Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Braided categories of bimodules from stated skein TQFTs

Francesco Costantino, Matthieu Faitg

TL;DR

The paper develops a general algebraic framework that internalizes half-braiding data inside a braided category $\mathcal{C}$ via half-braided algebras and hb-compatible bimodules, organizing them into a braided, balanced Morita-type category $\mathrm{Bim}^{\mathrm{hb}}_{\mathcal{C}}$. It then introduces the coend $\mathscr{L}$ to reinterpret hb-algebras as $\mathscr{L}$-linear algebras and hb-bimodules as $\mathscr{L}$-compatible objects, establishing an equivalence with the Drinfeld center $\mathcal{Z}(\mathcal{C})$ and a robust monoidal/braided structure. Specializing to $\mathcal{C}=\mathrm{Comod}-\mathcal{O}$, the theory yields stated skein algebras and bimodules that form a braided, balanced TQFT-like functor from the cobordism category $\mathrm{Cob}$ to $\mathrm{Bim}^{\mathrm{hb}}_{\mathcal{C}}$, with a direct link to Kerler–Lyubashenko’s TQFT in the finite-dimensional factorizable case: stated skeins encode endomorphisms of KL state spaces, thereby bridging skein-theoretic constructions with established topological quantum field theory frameworks. This provides a rich algebraic target for topological invariants and factorization-homology interpretations tied to ribbon/factorizable Hopf data and quantum moment maps.

Abstract

For each braided category $\mathcal{C}$ we show that, under mild hypotheses, there is an associated category of "half braided algebras" and their bimodules internal to $\mathcal{C}$ which is not only monoidal but even braided and balanced. We use this in the case where $\mathcal{C}$ is the category of modules over a ribbon Hopf algebra to interpret stated skeins as a TQFT, namely a braided balanced functor from a category of cobordisms to this category of algebras and their bimodules. Although our construction works in full generality, we relate in the special case of finite-dimensional ribbon factorizable Hopf algebras the stated skein functor to the Kerler-Lyubashenko TQFT by interpreting the former as the "endomorphisms" of the latter.

Braided categories of bimodules from stated skein TQFTs

TL;DR

The paper develops a general algebraic framework that internalizes half-braiding data inside a braided category via half-braided algebras and hb-compatible bimodules, organizing them into a braided, balanced Morita-type category . It then introduces the coend to reinterpret hb-algebras as -linear algebras and hb-bimodules as -compatible objects, establishing an equivalence with the Drinfeld center and a robust monoidal/braided structure. Specializing to , the theory yields stated skein algebras and bimodules that form a braided, balanced TQFT-like functor from the cobordism category to , with a direct link to Kerler–Lyubashenko’s TQFT in the finite-dimensional factorizable case: stated skeins encode endomorphisms of KL state spaces, thereby bridging skein-theoretic constructions with established topological quantum field theory frameworks. This provides a rich algebraic target for topological invariants and factorization-homology interpretations tied to ribbon/factorizable Hopf data and quantum moment maps.

Abstract

For each braided category we show that, under mild hypotheses, there is an associated category of "half braided algebras" and their bimodules internal to which is not only monoidal but even braided and balanced. We use this in the case where is the category of modules over a ribbon Hopf algebra to interpret stated skeins as a TQFT, namely a braided balanced functor from a category of cobordisms to this category of algebras and their bimodules. Although our construction works in full generality, we relate in the special case of finite-dimensional ribbon factorizable Hopf algebras the stated skein functor to the Kerler-Lyubashenko TQFT by interpreting the former as the "endomorphisms" of the latter.

