Lie Algebras Arising from Nichols Algebras of Diagonal Type

Abstract Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type with braiding matrix $\mathfrak{q}$, $\mathcal{L}_{\mathfrak{q}}$ the corresponding Lusztig as in [ 4], and $\operatorname{Fr}_{\mathfrak{q}}: \mathcal{L}_{\mathfrak{q}} \to U(\mathfrak{n}^{\mathfrak{q}})$ quantum Frobenius map 5]. We prove that Lie $\mathfrak{n}^{\mathfrak{q}}$ is either 0 or positive part semisimple $\mathfrak{g}^{\mathfrak{q}}$, which determined for each $\mathfrak{q}$ list 25].