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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 $\\mathcal{L}_{\\mathfrak{q}} # \\mathbf{k} \\mathbb{Z}^{\\theta}$. 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