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Non-factorizable ribbon Hopf Algebras

Quentin Faes, Maksymilian Manko

Abstract

Building on the work of Nenciu we provide examples of non-factorizable ribbon Hopf algebras, and introduce a stronger notion of non-factorizability. These algebras are designed to provide invariants of $4$-dimensional $2$-handlebodies up to 2-deformations. We prove that some of the invariants derived from these examples are invariants dependent only on the boundary or on the presentation of the fundamental group of the 2-handlebody.

Non-factorizable ribbon Hopf Algebras

Abstract

Building on the work of Nenciu we provide examples of non-factorizable ribbon Hopf algebras, and introduce a stronger notion of non-factorizability. These algebras are designed to provide invariants of -dimensional -handlebodies up to 2-deformations. We prove that some of the invariants derived from these examples are invariants dependent only on the boundary or on the presentation of the fundamental group of the 2-handlebody.

Paper Structure

This paper contains 32 sections, 47 theorems, 253 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the paper
  4. Preliminaries
  5. Notations for Hopf algebras
  6. Integrals and unimodularity
  7. Quasitriangular and ribbon structures
  8. Algebra generators and tuple notation
  9. The Nenciu construction
  10. Hopf algebra structure
  11. Examples
  12. Unimodular structure
  13. Quasitriangular structure
  14. Ribbon structure
  15. Examples revisited
  16. ...and 17 more sections

Key Result

Theorem 2.9

Let $H$ be a Hopf algebra. Any element $R \in H \otimes H$ satisfying (QT1)-(QT4) induces a Hopf algebra map where $H^{cop}$ is the coopposite Hopf algebra of $H$, defined above.

Figures (1)

  • Figure 1: A Kirby diagram with $k_1$ 1-handles isotoped to the bottom, $k_2$ (possibly knotted) 2-handles and $m$ crossings, and the image under the bead algorithm, where each $2$-handle is linked to at least one $1$-handle. The beads have been colored arbitrarily and not all labeled for a better readability.

Theorems & Definitions (132)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: radford_2012 Definition 10.2.3
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9: radford_1994, Section 2.1
  • Proposition 2.10
  • ...and 122 more