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Some Constructions on Quantum Principal Bundles

Gustavo Amilcar Saldaña Moncada

TL;DR

The paper develops a framework for universal differential calculus on quantum groups and quantum principal bundles, showing that the universal envelope $\\Gamma^{\\wedge}$ forms a graded differential $*$-Hopf algebra compatible with the coactions and adjoint actions, and specializing to classical compact matrix Lie groups recovers the exterior algebra $H\\otimes\\bigwedge\\mathfrak{g}^{\\#}_{\\mathbb{C}}$. It proves a key group isomorphism between convolution-invertible maps and graded left $\\Omega^{\\bullet}(B)$-module automorphisms of $\\Omega^{\\bullet}(P)$, giving constructive correspondences via $\\mathfrak{f}_{\\mathfrak{F}}$ and $\\mathfrak{F}_{\\mathfrak{f}}$, and shows how a finite set of maps $\\{T^{L}_k\\}$ can be obtained from corepresentations to realize translations from the total space to the base in the quantum setting. The work provides a concrete counterexample where the base forms are not generated by the base space, illustrating subtleties in the base calculus, and culminates with a Dunkl-operator-based approach to quantum principal bundles, demonstrating how Dunkl connections yield real, multiplicative covariant derivatives and extend classical gauge transformations to the quantum realm. Overall, the results advance the understanding of differential structures on quantum principal bundles and their links to representation theory and Dunkl operators, with implications for noncommutative geometry and quantum gauge theory.

Abstract

This paper works as an appendix of the paper titled Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group. 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.

Some Constructions on Quantum Principal Bundles

TL;DR

The paper develops a framework for universal differential calculus on quantum groups and quantum principal bundles, showing that the universal envelope forms a graded differential -Hopf algebra compatible with the coactions and adjoint actions, and specializing to classical compact matrix Lie groups recovers the exterior algebra . It proves a key group isomorphism between convolution-invertible maps and graded left -module automorphisms of , giving constructive correspondences via and , and shows how a finite set of maps can be obtained from corepresentations to realize translations from the total space to the base in the quantum setting. The work provides a concrete counterexample where the base forms are not generated by the base space, illustrating subtleties in the base calculus, and culminates with a Dunkl-operator-based approach to quantum principal bundles, demonstrating how Dunkl connections yield real, multiplicative covariant derivatives and extend classical gauge transformations to the quantum realm. Overall, the results advance the understanding of differential structures on quantum principal bundles and their links to representation theory and Dunkl operators, with implications for noncommutative geometry and quantum gauge theory.

Abstract

This paper works as an appendix of the paper titled Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group. Here, we are going to prove four statements in the theory of quantum principal bundles:: 1) The universal differential envelope --calculus of a matrix (compact) Lie group, for the classical bicovariant --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 from Remark 3.1 look in differential geometry.

Paper Structure

This paper contains 5 sections, 5 theorems, 155 equations.

Table of Contents

  1. Universal Differential Envelope $\ast$--calculus in Differential Geometry
  2. An example about the space of base forms.
  3. The Group Isomorphism
  4. The form of the maps $\{T^\mathrm{L}_k \}$
  5. Quantum Principal Bundles and Dunkl Operators

Key Result

Theorem 1.1

Let $\mathcal{T}$ be a complete set of mutually non--equivalent irreducible unitary (necessarily finite--dimensional) $\mathcal{G}$--corepresentations with $\delta^\mathbb{C}_\mathrm{triv}$$\in$$\mathcal{T}$ (the trivial corepresentation on $\mathbb{C}$). For any $\delta^V$$\in$$\mathcal{T}$ that co where $\{ e_i\}^{n_{V}}_{i=1}$ is an ortonormal basis of $V$ and $\{g^{V}_{ij}\}^{n_{V}}_{i,j=1}$$\

Theorems & Definitions (10)

  • Theorem 1.1
  • Example 1.2
  • Example 2.1
  • Remark 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • proof