  The Kuperberg invariant is a topological invariant of closed 3-manifolds
based on finite-dimensional Hopf algebras. In this paper, we initiate the
program of constructing 4-manifold invariants in the spirit of Kuperberg's
3-manifold invariant. We utilize a structure called a Hopf triplet, which
consists of three Hopf algebras and a bilinear form on each pair subject to
certain compatibility conditions. In our construction, we present 4-manifolds
by their trisection diagrams, a four-dimensional analog of Heegaard diagrams.
The main result is that every Hopf triplet yields a diffeomorphism invariant of
closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter
invariants and generalized dichromatic invariants, and conjecturally Kashaev's
invariant. As a starting point, we assume that the Hopf algebras involved in
the Hopf triplets are semisimple. We speculate that relaxing semisimplicity
will lead to even richer invariants.
