Advances in Geometry
Managing Editor: Grundhöfer, Theo / Joswig, Michael
Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard
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Quaternionic Hermitian spinor systems and compatibility conditions
In this paper we show that the systems introduced in [Eelbode, Adv. Appl. Clifford Algebr. 17: 635–649, 2007] and [Peña-Peña, Sabadini, Sommen, Complex Anal. Oper. Theory 1: 97–113, 2007] are equivalent, both giving the notion of quaternionic Hermitian monogenic functions. This makes it possible to prove that the free resolution associated to the system is linear in any dimension, and that the first cohomology module is nontrivial, thus generalizing the results in [Peña-Peña, Sabadini, Sommen, Complex Anal. Oper. Theory 1: 97–113, 2007]. Furthermore, exploiting the decomposition of the spinor space into 𝔰𝔭(m)-irreducibles, we find a certain number of “algebraic” compatibility conditions for the system, suggesting that the usual spinor reduction is not applicable.
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