The braided groups and braided matrices B(R) for the solution R of the Yang–Baxter equation associated to the quantum Heisenberg group are computed. It is also shown that a particular extension of the quantum Heisenberg group is dual to the Heisenberg universal enveloping algebra Uq(h), and this result is used to derive an action of Uq(h) on the braided groups. The various covariance properties are then demonstrated using the braided Heisenberg group as an explicit example. In addition, the braided Heisenberg group is found to be self-dual. Finally, a physical application to a system of n braided harmonic oscillators is discussed. An isomorphism is found between the n-fold braided and unbraided tensor products, and the usual ‘‘free’’ time evolution is shown to be equivalent to an action of a primitive generator of Uq(h) on the braided tensor product.