Let $mathcal{B}_{mathfrak{q}}$ be a finite-dimensional Nichols algebra ofdiagonal type corresponding to a matrix $mathfrak{q} in mathbf{k}^{hetaimes heta}$, where $mathbf{k}$ is an algebraically closed field ofcharacteristic 0. Let $mathcal{L}_{mathfrak{q}}$ be the Lusztig algebraassociated to $mathcal{B}_{mathfrak{q}}$, seehttp://arxiv.org/abs/1501.04518. We present $mathcal{L}_{mathfrak{q}}$ as anextension (as braided Hopf algebras) of $mathcal{B}_{mathfrak{q}}$ by$mathfrak Z_{mathfrak{q}}$ where $mathfrak Z_{mathfrak{q}}$ is isomorphicto the universal enveloping algebra of a Lie algebra $mathfrakn_{mathfrak{q}}$. We compute the Lie algebra $mathfrak n_{mathfrak{q}}$ when$heta = 2$.
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