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Poisson Lie symmetry and D-branes in WZW model on the Heisenberg Lie group $H_4$

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.