Although the report On a correspondence principle between discrete differential forms, graph structure and multi-vector calculus on symmetric lattices were never published in a journal, this report is itself a good recipee to start studying discrete counterparts of Dirac operators in interrelationship with differential forms and differential calculus.
The new report that I’m still writting is intend to clean up all of the aspects within Section 4 (pp. 16-21), with the goal to give a meaningful description for the so-called discrete monogenic solutions carrying the discrete Dirac operator, i.e. the null solutions of the operator defined viz formula (pp. 18). The main novelty of this approach against other approaches (see e.g discrete function theory based on Skew-Weyl relations) is the replacement of the so-called (Skew-)Weyl relations carrying a certain Heisenberg-type group of dimension by the canonical relations carrying the Lie algebra intriguing on the structure of the (symmetrized) multiplication operators of the type
Taking this alternative [but interesting] route, it will be possible to describe afterwards the discrete Clifford operators as a canonical realization of the orthosymplectic Lie algebra in a similar fashion of what was done in the above papers:
- N. Faustino, “Further results in discrete Clifford analysis, Progress in Analysis and Its Applications” – Proceedings of the 7th International ISAAC Congress, (M. Ruzhansky, J. Wirth eds.), World Scientific, pp. 205–211, (2010),
- N. Faustino, G. Ren, “(Discrete) Almansi type decompositions: an umbral calculus framework based on osp(1|2) symmetries.”, Mathematical Methods in the Applied Sciences, Volume 34, Issue 16, pages 1961–1979, November 2011.