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In the structure theory of quantized enveloping algebras, the algebraisomorphisms determined by Lusztig led to the first general construction of PBWbases of these algebras. Also, they have important applications to therepresentation theory of these and related algebras. In the present paper theDrinfel'd double for a class of graded Hopf algebras is investigated. Variousquantum algebras, including multiparameter quantizations of semisimple Liealgebras and of Lie superalgebras, are covered by the given definition. Forthese Drinfel'd doubles Lusztig maps are defined. It is shown that these mapsinduce isomorphisms between doubles of Nichols algebras of diagonal type.Further, the obtained isomorphisms satisfy Coxeter type relations in ageneralized sense. As an application, the Lusztig isomorphisms are used to givea characterization of Nichols algebras of diagonal type with finite arithmeticroot system.  Key words: Hopf algebra, quantum group, Weyl groupoid

Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type