Editing Dirac operator (section)

== Generalisations ==
In Clifford analysis, the operator {{nowrap|''D'' : ''C''<sup>∞</sup>('''R'''<sup>''k''</sup> ⊗ '''R'''<sup>''n''</sup>, ''S'') → ''C''<sup>∞</sup>('''R'''<sup>''k''</sup> ⊗ '''R'''<sup>''n''</sup>, '''C'''<sup>''k''</sup> ⊗ ''S'')}} acting on spinor valued functions defined by 
:<math>f(x_1,\ldots,x_k)\mapsto
\begin{pmatrix}
\partial_{\underline{x_1}}f\\
\partial_{\underline{x_2}}f\\
\ldots\\
\partial_{\underline{x_k}}f\\
\end{pmatrix}</math>
is sometimes called Dirac operator in ''k'' Clifford variables. In the notation, ''S'' is the space of spinors, <math>x_i=(x_{i1},x_{i2},\ldots,x_{in})</math> are ''n''-dimensional variables and <math>\partial_{\underline{x_i}}=\sum_j e_j\cdot \partial_{x_{ij}}</math> is the Dirac operator in the ''i''-th variable. This is a common generalization of the Dirac operator ({{nowrap|1=''k'' = 1}}) and the [[Dolbeault cohomology|Dolbeault operator]] ({{nowrap|1=''n'' = 2}}, ''k'' arbitrary). It is an [[invariant differential operator]], invariant under the action of the group {{nowrap|SL(''k'') × Spin(''n'')}}. The [[injective_resolution#Injective_resolutions|resolution]] of ''D'' is known only in some special cases.