Hopf algebras of finite Gelfand-Kirillov dimension

Abstract

We study Hopf algebras of finite Gelfand-Kirillov dimension. By analyzing the free pointed Hopf algebra F(t), we show the existence of certain Hopf subalgebras of pointed Hopf domains H whose GK-dimension are finite and no less than 2. Using Takeuchi's construction of free Hopf algebras, we study the GK-dimension of a pointed Hopf algebra H and its associated graded Hopf algebra grH. We also get some lower bounds of GK-dimension of Hopf algebras in terms of certain invariants of skew primitive elements. With the help of these results, we classify connected Hopf algebras of GK-dimension 3. We introduce the notion of coassociative Lie algebras, which generalize both the notions of Lie algebras and coalgebras (without counits). It turns out to be very useful in our classification of connected Hopf algebras of GK-dimension 4, which is accomplished in the last part of this paper.