Finite GK-dimensional Nichols algebras over the quaternion group $Q_8$

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over Q 8 ,the quaternion group . We find all the irreducible Yetter-Drinfeld modules V over Q 8 , and determine which Nichols algebras B(V ) of V are finite GK-dimensional, and which Nichols algebras B(V ) of V are finite dimensional.