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-...
Main Authors: | , |
---|---|
Format: | Article |
Language: | English |
Published: |
MDPI AG
2017-12-01
|
Series: | Axioms |
Subjects: | |
Online Access: | https://www.mdpi.com/2075-1680/6/4/32 |
_version_ | 1811229790682742784 |
---|---|
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. |
first_indexed | 2024-04-12T10:18:57Z |
format | Article |
id | doaj.art-2e5f3f43b3c3418f9923c68a6e3b6d1c |
institution | Directory Open Access Journal |
issn | 2075-1680 |
language | English |
last_indexed | 2024-04-12T10:18:57Z |
publishDate | 2017-12-01 |
publisher | MDPI AG |
record_format | Article |
series | Axioms |
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 |