Regular vs. classical M\"obius transformations of the quaternionic unit ball

The regular fractional transformations of the extended quaternionic space have been recently introduced as variants of the classical linear fractional transformations. These variants have the advantage of being included in the class of slice regular functions, introduced by Gentili and Struppa in 2006, so that they can be studied with the useful tools available in this theory. We first consider their general properties, then focus on the regular M\"obius transformations of the quaternionic unit ball B, comparing the latter with their classical analogs. In particular we study the relation between the regular M\"obius transformations and the Poincar\'e metric of B, which is preserved by the classical M\"obius transformations. Furthermore, we announce a result that is a quaternionic analog of the Schwarz-Pick lemma.Comment: 14 page

Prime ideals in nilpotent Iwasawa algebras

Let G be a nilpotent complete p-valued group of finite rank and let k be a field of characteristic p. We prove that every faithful prime ideal of the Iwasawa algebra kG is controlled by the centre of G, and use this to show that the prime spectrum of kG is a disjoint union of commutative strata. We also show that every prime ideal of kG is completely prime. The key ingredient in the proof is the construction of a non-commutative valuation on certain filtered simple Artinian rings

Regular Moebius transformations of the space of quaternions

Let H be the real algebra of quaternions. The notion of regular function of a quaternionic variable recently presented by G. Gentili and D. C. Struppa developed into a quite rich theory. Several properties of regular quaternionic functions are analogous to those of holomorphic functions of one complex variable, although the diversity of the quaternionic setting introduces new phenomena. This paper studies regular quaternionic transformations. We first find a quaternionic analog to the Casorati-Weierstrass theorem and prove that all regular injective functions from H to itself are affine. In particular, the group Aut(H) of biregular functions on H coincides with the group of regular affine transformations. Inspired by the classical quaternionic linear fractional transformations, we define the regular fractional transformations. We then show that each regular injective function from the Alexandroff compactification of H to itself is a regular fractional transformation. Finally, we study regular Moebius transformations, which map the unit ball B onto itself. All regular bijections from B to itself prove to be regular Moebius transformations.Comment: 12 page