Braided categories of bimodules from stated skein TQFTs

TL;DR

This paper constructs a braided and balanced category of bimodules from stated skein TQFTs, linking algebraic structures with topological quantum field theories and extending known TQFT frameworks.

Contribution

It introduces a new categorical framework of half braided algebras and bimodules within braided categories, connecting skein TQFTs to existing TQFT models for ribbon Hopf algebras.

Findings

Constructs a monoidal, braided, and balanced category of bimodules from braided categories.

Interprets stated skeins as a TQFT functor into categories of algebras and bimodules.

Relates the construction to the Kerler-Lyubashenko TQFT in the finite-dimensional case.

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.