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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Finite Weyl groupoids of rank three


Authors: M. Cuntz and I. Heckenberger
Journal: Trans. Amer. Math. Soc. 364 (2012), 1369-1393
MSC (2010): Primary 20F55, 16T30, 52C30
Published electronically: September 2, 2011
MathSciNet review: 2869179
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Abstract: We continue our study of Cartan schemes and their Weyl groupoids and obtain a complete list of all connected simply connected Cartan schemes of rank three for which the real roots form a finite irreducible root system. We achieve this result by providing an algorithm which determines all the root systems and eventually terminates: Up to equivalence there are exactly 55 such Cartan schemes, and the number of corresponding real roots varies between $ 6$ and $ 37$. We identify those Weyl groupoids which appear in the classification of Nichols algebras of diagonal type.


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Additional Information

M. Cuntz
Affiliation: Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, D-67653 Kaiserslautern, Germany
Email: cuntz@mathematik.uni-kl.de

I. Heckenberger
Affiliation: Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans- Meerwein-Straße, D-35032 Marburg, Germany
Email: heckenberger@mathematik.uni-marburg.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05368-7
PII: S 0002-9947(2011)05368-7
Received by editor(s): December 4, 2009
Received by editor(s) in revised form: March 22, 2010, March 25, 2010, and April 7, 2010
Published electronically: September 2, 2011
Additional Notes: The second author was supported by the German Research Foundation (DFG) via a Heisenberg fellowship
Article copyright: © Copyright 2011 American Mathematical Society