Factorization of Graded Traces on Nichols Algebras

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-...

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Main Authors: Simon Lentner, Andreas Lochmann
Format: Article
Language:English
Published: MDPI AG 2017-12-01
Series:Axioms
Subjects:
Online Access:https://www.mdpi.com/2075-1680/6/4/32
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author Simon Lentner
Andreas Lochmann
author_facet Simon Lentner
Andreas Lochmann
author_sort Simon Lentner
collection DOAJ
description 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.
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spelling doaj.art-2e5f3f43b3c3418f9923c68a6e3b6d1c2022-12-22T03:37:08ZengMDPI AGAxioms2075-16802017-12-01643210.3390/axioms6040032axioms6040032Factorization of Graded Traces on Nichols AlgebrasSimon Lentner0Andreas Lochmann1Department of Mathematics, University Hamburg, Bundesstraße 55, D-20146 Hamburg, GermanyFaculty of Mathematics and Informatics, Philipps-University Marburg, Hans-Meerwein-Straße, D-35032 Marburg, GermanyA 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.https://www.mdpi.com/2075-1680/6/4/32Nichols algebraHilbert seriesgraded traces
spellingShingle Simon Lentner
Andreas Lochmann
Factorization of Graded Traces on Nichols Algebras
Axioms
Nichols algebra
Hilbert series
graded traces
title Factorization of Graded Traces on Nichols Algebras
title_full Factorization of Graded Traces on Nichols Algebras
title_fullStr Factorization of Graded Traces on Nichols Algebras
title_full_unstemmed Factorization of Graded Traces on Nichols Algebras
title_short Factorization of Graded Traces on Nichols Algebras
title_sort factorization of graded traces on nichols algebras
topic Nichols algebra
Hilbert series
graded traces
url https://www.mdpi.com/2075-1680/6/4/32
work_keys_str_mv AT simonlentner factorizationofgradedtracesonnicholsalgebras
AT andreaslochmann factorizationofgradedtracesonnicholsalgebras