On a non-commutative sixth $q$-Painlev\'e system: from discrete system to surface theory

In this paper, we describe the non-commutative formal geometry underlying a certain class of discrete integrable systems. Our main example is a non-commutative analog, labeled $q$-P$(A_3)$, of the sixth $q$-Painlev\'e equation. The system $q$-P$(A_3)$ is constructed by postulating an extended birational representation of the extended affine Weyl group $\widetilde{W}$ of type $D_5^{(1)}$ and by selecting the same translation element in $\widetilde{W}$ as in the commutative case. Starting from this non-commutative discrete system, we develop a non-commutative version of Sakai$'$s surface theory, which allows us to derive the same birational representation that we initially postulated. Moreover, we recover the well-known cascade of multiplicative discrete Painlev\'e equations rooted in $q$-P$(A_3)$ and establish a connection between $q$-P$(A_3)$ and the non-commutative $d$-Painlev\'e systems introduced in I. Bobrova. Affine Weyl groups and non-Abelian discrete systems: an application to the $d$-Painlev\'e equations.