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Open AccessArticle

Factorization of Graded Traces on Nichols Algebras

Department of Mathematics, University Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany
Faculty of Mathematics and Informatics, Philipps-University Marburg, Hans-Meerwein-Straße, D-35032 Marburg, Germany
Author to whom correspondence should be addressed.
Axioms 2017, 6(4), 32;
Received: 27 October 2017 / Revised: 23 November 2017 / Accepted: 25 November 2017 / Published: 4 December 2017
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang–Baxter Equations 2017)
PDF [506 KB, uploaded 5 December 2017]


A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system and a Poincare-Birkhoff-Witt (PBW) basis basis, but, for nondiagonal examples (e.g., Fomin–Kirillov algebras), this is an ongoing surprise. In this article, we discuss this phenomenon and observe that it continues to hold for the graded character of the involved group and for automorphisms. First, we discuss thoroughly the diagonal case. Then, we prove factorization for a large class of nondiagonal Nichols algebras obtained by the folding construction. We conclude empirically by listing all remaining examples, which were in size accessible to the computer algebra system GAP and find that again all graded characters factorize. View Full-Text
Keywords: Nichols algebra; Hilbert series; graded traces Nichols algebra; Hilbert series; graded traces
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Lentner, S.; Lochmann, A. Factorization of Graded Traces on Nichols Algebras. Axioms 2017, 6, 32.

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