  The theory of Nichols algebras of diagonal type is known to be closely
related to that of semisimple 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 semisimple Lie algebra. They
give rise to the definition of a Brandt 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.
  Key Words: Brandt groupoid, Hopf algebra, pseudo-reflections, Weyl group
