Classification Of Hopf Algebras Of GK-dimension One

As a natural form of non-commutative algebraic group theory, infinite dimensional Hopf algebras have been studied intensively and substantial progress has been made in classifying infinite dimensional noetherian Hopf algebras of low GK-dimension in recent years. Infinite dimensional Hopf algebras and finite dimensional Hopf algebras share exquisite properties in many aspects. It is well-known that there are only two connected algebraic groups of dimension one:k+ and k×. This fact makes us believe that there should be a complete classification of affine prime regular Hopf algebras of GK-dimension one. In this thesis, we finish the classification based on previous studies. Concretely, Lu-Wu-Zhang [24] introduced the notion of homological integral of Hopf algebras and initiated the classification of Hopf algebras of GK-dimension one. Brown and Zhang [12] made further efforts in this direction and classified all affine prime regular Hopf algebras H of GK-dimension one under a hypothesis. We construct a new class of Hopf algebras D(m,d,ζ) and finish the classification of affine prime regular Hopf algebras of GK-dimension one. Further, properties of these new Hopf algebras are studied detailed.