Hongdi Huang, Department of Pure Mathematics, University of Waterloo
"On Hopf Ore Extensions and Zariski Cancellation Problems"
I will talk about the topics from my thesis, i.e., Hopf Ore extensions and the Zariski Cancellation problem for noncommutative rings. We improve upon the existing conditions for when $T=R[x; \sigma, \delta]$ is a Hopf Ore extension of a Hopf algebra $R$ by studying the form of coproduct $\Delta(x)$. This is originally motivated by a question asked by Panov; that is, given a Hopf algebra $R$, for which automorphisms $\sigma$ and $\sigma$-derivations $\delta$ does the Ore extension $T=R[x; \sigma, \delta]$ have a Hopf algebra structure extending the given Hopf algebra structure on $R$? On the other hand, we consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand-Kirillov dimension one and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand-Kirillov dimension one need not be cancellative when the base field has a positive characteristic, giving a counterexample to a conjecture of Tang et al. In the light of the Zariski Cancellation problem, we prove a skew analogue of the result of Abhyankar-Eakin-Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings.
MC 5403