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

Mathematische Zeitschrift volume 259, pages 255–276 (2008)Cite this article

Abstract

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 general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by simple reflections and Coxeter relations. In a broad sense this answers a question of Serganova. Our weak version of the exchange condition allows us to prove Matsumoto’s theorem. Therefore the word problem is solved for the groupoid.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Leipzig, Johannisgasse 26, 04103, Leipzig, Germany

    István Heckenberger

  2. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, 560-0043, Japan

    Hiroyuki Yamane

Authors
  1. István Heckenberger
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  2. Hiroyuki Yamane
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Correspondence to István Heckenberger.

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Heckenberger, I., Yamane, H. A generalization of Coxeter groups, root systems, and Matsumoto’s theorem. Math. Z. 259, 255–276 (2008). https://doi.org/10.1007/s00209-007-0223-3

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  • DOI: https://doi.org/10.1007/s00209-007-0223-3

Mathematics Subject Classification (2000)

  • 20F55
  • 20F10
  • 17B20
  • 16W30