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Discrete connections and difference linear equations. (English. Russian original) Zbl 1109.39020
Geometric topology and set theory. Collected papers. Dedicated to the 100th birthday of Professor Lyudmila Vsevolodovna Keldysh. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica. Proceedings of the Steklov Institute of Mathematics 247, 168-183 (2004); translation from Tr. Mat. Inst. Steklova 247, 186-201 (2004).
The author develops a nonstandard discrete analogue of the theory of differential-geometric \(GL_{n}\)-connections on triangulated manifolds. The connection is interpretated as a first-order linear difference equation for scalar functions of vertices in simplicial complexes. This equation is called the “triangle equation”. The theory is related to the discretization of some completely integrable systems. After introducing the concept of discrete differential-geometric connection and framed abelian holonomy, the author develops the notions of nonabelian holonomy and curvature. He then proves that, for \(n \geq 3,\) the data of the framed abelian holonomy completely define all coefficients of the connection up to an abelian gauge transformation. The case of \(n = 2\) is considered, before returning to the multidimensional case.
For the entire collection see [Zbl 1087.55002].
39A12 Discrete version of topics in analysis
39A10 Additive difference equations
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
57M50 General geometric structures on low-dimensional manifolds
57Q15 Triangulating manifolds
53C05 Connections (general theory)