We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based on Kharchenko’s theory of PBW bases of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.

Let $𝔤$ be a simple Lie algebra and ${\mathrm{𝔄𝔟}}^o$ the poset of non-trivial abelian ideals of a fixed Borel subalgebra of $𝔤$. In [8], we constructed a partition ${\mathrm{𝔄𝔟}}^o={\bigsqcup }_{\mu }{\mathrm{𝔄𝔟}}_{\mu }$ parameterised by the long positive roots of $𝔤$ and studied the subposets ${\mathrm{𝔄𝔟}}_{\mu }$. In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation and to the centralisers of abelian ideals. We also prove that the poset of positive roots of $𝔤$ is a join-semilattice.