Hopf algebras under finiteness conditions

This is a brief survey of some recent developments in the study of infinite dimensional Hopf algebras which are either noetherian or have finite Gelfand-Kirillov dimension. A number of open questions are listed.Comment: Comments welcom

Unimodular graded Poisson Hopf algebras

Let $A$ be a Poisson Hopf algebra over an algebraically closed field of characteristic zero. If $A$ is finitely generated and connected graded as an algebra and its Poisson bracket is homogeneous of degree $d \geq 0$, then $A$ is unimodular; that is, the modular derivation of $A$ is zero. This is a Poisson analogue of a recent result concerning Hopf algebras which are connected graded as algebras.Comment: 14 pages; preliminary version, comments welcom

Existence of Hopf subalgebras of GK-dimension two

Let $H$ be a pointed Hopf algebra over an algebraically closed field of characteristic zero. If $H$ is a domain with finite Gelfand-Kirillov dimension greater than or equal to two, then $H$ contains a Hopf subalgebra of Gelfand-Kirillov dimension two.Comment: Accepted by Journal of Pure and Applied algebra, 10.1016/j.jpaa.2011.04.01

Quantum homogeneous spaces of connected Hopf algebras

Let H be a connected Hopf k-algebra of finite Gel'fand-Kirillov dimension over an algebraically closed field k of characteristic 0. The objects of study in this paper are the left or right coideal subalgebras T of H. They are shown to be deformations of commutative polynomial k-algebras. A number of well-known homological and other properties follow immediately from this fact. Further properties are described, examples are considered, invariants are constructed and a number of open questions are listed.Comment: 26 pages; comments welcom