Affine Weyl groups and non-Abelian discrete systems: an application to the $d$-Painlev\'e equations

A non-abelian generalisation of a birational representation of affine Weyl groups and their application to the discrete dynamical systems is presented. By using this generalisation, non-commutative analogs for the discrete systems of $A_n^{(1)}$, $n \geq 2$ type and of $d$-Painlev\'e equations with an additive dynamic were derived. A coalescence cascade of the later is also discussed.