Invariants of 4-manifolds from Hopf algebras

The Kuperberg invariant is a quantum invariant of 3-manifolds based on any finite dimensional Hopf algebras, which is also related to the Turaev-Viro-Barrett-Westbury invariant. We give a construction of an invariant of 4-manifolds that can be viewed as a 4-dimensional analog of the Kuperberg invariant. The algebraic data for the construction is what we call a Hopf triple which consists of three Hopf algebras and a bilinear form on each pair of the Hopf algebras satisfying certain compatibility conditions. Quasi-triangular Hopf algebras are special examples of Hopf triples. Trisection diagrams of 4-manifolds proposed by Gay and Kirby will be used in the construction of the invariant. Some interesting properties of the invariant will also be discussed.