A generalization of Coxeter groups, root systems, and Matsumotos 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 general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by reflections and Coxeter relations. This answers a question of Serganova. Our weak version of the exchange condition allows us to prove Matsumotos theorem. Therefore the word problem is solved for the groupoid.