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We present an invariant of connected and oriented closed 3-manifolds based ona coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant canbe understood as the generalization to Weak Hopf Algebras of theHennings-Kauffman-Radford evaluation of an unoriented framed link using a dualquantum-trace. This quantum trace satisfies conditions that render the linkevaluation invariant under Kirby moves. If H is a suitable finite-dimensionalHopf algebra (not weak), our invariant reduces to the Kauffman-Radfordinvariant for the dual of H. If H is the Weak Hopf Algebra Tannaka-Kreinreconstructed from a modular category C, our invariant agrees with theReshetikhin-Turaev invariant. In particular, the proof of invariance of theReshetikhin-Turaev invariant becomes as simple as that of the Kauffman-Radfordinvariant. Modularity of C is only used once in order to show that theinvariant is non-zero; apart from this, a fusion category with ribbon structurewould be sufficient. Our generalization of the Kauffman-Radford invariant for aWeak Hopf Algebra H and the Reshetikhin-Turaev invariant for its category offinite-dimensional comodules C=M^H always agree by construction. There is noneed to consider a quotient of the representation category modulo 'negligiblemorphisms' at any stage, and our construction contains the Reshetikhin-Turaevinvariant for an arbitrary modular category C, whether its relationship withsome quantum group is known or not.

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