Integrable non-commutative equations on quad-graphs. The consistency approach

Abstract

We extend integrable systems on quad-graphs, such as the Hirota equation and the cross-ratio equation, to the non-commutative context, when the fields take values in an arbitrary associative algebra. We demonstrate that the three-dimensional consistency property remains valid in this case. We derive the non-commutative zero curvature representations for these systems, based on the latter property. Quantum systems with their quantum zero curvature representations are particular cases of the general non-commutative ones.

LaTeX2e, 16 pp

Cited by in corpus (8)

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• Non-commutative lattice modified Gel'fand-Dikii systems
• The pentagon relation and incidence geometry
• The Coxeter relations and KP map for non-commuting symbols
• Hirota equation and the quantum plane
• Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables