On noncommutative leapfrog map

Abstract

We investigate the integrability of the noncommutative leapfrog map in this paper. First, we derive the explicit formula for the noncommutative leapfrog map and corresponding discrete zero-curvature equation by employing the concept of noncommutative cross-ratio. Then we revisit this discrete map, as well as its continuous limit, from the perspective of noncommutative Laurent bi-orthogonal polynomials. Finally, the Poisson structure for this discrete noncommutative map is formulated with the help of a noncommutative network. Through these constructions, we aim to enhance our understanding of the integrability properties of the noncommutative leapfrog map and its related mathematical structures.