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The Weyl groupoid of a Nichols algebra of diagonal type

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Abstract

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 algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semi-simple Lie algebra. They give rise to the definition of a groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braiding.

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References

  1. Andruskiewitsch, N., Schneider, H.-J.: Finite quantum groups and Cartan matrices. Adv. Math. 154, 1–45 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andruskiewitsch, N., Schneider, H.-J.: Pointed Hopf algebras. In: New Directions in Hopf Algebras. Ser. MSRI Publications, vol. 43. Cambridge University Press 2002

  3. Bourbaki, N.: Groupes et algèbres de Lie, Ch. 4, 5 et 6. Éléments de mathematique. Paris: Hermann 1968

  4. Clifford, A.H., Preston, G.B.: The algebraic theory of semigroups. Ser. Mathematical Surveys, vol. 7. Providence: Am. Math. Soc. 1961

  5. Heckenberger, I.: Finite dimensional rank 2 Nichols algebras of diagonal type. I: Examples. Preprint math.QA/0402350

  6. Heckenberger, I.: Finite dimensional rank 2 Nichols algebras of diagonal type. II: Classification. Preprint math.QA/0404008

  7. Hiller, H.: Geometry of Coxeter groups. Ser. Research Notes in Mathematics, vol. 54. Boston, MA, London: Pitman 1982

  8. Joseph, A.: A generalization of the Gelfand–Kirillov conjecture. Am. J. Math. 99, 1151–1165 (1977)

    MATH  MathSciNet  Google Scholar 

  9. Kharchenko, V.: A quantum analog of the Poincaré–Birkhoff–Witt theorem. Alg. Logic 38, 259–276 (1999)

    MATH  MathSciNet  Google Scholar 

  10. Rosso, M.: Quantum groups and quantum shuffles. Invent. Math. 133, 399–416 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Schauenburg, P.: A characterization of the Borel-like subalgebras of quantum enveloping algebras. Commun. Alg. 24, 2811–2823 (1996)

    MATH  MathSciNet  Google Scholar 

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  1. Mathematisches Institut, Universität Leipzig, 04109, Leipzig, Germany

    I. Heckenberger

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  1. I. Heckenberger
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Correspondence to I. Heckenberger.

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Mathematics Subject Classification (2000)

17B37, 16W35

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Heckenberger, I. The Weyl groupoid of a Nichols algebra of diagonal type. Invent. math. 164, 175–188 (2006). https://doi.org/10.1007/s00222-005-0474-8

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  • DOI: https://doi.org/10.1007/s00222-005-0474-8

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