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 $\mathfrak{q}$, let ${\mathcal{L}}_{\mathfrak{q}}$ be the corresponding Lusztig algebra as in arXiv:1501.04518 and let ${\mathrm{Fr}}_{\mathfrak{q}}:{\mathcal{L}}_{\mathfrak{q}}\to U({\mathfrak{n}}^{\mathfrak{q}})$ be the corresponding quantum Frobenius map as in arXiv:1603.09387. We prove that the finite-dimensional Lie algebra ${\mathfrak{n}}^{\mathfrak{q}}$ is either 0 or else the positive part of a semisimple Lie algebra ${\mathfrak{g}}^{\mathfrak{q}}$ which is determined for each $\mathfrak{q}$ in the list of arXiv:math/0605795.

arXiv commentv2: some references were added. v3: 27 pages, we added an appendix with explicit computations in type B2. Accepted in IMRN