On finite GK-dimensional Nichols algebras over abelian groups

ANDRUSKIEWITSCH, NICOLÁS; ANGIONO, IVÁN; ISTVÁN HECKENBERGER

MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY (AMS)

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2021 vol. 271

0065-9266

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, $GK$ 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 algebrahas finite $GK$ 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 aNichols algebra of diagonal type has finite $GK$ 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 whoseNichols algebra has finite $GK$. Consequently we present several new examples of Nichols algebras with finite $GK$,including two not in the class alluded to above. We determine which among these Nichols algebras are domains.