The Heisenberg double and the pentagon relation

It is shown that the Heisenberg double of an arbitrary Hopf algebra has a canonical element satisfying the pentagon relation. The structure of the underlying algebras can be recovered by a given invertible constant solution of the pentagon relation. The Drinfeld double is representable as a subalgebra in the tensor square of the Heisenberg double. This offers a possibility of expressing solutions of the Yang–Baxter relation in terms of solutions of the pentagon relation.