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

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

2011-05-02

arxiv.org/abs/1105.0434v2

Mathematics

Quantum Algebra

31 pages; TikZ figures

Scientific paper

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.

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