Algebraic Analysis of Differential Equations
URI (for links/citations):https://elib.sfu-kras.ru/handle/2311/827
Osetrova, Tatyana A.
(Tatyana A.Osetrova: Institute of Mathematics Siberian Federal University Svobodny 79, Krasnoyarsk, 660041, Russia, e-mail: firstname.lastname@example.org; Nikolai Tarkhanovy: Institut fur Mathematik Universitat Potsdam Am Neuen Palais 10, 14469 Potsdam, Germany, e-mail: email@example.com)
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.