Paper Structure

This paper contains 32 sections, 52 theorems, 190 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Drinfeld center of $\mathcal{C}$
  5. Algebras and (bi)modules
  6. The Morita category $\mathrm{Bim}_{\mathcal{C}}$
  7. Twisting of bimodules by algebra morphisms
  8. The Morita category of half-braided algebras
  9. Half-braided algebras
  10. (Bi)modules over half-braided algebras
  11. The monoidal category of hb-compatible bimodules
  12. The braiding on $\mathrm{Bim}^{\mathrm{hb}}_{\mathcal{C}}$.
  13. A braided functor $\overline{\mathcal{C}} \to \mathrm{Bim}_{\mathcal{C}}^{\mathrm{hb}}$
  14. The $\mathscr{L}$-linear Morita category
  15. The coend $\mathscr{L}$
  16. ...and 17 more sections

Key Result

Theorem 1.1

If $\mathcal{U}$ is finite dimensional and factorizable there exists a ribbon monoidal functor $KL: \mathrm{Cob}^{\sigma}\to \mathcal{U}\text{-}\mathrm{mod}$ sending the once punctured torus to the adjoint representation of $\mathcal{U}$.

Figures (8)

  • Figure 1: On the left we schematize the tensor product of two marked surfaces or marked manifolds $M_1$ and $M_2$ (the two points visible in the boundary are the boundaries of the single marking $\mathcal{N}$ going from the marked point of $S^-$ to the one of $S^+$, assuming that we look the cobordism from the top and that $\mathcal{N}$ is slightly bent). On the right we exhibit the braiding $c_{S_1,S_2}$; the marking of the two surfaces is running vertical on the dotted cylinders and that of the cobordism is depicted in front.
  • Figure 2: On the left we represent a neighbourhood of the side boundary of a $3$-manifold $M$, and make a $+1$ surgery along the knot parallel to the core of the side boundary. This gives a manifold diffeomorphic to the one depicted at its right. Similarly the last two cobordisms are diffeomorphic. When the manifold $M$ is $S\times [-1,1]$ then the cobordism depicted on the left is the balancing $\tau_S$, the one on the right is $\tau_S^{-1}$.
  • Figure 3: Graphical proof of the fact that $\tau$ is indeed a balancing: on the left the manifold contains a $+1$-surgery knot around the core of the side boundary. On the right a slam-dunk move is applied; equivalently the whole cylinders representing $S_1$ and $S_2$ are slid over the knot and so that the knot becomes contained in a ball and can be removed. During the sliding $S_1$ and $S_2$ acquire a full twist encoded by the two unknots with framing $+1$ in the bottom of the right figure.
  • Figure 4: The basic skein relation: in the drawing we picked a ribbon graph represented by a single coupon; in general one can push off the surface any graph $G$ and use $f=\mathrm{RT}(G)$ to compute the state of the resulting graph. Here the strands are decorated by objects of $\mathcal{C}^{fin}$ and oriented (not indicated on the picture). The horizontal oriented segment represents the marking $\mathcal{N}$ of $M$. We use the convention that marking each boundary point by a state $s_i$ (as on the left-hand side) is the same thing as globally marking the boundary points by one state $s_1 \otimes \ldots \otimes s_k$ (as on the right-hand side); this can be formalized by the use of identity coupons.
  • Figure 5: The $\mathscr{L}$-linear structure $\mathfrak{d} : \mathscr{L} \to \mathcal{S}_{\mathcal{O}}(S)$ on stated skein algebras (as proved in Prop. \ref{['prop:skeinLlinear']}). The two points visible in the boundary are the extremities of the single marking $\mathcal{N}$ ending to the marked points of $S \times \{-1\}$ and $S \times \{1\}$. We slightly bend $\mathcal{N}$ and project on $S$ in order to be able to draw diagrams.
  • ...and 3 more figures

Theorems & Definitions (126)

  • Theorem 1.1: KL, following BD
  • Theorem 1.2: see Thm. \ref{['thBraided']} and \ref{['thmBalanceBim']}
  • Theorem 1.3: Thm. \ref{['teo:monoidalfunctor']}
  • Theorem 1.4: See Thm. \ref{['teo:commutativediagram']} for a precise statement
  • Definition 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Definition 3.1
  • ...and 116 more