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arXiv · 1509.03129

On the classification of multidimensionally consistent 3D maps

Matteo Petrera·Yuri B. Suris
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Abstract

We classify multidimensionally consistent maps given by (formal or convergent) series of the following kind: $$ T_k x_{ij}=x_{ij} + \sum_{m=2}^\infty A_{ij ; \, k}^{(m)}(x_{ij},x_{ik},x_{jk}), $$ where $A_{ij;\, k}^{(m)}$ are homogeneous polynomials of degree $m$ of their respective arguments. The result of our classification is that the only non-trivial multidimensionally consistent map in this class is given by the well known symmetric discrete Darboux system $$ T_k x_{ij}=\frac{x_{ij}+x_{ik}x_{jk}}{\sqrt{1-x_{ik}^2}\sqrt{1-x_{jk}^2}}. $$

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Matteo Petrera, Yuri B. Suris. 2015-09-10. On the classification of multidimensionally consistent 3D maps. https://doi.org/10.1007/s11005-017-0976-5

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