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The Dirac complex on abstract vector variables: megaforms. (English) Zbl 1078.30044

The authors introduced in several articles [see e.g. I. Sabadini, F. Sommen, D. C. Struppa and P. van Lancker, Math. Z. 239, No. 2, 293–320 (2002; Zbl 1078.30043), reviewed below], abstract vector variables over a ring, here \(\mathbb{R}\). For systems of hypercomplex operators one may ask for the syzygies, i.e. the compatibility conditions for the related inhomogeneous equations. Several examples show how one can calculate these syzygies, getting the conditions for solvability. Especially Dirac operators are dealt with, and the question of ’exceptional’ syzygies is solved. The paper opens a wide field of research in a very general setting.

MSC:

30G35 Functions of hypercomplex variables and generalized variables
35Q40 PDEs in connection with quantum mechanics
16E05 Syzygies, resolutions, complexes in associative algebras
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