  In the structure theory of quantized enveloping algebras, the algebra
isomorphisms determined by Lusztig led to the first general construction of PBW
bases of these algebras. Also, they have important applications to the
representation theory of these and related algebras. In the present paper the
Drinfel'd double for a class of graded Hopf algebras is investigated. Various
quantum algebras, including multiparameter quantizations of semisimple Lie
algebras and of Lie superalgebras, are covered by the given definition. For
these Drinfel'd doubles Lusztig maps are defined. It is shown that these maps
induce isomorphisms between doubles of Nichols algebras of diagonal type.
Further, the obtained isomorphisms satisfy Coxeter type relations in a
generalized sense. As an application, the Lusztig isomorphisms are used to give
a characterization of Nichols algebras of diagonal type with finite arithmetic
root system.
  Key words: Hopf algebra, quantum group, Weyl groupoid
