  We consider the three-dimensional associative algebra H consisting of the 3x3
strictly upper triangular matrices whose the commutator is the Heisenberg Lie
algebra. We determine the solutions of the Yang-Baxter associative equation in
H. For the antisymmetric solutions, the corresponding bialgebraic structures,
double constructions of Frobenius algebras and properties are given
explicitly.Besides, we determine some related compatible dendriform algebras
and solutions of their D-equations. Using symmetric solutions of these
equations, we build the double constructions of related Connes cocycles.
Finally, we compute solutions of the three-dimensional non decomposable
associative Yang-Baxter equation and build the double constructions of
associated Frobenius algebras.
