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Reduction of Quantum Principal Bundles over non affine bases

Rita Fioresi, Emanuele Latini, Chiara Pagani

Abstract

In this paper we develop the theory of reduction of quantum principal bundles over projective bases. We show how the sheaf theoretic approach can be effectively applied to certain relevant examples as the Klein model for the projective spaces; in particular we study in the algebraic setting the reduction of the principal bundle $\mathrm{GL}(n) \to \mathrm{GL}(n)/P= \mathbf{P}^{n-1}(\mathbb{C})$ to the Levi subgroup $G_0$ inside the maximal parabolic subgroup $P$ of $\mathrm{GL}(n)$. We characterize reductions in the sheaf theoretic setting.

Reduction of Quantum Principal Bundles over non affine bases

Abstract

In this paper we develop the theory of reduction of quantum principal bundles over projective bases. We show how the sheaf theoretic approach can be effectively applied to certain relevant examples as the Klein model for the projective spaces; in particular we study in the algebraic setting the reduction of the principal bundle to the Levi subgroup inside the maximal parabolic subgroup of . We characterize reductions in the sheaf theoretic setting.
Paper Structure (18 sections, 19 theorems, 127 equations)

This paper contains 18 sections, 19 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. The classical setting
  3. Example of a reduction
  4. Quantum Principal Bundles and Reductions
  5. Hopf Galois Extensions.
  6. Quantum reductions.
  7. Examples of Reductions
  8. Multiparametric deformations of $\mathrm{GL}(n)$.
  9. The Takhtajan-Sudbery algebra.
  10. The Quantum ringed space $(\mathbf{P}^{n-1}, \mathcal{O}_{\mathbf{P}^n})$.
  11. The Quantum Principal bundle $\mathcal{F}$.
  12. Quantum reduction of $\mathcal{F}$
  13. The $n=2$ case: quantum reduction from $\widetilde{\mathcal{O}}_q(\mathrm{GL}(2))$.
  14. Quantum sheaf algebraic Reductions
  15. Quantum affine algebraic Reductions
  16. ...and 3 more sections

Key Result

Proposition 2.3

A principal $P$--bundle $\xi=(E, \pi, M)$ is reducible to a principal $K$--bundle $\xi_0=(E_0, \pi_0, M)$ if and only if the bundle $\xi_K := (E \setminus K ,\pi_K, M)$, with ${\pi}_K$ being the projection induced by $\pi$ on $E \setminus K$, admits a global section.

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Definition 3.5
  • Remark 3.6
  • Example 4.1
  • ...and 35 more