Basic quantum algebra

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 $F$.