A noncommutative discrete potential KdV lift

In this paper, we construct a Grassmann extension of a Yang-Baxter map which first appeared in the work of Kouloukas and Papageorgiou [J. Phys. A: Math. Theor. 42, 404012 (2009)] and can be considered as a lift of the discrete potential Korteweg-de Vries (dpKdV) equation. This noncommutative extension satisfies the Yang-Baxter equation, and it admits a 3 × 3 Lax matrix. Moreover, we show that it can be squeezed down to a novel system of lattice equations which possesses a Lax representation and whose bosonic limit is the dpKdV equation. Finally, we consider commutative analogs of the constructed Yang-Baxter map and its associated quad-graph system, and we discuss their integrability.