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Projective Versions of Spatial Partition Quantum Groups

Nicolas Faroß

Abstract

We generalize categories of spatial partitions in the sense of Cébron-Weber by introducing new base partitions. This allows us to construct additional examples of free orthogonal quantum groups but yields the same class of spatial partition quantum groups as before. Further, we use these new base partitions to show that the class of spatial partition quantum groups is closed under taking projective versions and in particular contains the projective version of all easy quantum groups. As an application, we determine the quantum groups corresponding to the categories of all spatial pair partitions and give explicit descriptions of the projective versions of easy quantum groups in terms of spatial partitions.

Projective Versions of Spatial Partition Quantum Groups

Abstract

We generalize categories of spatial partitions in the sense of Cébron-Weber by introducing new base partitions. This allows us to construct additional examples of free orthogonal quantum groups but yields the same class of spatial partition quantum groups as before. Further, we use these new base partitions to show that the class of spatial partition quantum groups is closed under taking projective versions and in particular contains the projective version of all easy quantum groups. As an application, we determine the quantum groups corresponding to the categories of all spatial pair partitions and give explicit descriptions of the projective versions of easy quantum groups in terms of spatial partitions.

Paper Structure

This paper contains 17 sections, 23 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compact matrix quantum groups
  4. Representation categories and rigidity
  5. Woronowicz Tannaka-Krein duality
  6. Combinatorics of spatial partitions
  7. Spatial partitions
  8. Categories of spatial partitions
  9. Spatial partition functors
  10. Permuting levels
  11. Flattening partitions
  12. From spatial partitions to quantum groups
  13. Spatial partition quantum groups
  14. Presentations of spatial partitions quantum groups
  15. Permuting levels
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $G$ be a spatial partition quantum group defined by any pair of spatial base partitions satisfying the conjugate equations. Then $G$ is equivalent to a spatial partition quantum group in the sense of Cébron-Weber defined by the base partitions ${ \pgfpicture \pgfmathmin{0}{} \pgfmathmax{0}{} \pg

Theorems & Definitions (73)

  • Theorem 1.1: \ref{['corr:assume-base-partitions']}
  • Theorem 1.2: \ref{['corr:proj-spatial-closed']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 63 more