  We show that the WZW model on the Heisenberg Lie group $H_4$ has Poisson-Lie
symmetry only when the dual Lie group is ${ A}_2 \oplus 2{ A}_1$. In this way,
we construct the mutual T-dual sigma models on Drinfel'd double generated by
the Heisenberg Lie group $H_4$ and its dual pair, ${ A}_2 \oplus 2{ A}_1$, as
the target space in such a way that the original model is the same as the $H_4$
WZW model. Furthermore, we show that the dual model is conformal up to two
loops order. Finally, we discuss $D$-branes and the worldsheet boundary
conditions defined by a gluing matrix on the $H_4$ WZW model. Using the duality
map obtained from the canonical transformation description of the Poisson-Lie
T-duality transformations for the gluing matrix which locally defines the
properties of the $D$-brane, we find two different cases of the gluing matrices
for the WZW model based on the Heisenberg Lie group $H_4$ and its dual model.
