A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems

By I. Angiono. 2010

89 citations recorded by the source corpus.

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 in Kharchenko's theory of PBW basis of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for such PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f3a3a625e47ddc4a4ec12c72e905839892fee409

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Citations in the corpus

Hopf Algebras and Root Systems (2020, methodology): “Angiono in [ Ang15 ] (using the results on right coideal subalgebras in Corol-lary 14.6.8) and [ Ang13 ] found a celebrated presentation of the Nichols algebras appearing in [ Hec09 ] in terms of generators and relations, where the ground ﬁeld is algebraically closed of characteristic 0.”
Nichols algebras of diagonal type (2011, methodology): “ Next we recall the deﬁnitions of Weyl groupoid, the associated root systems and some properties thereof [HS, HY]. We close this Section stating a general presentation of Nichols algebras coming from [A2]. 1.1. Lyndon words and PBW bases for Nichols algebras of diagonal type. To begin with, we recal”