Pointed and ambiskew Hopf algebras.
PhD thesis, University of Glasgow.
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This thesis is concerned with properties of pointed Hopf algebras: that is, Hopf algebras whose coradicals are the group algebras of their grouplike elements. These have been fruitfully studied via their associated graded Hopf algebras with respect to the coradical filtration. In fact, the associated graded Hopf algebra gr H of a pointed Hopf algebra H can be decomposed into a braided graded Hopf algebra of coinvariants adjoined to a group algebra by a process called bosonisation.
Chapter 1 consists of background material, which fully explains the process outlined above.
In Chapter 2, we outline and discuss the main results of Kharchenko in , which gives a PBW-basis for a certain class of associated graded Hopf algebras gr H of pointed Hopf algebras H. The hypotheses on gr H are that its grouplikes form an abelian group that acts on the braided Hopf algebra of coinvariants diagonalisably - that is, by multiplication by scalars, which are called the braiding coefficients. In Theorem 2.4.1, we give an expanded proof of [32, Corollary 5].
This provides a tool which we use in Chapter 3 to show that the ordering of the PBW-generators in Kharchenko's PBW-basis for gr H may be permuted in the case where there are only a finite number of generators. We then use this in order to prove that gr H, and hence H, satisfy certain homological properties.
In Chapter 4, we prove a result giving sufficient conditions on the braiding coefficients for the braided Hopf algebra of coinvariants to be a free algebra, thus answering a question of Andruskiewitsch and Schneider in .
Chapter 5 switches the focus to a type of skew-polynomial algebras called ambiskew polynomial algebras, defined over a base algebra R. We drop the hypothesis that R is commutative, which was generally assumed in previous work on these algebras. We then give necessary and sufficient conditions for a Hopf algebra structure on R to be extended to a Hopf algebra structure on the ambiskew polynomial algebra, generalising work of Hartwig in . We also calculate explicitly their coradical filtration, which gives as a corollary some theorems of Montgomery  and Boca  on the coradical filtration of U_q(sl_2). Finally, we consider some homological and ring-theoretic properties of ambiskew polynomial algebras.
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