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Graded spherical skein 2+1-G-HQFT and modified Turaev-Viro invariants

Francesco Costantino, Nathan Geer, Benjamin Haïoun, Bertrand Patureau-Mirand

Abstract

For G a group, we present a G-graded version of chromatic maps and skein modules and use them to define a 2+1-G-HQFT out of a G-chromatic category. The construction applies to the representations of unrestricted quantum groups at root of unity and recovers the modified Turaev-Viro 3-dimensional invariants.

Graded spherical skein 2+1-G-HQFT and modified Turaev-Viro invariants

Abstract

For G a group, we present a G-graded version of chromatic maps and skein modules and use them to define a 2+1-G-HQFT out of a G-chromatic category. The construction applies to the representations of unrestricted quantum groups at root of unity and recovers the modified Turaev-Viro 3-dimensional invariants.
Paper Structure (32 sections, 42 theorems, 99 equations, 6 figures)

This paper contains 32 sections, 42 theorems, 99 equations, 6 figures.

Table of Contents

  1. Preliminaries
  2. Pivotal categories
  3. ${\mathbb K}$--categories
  4. $\mathsf{G}$-graded pivotal ${\mathbb K}$-category
  5. M-traces on ideals in pivotal categories
  6. Invariants of colored ribbon graphs
  7. $\mathsf{G}$-decorated cobordisms
  8. $\mathsf{G}$-representations
  9. The category of $\mathsf{G}$-decorated cobordisms
  10. (Graded) admissible skein modules
  11. Definition of admissible skein modules
  12. Some properties of admissible skein modules
  13. Definition of graded admissible skein modules
  14. $\mathsf{G}$-representations as a skein module
  15. Finite dimensionality
  16. ...and 17 more sections

Key Result

Theorem 1

For any equivalence between $\widetilde{\Sigma}=(\Sigma,\rho,Y)$ and $\widetilde{\Sigma}'=(\Sigma,\rho',Y')$ (as in Definition def:equivalence) there exists a canonical isomorphism between ${\matheusm{S}}(\widetilde{\Sigma})$ and ${\matheusm{S}}(\widetilde{\Sigma}')$.

Figures (6)

  • Figure 1: The morphisms $e_{V,P}$ and $e'_{V,P}$.
  • Figure 2: Exemple of intersection of a path $\gamma$ with a skein ${T}$.
  • Figure 3: Locally creating an edge colored by $V$.
  • Figure 4: Chromatic modification
  • Figure 5: Reversing the orientation of a red circle
  • ...and 1 more figures

Theorems & Definitions (108)

  • Theorem : Partial restatement of Theorem \ref{['P:SkeinRepCobe']}
  • Theorem : Proposition \ref{['chrom-cat1']}
  • Theorem : Theorem \ref{['T:main']}
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Definition 2.1
  • ...and 98 more