# Quadratic differential algebras generated by Euclidean spaces

## TL;DR

This paper introduces a new class of quadratic differential algebras generated by Euclidean spaces, generalizing exterior calculus on polynomial functions and exploring their algebraic structures and connections with Koszul complexes.

## Contribution

It defines quadratic differential algebras generated by Euclidean spaces and investigates their structure and relation to Koszul complexes, extending classical differential calculus.

## Key findings

- Established a framework for quadratic differential algebras generated by Euclidean spaces.
- Connected these algebras with Koszul complexes of quadratic algebras.
- Generalized exterior polynomial differential forms to arbitrary quadratic algebras.

## Abstract

We define a class of quadratic differential algebras which are generated as differential graded algebras by the elements of an Euclidean space. Such a differential algebra is a differential calculus over the quadratic algebra of its elements of differential degree zero. This generalizes for arbitrary quadratic algebras the differential graded algebra of exterior polynomial differential forms for the algebra of polynomial functions on $\mathbb R^n$. We investigate the structure of these differential algebras and their connection with the Koszul complexes of quadratic algebras.