Differential calculus on Hopf-Galois extension via the Durdevic braiding

TL;DR

The paper develops a framework for first-order differential calculi on Hopf-Galois extensions that respect the Đurđević braiding by introducing $\sigma$-generated calculi. It constructs such calculi from a universal calculus, a strong connection, and a vertical ideal, proving existence for arbitrary principal comodule algebras and demonstrating the descent of connection 1-forms and vertical maps under braiding. It further clarifies obstructions to $\sigma$-generation, showing that the defining relations must be vertical-data–driven and that the class is broader than bicovariant, complete calculi. An explicit Podleś sphere example illustrates the construction and notes how principality may be established or obstructed in higher rank cases. Overall, the approach provides a minimal, braided-first-order counterpart to Đurđević’s complete calculus, applicable in broad noncommutative settings and offering explicit descent properties for gauge-theoretic data.

Abstract

We introduce a class of first-order differential calculus on principal comodule algebras generated by the Durdevic braiding $σ$ and a chosen vertical ideal. Starting from the universal calculus and a strong connection, we construct $σ$-generated calculus and prove their existence for arbitrary principal comodule algebras. We show that, in this setting, connection $1$-forms and vertical maps descend to the quotient calculus and are compatible with the induced braided symmetry. We also compare this framework with Durdevic's complete differential calculus.

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5: durdevic1996quantum2, Proposition 2.1
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof