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# Flashback-“On a correspondence principle between discrete differential forms, graph structure and multi-vector calculus on symmetric lattices”

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 $\partial_X$ defined viz formula  $(48)$  (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 $2n+1$ by the canonical relations carrying the Lie algebra $\mathfrak{sl}(2n,\mathbb{R})$ intriguing on the structure of the (symmetrized) multiplication operators of the type

• $f(x) \mapsto (x_j+h/2) f(x+h)$,
• $f(x) \mapsto (x_j-h/2) f(x-h)$,
• $f(x) \mapsto x_j f(x)$.

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 $\mathfrak{osp}(1|2)$ in a similar fashion of what was done in the above papers: