Let $\mathcal{ B}_\mathfrak{ q}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{ q}$. We consider the graded dual $\mathcal{ L}_{ \mathfrak{ q}}$ of the distinguished pre-Nichols algebra $\widetilde{ \mathcal{ B}}_{ \mathfrak{ q}}$ from [A3] and the divided powers algebra $\mathcal{ U}_{ \mathfrak{ q}}$, a suitable Drinfeld double of $\mathcal{ L}_{ \mathfrak{ q}} # \mathbf{ k} \mathbb{ Z}^{ \theta}$. We provide basis and presentations by generators and relations of $\mathcal{ L}_{ \mathfrak{ q}}$ and $\mathcal{ U}_{ \mathfrak{ q}}$, and prove that they are noetherian and have finite Gelfand-Kirillov dimension.
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