Finite GK-Dimensional Nichols Algebras Over the Infinite Dihedral Group

Abstract

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over Q8,the quaternion group . We find all the irreducible Yetter-Drinfeld modules V over Q8, and determine which Nichols algebras B(V ) of V are finite GK-dimensional, and which Nichols algebras B(V ) of V are finite dimensional. Contents 1. Introduction 1 2. Preliminaries 2 2.1. Notations 2 2.2. Yetter-Drinfeld modules and Nichols algebras 2 2.3. The quaternion group Q 8 3 2.4. Gelfand-Kirillov dimension 4 3. The Nichols algebras B(O 1 , ρ) 4 4. The Nichols algebras B(O x , φ) 6 5. The Nichols algebras B(O x 2 , ρ) 8 6. The Nichols algebras B(O y , φ) 10 7. The Nichols algebras B(O xy , φ) 12 8. Acknowledgments 14 References 14