Weak Hopf Algebras unify the Hennings-Kauffman-Radford and the Reshetikhin-Turaev invariant

TL;DR

This paper introduces a new 3-manifold invariant based on coribbon Weak Hopf Algebras, unifying and generalizing the Kauffman-Radford and Reshetikhin-Turaev invariants through a simplified, category-independent approach.

Contribution

It extends the Hennings-Kauffman-Radford evaluation to Weak Hopf Algebras, unifies it with Reshetikhin-Turaev invariants, and simplifies their proofs without relying on negligible morphisms.

Findings

Invariant reduces to known invariants for special cases

Invariant is non-zero for modular categories

Construction does not require quotienting by negligible morphisms

Abstract

We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. This quantum trace satisfies conditions that render the link evaluation invariant under Kirby moves. If H is a suitable finite-dimensional Hopf algebra (not weak), our invariant reduces to the Kauffman-Radford invariant for the dual of H. If H is the Weak Hopf Algebra Tannaka-Krein reconstructed from a modular category C, our invariant agrees with the Reshetikhin-Turaev invariant. In particular, the proof of invariance of the Reshetikhin-Turaev invariant becomes as simple as that of the Kauffman-Radford invariant. Modularity of C is only used once in order to show that the invariant is non-zero; apart from this, a fusion category with ribbon structure would be sufficient. Our generalization of the Kauffman-Radford invariant for a Weak Hopf Algebra H and the Reshetikhin-Turaev invariant for its category of finite-dimensional comodules C=M^H always agree by construction. There is no need to consider a quotient of the representation category modulo 'negligible morphisms' at any stage, and our construction contains the Reshetikhin-Turaev invariant for an arbitrary modular category C, whether its relationship with some quantum group is known or not.