Lie Algebras Arising from Nichols Algebras of Diagonal Type

Abstract

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