Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 019, 23 pages      arXiv:2204.05720      https://doi.org/10.3842/SIGMA.2023.019

Higher Braidings of Diagonal Type

Michael Cuntz and Tobias Ohrmann
Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Welfengarten 1, D-30167 Hannover, Germany

Received May 30, 2022, in final form March 27, 2023; Published online April 06, 2023

Abstract
Heckenberger introduced the Weyl groupoid of a finite-dimensional Nichols algebra of diagonal type. We replace the matrix of its braiding by a higher tensor and present a construction which yields further Weyl groupoids. Abelian cohomology theory gives evidence for the existence of a higher braiding associated to such a tensor.

Key words: Nichols algebra; braiding; Weyl groupoid.

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References

  1. Cuntz M., Frieze patterns as root posets and affine triangulations, European J. Combin. 42 (2014), 167-178, arXiv:1307.7986.
  2. Cuntz M., Heckenberger I., Weyl groupoids of rank two and continued fractions, Algebra Number Theory 3 (2009), 317-340, arXiv:0807.0124.
  3. Cuntz M., Heckenberger I., Weyl groupoids with at most three objects, J. Pure Appl. Algebra 213 (2009), 1112-1128, arXiv:0805.1810.
  4. Cuntz M., Heckenberger I., Finite Weyl groupoids, J. Reine Angew. Math. 702 (2015), 77-108, arXiv:1008.5291.
  5. Cuntz M., Lentner S., A simplicial complex of Nichols algebras, Math. Z. 285 (2017), 647-683, arXiv:1503.08117.
  6. Eilenberg S., MacLane S., Cohomology theory of Abelian groups and homotopy theory. I, Proc. Nat. Acad. Sci. USA 36 (1950), 443-447.
  7. Eilenberg S., MacLane S., Cohomology theory of Abelian groups and homotopy theory. II, Proc. Nat. Acad. Sci. USA 36 (1950), 657-663.
  8. Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor categories, Math. Surveys Monogr., Vol. 205, Amer. Math. Soc., Providence, RI, 2015.
  9. Heckenberger I., The Weyl groupoid of a Nichols algebra of diagonal type, Invent. Math. 164 (2006), 175-188, arXiv:math.QA/0411477.
  10. Heckenberger I., Classification of arithmetic root systems, Adv. Math. 220 (2009), 59-124, arXiv:math.QA/0605795.
  11. Joseph A., A generalization of the Gelfand-Kirillov conjecture, Amer. J. Math. 99 (1977), 1151-1165.
  12. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  13. Laugwitz R., Walton C., Constructing non-semisimple modular categories with relative monoidal centers, Int. Math. Res. Not. 2022 (2022), 15826-15868, arXiv:2010.11872.
  14. MacLane S., Cohomology theory of Abelian groups, in Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, Vol. 2, Amer. Math. Soc., Providence, R.I., 1952, 8-14.
  15. Rosso M., Quantum groups and quantum shuffles, Invent. Math. 133 (1998), 399-416.

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