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Lie algebras arising from Nichols algebras of diagonal type (1911.06586v4)
Published 15 Nov 2019 in math.QA and math.RA
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 $\operatorname{Fr}{\mathfrak{q}}: \mathcal{L}{\mathfrak{q}} \to U(\mathfrak{n}{\mathfrak{q}})$ 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.