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

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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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