The Weyl groupoid of a Nichols algebra of diagonal type

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. Mathematics Subject Classification (2000) 17B37, 16W35