English

Abstract:
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 H-modules, by twisting the structure using the universal R-matrix. 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.