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Nichols algebras are fundamental objects in the construction of quantizedenveloping algebras and in the classification of pointed Hopf algebras bylifting method of Andruskiewitsch and Schneider. Arithmetic root systems areinvariants of Nichols algebras of diagonal type with a certain finitenessproperty. In the present paper, all rank 4 Nichols algebras of diagonal typewith a finite arithmetic root system over fields of arbitrary characteristicare classified. Our proof uses the classification of the finite arithmetic rootsystems of rank 4.

Rank 4 finite-dimensional Nichols algebras of diagonal type in positive characteristic