
The Hopf algebra object in the (braided monoidal) category Cob of
cobordisms of surfaces with boundary parametrized by a circle was
discovered by Crane and Yetter and by Kerler independently.
Transmutation, introduced by Majid, is the process of transforming a
quasitriangular Hopf algebra H into a Hopf algebra object in the
braided monoidal category of left Hmodules, by twisting the
structure using the universal Rmatrix. For a finite dimensional
ribbon Hopf algebra H, we construct the representation of Cob
(hence a $3$manifold invariant), which seems closely related to the
Hennings invariant. For more general ribbon Hopf algebra H, we
construct a representation of a subcategory of Cob, and explain the
relationships between this representation and a natural generalization
of Lawrence's universal link invariant.
