Noncommutative Galois Extension and Graded q-Differential Algebra

TL;DR

This paper demonstrates that semi-commutative Galois extensions of unital associative algebras can be structured as graded q-differential algebras, enabling the development of noncommutative differential calculus with applications to quaternions.

Contribution

It introduces a novel connection between semi-commutative Galois extensions and graded q-differential algebras, expanding the framework of noncommutative differential calculus.

Findings

Galois extensions can be endowed with graded q-differential algebra structure

Development of higher order noncommutative differential calculus

Application to quaternion algebra as a semi-commutative Galois extension

Abstract

We show that a semi-commutative Galois extension of a unital associative algebra can be endowed with the structure of a graded q-differential algebra. We study the first and higher order noncommutative differential calculus of semi-commutative Galois extension induced by the graded q-differential algebra. As an example we consider the quaternions which can be viewed as the semi-commutative Galois extension of complex numbers.