Weyl groupoids with at most three objects

Abstract

a b s t r a c tWe adapt the generalization of root systems by the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we completely classify all finite Weyl groupoids with at most three objects. The classification yields the result that there exist infinitely many ''standard'', but only 9 ''exceptional'' cases.

Weyl groupoids of rank two and continued fractions

“…For further details and results we refer to [Heckenberger and Yamane 2008] and [Cuntz and Heckenberger 2008].…”

“…The theory includes and extends the theory of crystallographic Coxeter groups, but contains even such examples which do not seem to be related to Nichols algebras of diagonal type. In this paper we use the language and some structural and classification results achieved in [Cuntz and Heckenberger 2008]; see Section 2 for the most essential definitions and facts.…”

Virasoro Central Charges for Nichols Algebras

“…On the Nichols algebra side, the Weyl groupoid action is defined as follows [10,19,20,23]. There exists a generalized Cartan matrix (a i, j ) 1≤i, j≤θ such that a i,i = 2 and…”

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“…The theory includes and extends the theory of crystallographic Coxeter groups, but contains even such examples which do not seem to be related to Nichols algebras of diagonal type. In this paper we use the language and some structural and classification results achieved in [Cuntz and Heckenberger 2008]; see Section 2 for the most essential definitions and facts. For the classification of Nichols algebras of diagonal type it is crucial to be able to decide whether a given Cartan scheme (a categorical generalization of the notion of a generalized Cartan matrix; see Definition 2.1) admits a finite root system.…”

“…It relies on a relationship between Cartan schemes of rank two and continued fractions [Perron 1929]. Instead of giving a complete list of Cartan schemes of rank two admitting a finite root system (which is then unique by a result in [Cuntz and Heckenberger 2008]), we present an algorithm in Theorem 6.19. It works with very elementary operations on sequences of positive integers and transforms any Cartan scheme into another one, for which the answer is known.…”