Pre-Nichols algebras of nondiagonal objects have infinite GK dimension, so finite GK-dimensional Nichols algebras are exactly the diagonal ones with finite arithmetic root systems.
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On the classification of finite GK-dimensional pre-Nichols algebras and quasi-quantum groups
Pre-Nichols algebras of nondiagonal objects have infinite GK dimension, so finite GK-dimensional Nichols algebras are exactly the diagonal ones with finite arithmetic root systems.