Geodesic
Algebraic Analysis of Differential Equations
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 1 (2008) no. 4, pp. 391-398

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Given any algebra over a field with a finite number of generators, we define a first order partial differential operator acting on functions taking their values in the algebra. While being not canonical, the construction is fairly natural. We call this differential operator Dirac operator related to the algebra, and show some examples. Conversely, to each homogeneous first order differential operator one assigns an algebra which absorbs formal properties of the operator.
Keywords: normed algebras. 
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Tatyana A. Osetrova; Nikolai N. Tarkhanov. Algebraic Analysis of Differential Equations. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 1 (2008) no. 4, pp. 391-398. http://geodesic.mathdoc.fr/item/JSFU_2008_1_4_a3/