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Resolution of the \({\varvec{k}}\)-Dirac operator

  • Tomáš Salač
Article

Abstract

This is the second part in a series of two papers. The k-Dirac complex is a complex of differential operators which are naturally associated to a particular |2|-graded parabolic geometry. In this paper we will consider the k-Dirac complex over the homogeneous space of the parabolic geometry and as a first result, we will prove that the k-Dirac complex is formally exact (in the sense of formal power series). Then we will show that the k-Dirac complex descends from an affine subset of the homogeneous space to a complex of linear and constant coefficient differential operators and that the first operator in the descended complex is the k-Dirac operator studied in Clifford analysis. The main result of this paper is that the descended complex is locally exact and thus it forms a resolution of the k-Dirac operator.

Keywords

Resolution of overdetermined system k-Dirac operator k-Dirac complex Invariant differential complexes 

Mathematics Subject Classification

Primary 35N05 58J10 Secondary 58A20 

Notes

Acknowledgements

The author is grateful to Vladimír Souček for his support and many useful conversations. The author would also like to thank to Lukáš Krump for the possibility of using his package for Young diagram. The author wishes to thank to the unknown referee for many helpful suggestions which considerably improved and simplified the current manuscript.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Charles UniversityPragueCzech Republic

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