Complexes of Dirac operators in Clifford algebras. (English) Zbl 1078.30045

Sabadini, Irene; Struppa, Daniele C.; Sommen, Frank; Van Lancker, Peter

Math. Z. 239, No. 2, 293-320 (2002).

In this paper the following is studied: The Cauchy-Fueter operator in the quaternionic case was obtained by the Moisil-Theodorescu product (the case of the Dirac operator in the Clifford algebra \({\mathcal C}_3\). Thus it is motivated the choice of Dirac operators in the Clifford algebra \({\mathcal C}_m\) only in the case when \(m>4\) or \(m=4\).
Another case following directly from the Cauchy-Fueter case is the case of \(n\)-Dirac operators systems in \({\mathcal C}_3\). Any procedure reduces the number of equations (without changing the nature of non-homogeneous equations).
Seven methods of action of Dirac operator on a vector variable function are exposed. These methods use the facts that: the Dirac operator is a vector; Weyl equation; the case of \(m\) is even; the spinor formalism for \(m=2n\); the 4-dimensional and the 6-dimensional cases of Dirac equation; the Witt basis using the Clifford algebra conjugate.
The Fischer decomposition is discussed for general Clifford polynomials of several vector variables. Rough-Fischer decomposition lemmas and theorems were formulated for all Clifford polynomials.
In the case when the dimension \(m\) of the space exceeds \(2l-2\), \(l\) being the number of vector variables in the Cauchy-Dirac complex, the syzygies remain the same for all higher dimensions and they are “purely radial”, i.e. they are polynomials belonging to the algebra generated by the vector variables alone.
In section 4, the case of 3 and 4 operators acting in the Clifford algebra \({\mathcal C}_m\) \((m>4\) or \(m=4\)) is discussed, proving the authors’ own results.
The made running CoCoA version 3.7 on a Digital Alpha Server is being used. The resolution for the number of syzygies mimicks the quaternionic case.
The case of \({\mathcal C}_5\) is being treated in two different ways. The syzygies come from the radial algebra. The authors gave several propositions refering to the compatibility conditions for each step of the proposed programme.
In the case of 3 Dirac operators in \({\mathcal C}_m\) \((m>5\) or \(m=5)\) there can only be radial syzygies (Proposition 5.1). If the dimension \(m>2k\) or \(m=2k\), the resolution of Dirac complex can only have syzygies consisting in radial polynomials (Theorem 5.2).
The case of systems of 4 Dirac operators in \({\mathcal C}_m\) \((m>4\) or \(m=4)\) is more complicated from a computational point of view. Using CoCoA, it was possible to partially compute the resolutions they are interested in.

Reviewer: Lidia Elena (Baia Mare)

MSC:

30G35

Functions of hypercomplex variables and generalized variables

Keywords:

Moisil-Theodorescu product; Dirac operators; Clifford algebras; quaternion; complex of Cauchy-Fueter operators; syzygy operator; syzygy polynomial

Software:

CoCoA