Invariants of flat connections on 4-manifolds from Hopf group-algebras

For a given group G, we construct an invariant of flat G-connections on 4-manifolds from a finite type involutory quasitriangular Hopf G-algebra. Hopf G-algebras are generalizations of Hopf algebras, equipped with gradings by G. In our construction, we color the dotted components of a Kirby diagram with elements of G and employ the Hennings-type procedure. When G is finite, we also define an invariant of 4-manifolds by summing the invariants over all flat G-connections.