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Differential forms and/or multi-vector functions. (English) Zbl 1105.58002
For an open subset \(\Omega\) of the Euclidean space \(\mathbb R^{n+1}\), let \(\Lambda(\Omega)\) be the Cartan algebra of smooth differential forms endowed with the exterior differentiation operator \(d\). The Hodge co-differentiation \(d^*\) leads to the differential operator \(D=d+d^*\) by means of which the harmonic \(r\)-forms are characterized as smooth differential \(r\)-forms \(\omega^r\) satisfying \(D\omega^r=0\). The graded associative algebra \(\mathbb R_{0,m+1}=\sum_{r=0}^{m+1}\oplus\mathbb R^r_{0,m+1}\) is the Clifford algebra, if \(\mathbb R^r_{0,m+1}\) is the space of \(r\)-vectors for the vector space \(\mathbb R^{0,m+1}\) equipped with an anti-Euclidean metric. Let \(\varepsilon(\Omega)\) be the algebra of smooth \(r\)-vector functions with values in the Clifford algebra. A fundamental operator on \(\varepsilon(\Omega)\) is the Dirac operator \(\partial\), by means of which monogenic functions are characterized as the smooth functions satisfying \(\partial f=0\).
In this paper, the authors show that the spaces \(\Lambda(\Omega)\) and \(\varepsilon(\Omega)\) are isomprphic. Also, they present the Poincaré Lemma and the Dual Poincaré Lemma for both \(\Lambda(\Omega)\) and \(\varepsilon(\Omega)\). Moreover, the paper includes the interesting discussion on comparison of two mathematical languages: the language of differential forms and that of Clifford algebra-valued multi-vector functions. The authors conclude that one language does not substitute the other, or there is no preference one language over the other. On the contrary, they assert that differential forms and multi-vector functions, despite the natural identification given, are quite different mathematical objects whose use is imposed by the mathematical context.

58A10 Differential forms in global analysis
30G35 Functions of hypercomplex variables and generalized variables
Poincaré lemma