Some Constructions on Quantum Principal Bundles

TL;DR

This paper establishes foundational results in quantum principal bundle theory, including differential calculus structures, examples of non-generating base forms, and isomorphism characterizations, contributing to the mathematical understanding of quantum gauge theories.

Contribution

It proves four key statements about quantum principal bundles, clarifying differential calculus, examples, and isomorphisms, advancing the mathematical framework of quantum geometry.

Findings

Universal differential envelope equals algebra of forms for classical bicovariant calculus

Existence of quantum principal bundle with non-generated base forms

Isomorphism between convolution-invertible maps and covariant module isomorphisms

Abstract

This paper works as an appendix of the paper titled Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group and for paper titled Yang-Mills-Connes Theory and Quantum Principal SU(N)-Bundles. Here, we are going to prove four statements in the theory of quantum principal bundles:: 1) The universal differential envelope $\ast$--calculus of a matrix (compact) Lie group, for the classical bicovariant $\ast$--First Order Differential Calculus, is the algebra of differential forms. 2) An example of a quantum principal bundle in which the space of base forms is not generated by the base space. 3) The group isomorphism between convolution-invertible maps and covariant left module isomorphisms at the level of differential calculus 4) The way the maps $\{T^V_k \}$ from Remark 3.1 look in differential geometry.