algebra endowed with Hopf algebra structure

Applications to Riemannian Geometry and General Relativity

Abstract

We review the language of differential forms and their applications to Riemannian Geometry with an orientation to General Relativity. Working with the principal algebraic and differential operations on forms, we obtain the structure equations and their symmetries in terms of a new product (the co-multiplication). It is showen how the Cartan - Grassmann algebra can be endowed with the structure of a Hopf algebra.