Automorphisms of associative algebras and noncommutative geometry

TL;DR

This paper investigates differential calculi defined by automorphisms of associative algebras, illustrating their application to quantum planes and groups, and introduces geometric structures like metrics and connections.

Contribution

It introduces a class of differential calculi based on automorphisms and applies them to quantum geometries, providing new tools for noncommutative geometry.

Findings

Differential calculi on quantum plane and quantum group GLpq(2) are recovered.

Geometric structures such as metrics and linear connections are constructed.

The approach unifies various examples of noncommutative differential calculus.

Abstract

A class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra. Several examples are presented. In particular, differential calculi on the quantum plane, the $h$-deformed plane and the quantum group GLpq(2) are recovered in this way. Geometric structures like metrics and compatible linear connections are introduced.