k-Dirac Complexes

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2018
• Main Author: Salač, T.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2018
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/209452
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: k-Dirac Complexes / T. Salač // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ.

This is the first paper in a series of two papers. In this paper, we construct complexes of invariant differential operators that live on homogeneous spaces of |2|-graded parabolic geometries of some particular type. We call them k-Dirac complexes. More explicitly, we will show that each k-Dirac complex arises as the direct image of a relative BGG sequence, and so this fits into the scheme of the Penrose transform. We will also prove that each k-Dirac complex is formally exact, i.e., it induces a long exact sequence of infinite (weighted) jets at any fixed point. In the second part of the series, we use this information to show that each k-Dirac complex is exact at the level of formal power series at any point and that it descends to a resolution of the k-Dirac operator studied in Clifford analysis.