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Abstract

We contribute to the classification of Hopf algebras with finite
Gelfand-Kirillov dimension, $\operatorname{GKdim}$ for short, through the study
of Nichols algebras over abelian groups. We deal first with braided vector
spaces over $\mathbb Z$ with the generator acting as a single Jordan block and
show that the corresponding Nichols algebra has finite $\operatorname{GKdim}$
if and only if the size of the block is 2 and the eigenvalue is $\pm 1$; when
this is 1, we recover the quantum Jordan plane. We consider next a class of
braided vector spaces that are direct sums of blocks and points that contains
those of diagonal type. We conjecture that a Nichols algebra of diagonal type
has finite $\operatorname{GKdim}$ if and only if the corresponding generalized
root system is finite. Assuming the validity of this conjecture, we classify
all braided vector spaces in the mentioned class whose Nichols algebra has
finite $\operatorname{GKdim}$. Consequently we present several new examples of
Nichols algebras with finite $\operatorname{GKdim}$, including two not in the
class alluded to above. We determine which among these Nichols algebras are
domains.