Finite Weyl groupoids of rank three

Finite Weyl groupoids of rank three

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 sy...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society Vol. 364; no. 3; pp. 1369 - 1393
Main Authors: CUNTZ, M., HECKENBERGER, I.
Format: Journal Article
Language:English
Published: American Mathematical Society 01-03-2012
Subjects:
Algebra
Algorithms
Hyperplanes
Mathematical theorems
Morphisms
Normal vectors
Polygons
Root systems
Vertices
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-2011-05368-7