Premium
Sheaves of slice regular functions
Author(s) -
Colombo Fabrizio,
Sabadini Irene,
Struppa Daniele C.
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201000149
Subject(s) - mathematics , sheaf , cohomology , pure mathematics , invariant (physics) , sheaf cohomology , quaternion , open set , algebra over a field , differential operator , geometry , group cohomology , mathematical physics
Slice regular functions have been introduced in 20 as solutions of a special partial differential operator with variable coefficients. As such they do not naturally form a sheaf. In this paper we use a modified definition of slice regularity, see 21, to introduce the sheaf of slice regular functions with values in in the algebra of quaternions and, more in general, in a Clifford algebra and we study its cohomological properties. We show that the first cohomology group with coefficients in the sheaf of slice regular functions vanishes for any open set in the space of quaternions (resp. the space of paravectors in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+1}$\end{document} ). However, we prove that not all the open sets are domains of slice regularity but only those special sets which are axially symmetric, i.e., invariant with respect to rotations that fix the real axis.