Weyl groupoids of rank two and continued fractions

@article{Cuntz2008WeylGO,
  title={Weyl groupoids of rank two and continued fractions},
  author={Michael Cuntz and Istv{\'a}n Heckenberger},
  journal={arXiv: Group Theory},
  year={2008}
}
A relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two is found. This allows to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for the entries of the Cartan matrices. Key words: Cartan matrix, continued fraction, Nichols algebra, Weyl groupoid 

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References

SHOWING 1-10 OF 13 REFERENCES

Weyl groupoids with at most three objects

The Weyl groupoid of a Nichols algebra of diagonal type

The theory of Nichols algebras of diagonal type is known to be closely related to that of semi-simple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols

A generalization of Coxeter groups, root systems, and Matsumoto’s theorem

The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a

The Weyl–Brandt groupoid of a Nichols algebra of diagonal type

  • Mathematics
The theory of Nichols algebras of diagonal type is known to be closely related to that of semisimple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols

Classification of arithmetic root systems

On the classification of finite-dimensional pointed Hopf algebras

We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elementsG.A/ is abelian such

Hopf algebras and their actions on rings

Definitions and examples Integrals and semisimplicity Freeness over subalgebras Action of finite-dimensional Hopf algebras and smash products Coradicals and filtrations Inner actions Crossed products

The Nichols algebra of a semisimple Yetter-Drinfeld module

We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a "reflection"

Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of Orderp3

Abstract We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebraA, consider its coradical

Infinite-dimensional Lie algebras

1. Basic concepts.- 1. Preliminaries.- 2. Nilpotency and solubility.- 3. Subideals.- 4. Derivations.- 5. Classes and closure operations.- 6. Representations and modules.- 7. Chain conditions.- 8.