Differential calculus on Hopf--Galois extension via the Durdevi\'c braiding

TL;DR

This paper develops a new class of differential calculi on principal comodule algebras using the Durdević braiding, with applications to quantum spaces.

Contribution

It introduces $\sigma$--generated calculi on principal comodule algebras and proves their existence and properties, extending differential calculus in quantum geometry.

Findings

Constructed $\sigma$--generated calculi for arbitrary principal comodule algebras.

Proved the descent of universal vertical maps and connection 1-forms under compatibility conditions.

Provided explicit examples from quantum projective and lens spaces.

Abstract

We introduce a class of right $H$--covariant first--order differential calculi on principal comodule algebras generated by the Durdevi\'c braiding $\sigma$ and a chosen vertical ideal. Starting from the universal calculus, a strong connection, and a right $H$--colinear splitting map, we construct $\sigma$--generated differential calculi and prove their existence for arbitrary principal comodule algebras. We show that, in this framework, universal vertical maps and connection $1$--forms descend naturally to the quotient calculus under suitable compatibility conditions. We further develop a functorial formulation of $\sigma$--generated calculi and establish a universal factorization property for the associated quotient calculi. Finally, we present explicit examples arising from quantum projective spaces and quantum lens spaces.