Basic quantum algebra

TL;DR

This paper introduces quantum algebra from a geometric perspective, focusing on quantum spaces that are independent of the underlying field, aiming to unify various classes of quantum spaces.

Contribution

It identifies and emphasizes quantum spaces that are invariant across different fields, providing a unified geometric framework for quantum algebra.

Findings

Quantum spaces independent of the field are characterized.

A geometric approach to quantum algebra is developed.

The paper highlights the importance of field-independent quantum spaces.

Abstract

This is an introduction to quantum algebra, from a geometric perspective. The classical spaces $X$, such as the Lie groups, homogeneous spaces, or more general manifolds, are described by various algebras $A$, defined over various fields $F$. These algebras $A$ satisfy a commutativity type condition, and the general idea is that of lifting this condition, and calling quantum spaces the underlying space-like objects $X$. One problem comes from the fact that different fields $F$ lead, via different algebras $A$, to different classes of quantum spaces $X$. Our aim here is to identify and put at the center of the presentation those quantum spaces $X$ which do not depend on the choice of $F$.