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We contribute to the classification of Hopf algebras with finiteGelfand-Kirillov dimension, $\operatorname{GKdim}$ for short, through the studyof Nichols algebras over abelian groups. We deal first with braided vectorspaces over $\mathbb Z$ with the generator acting as a single Jordan block andshow that the corresponding Nichols algebra has finite $\operatorname{GKdim}$if and only if the size of the block is 2 and the eigenvalue is $\pm 1$; whenthis is 1, we recover the quantum Jordan plane. We consider next a class ofbraided vector spaces that are direct sums of blocks and points that containsthose of diagonal type. We conjecture that a Nichols algebra of diagonal typehas finite $\operatorname{GKdim}$ if and only if the corresponding generalizedroot system is finite. Assuming the validity of this conjecture, we classifyall braided vector spaces in the mentioned class whose Nichols algebra hasfinite $\operatorname{GKdim}$. Consequently we present several new examples ofNichols algebras with finite $\operatorname{GKdim}$, including two not in theclass alluded to above. We determine which among these Nichols algebras aredomains.

On finite GK-dimensional Nichols algebras over abelian groups