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\(k\)-Dirac operator and the Cartan-Kähler theorem. (English) Zbl 1313.58001

Summary: We apply the Cartan-Kähler theorem for the \(k\)-Dirac operator studied in Clifford analysis and to the parabolic version of this operator. We show that for \(k=2\) the tableaux of the first prolongations of these two operators are involutive. This gives us a new characterization of the set of initial conditions for the 2-Dirac operator.

MSC:

58A17 Pfaffian systems
53C27 Spin and Spin\({}^c\) geometry
58A15 Exterior differential systems (Cartan theory)
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