Abstract

Skew polynomial (Ore) extension is a classical construction sending a ring R to a deformed polynomial ring R[x; σ, δ]. This construction encompasses many very natural rings of interest, including group algebras of polycyclic groups, universal enveloping algebras of soluble Lie algebras and many quantum algebras, and in most cases of interest they are well studied and well understood. By contrast, skew power series extensions R[[x; σ, δ]] are much more modern and more poorly understood objects, not least because there are multiple competing contexts in which they can be studied. I will give a brief overview of the current state of our knowledge in these various contexts, before specialising to the case relevant to Iwasawa algebras (completed group rings of certain profinite groups) and describing my ongoing work in this area. Much of this work is joint with Adam Jones.