We study some aspects of noncommutative differential geometry on a finite Weyl group in the sense of S. Woronowicz, K. Bresser et al., and S. Majid. For any finite Weyl group $W$ we consider the subalgebra generated by flat connections in the left-invariant exterior differential algebra of $W$.
For root systems of type $A$ and $D$ we describe a set of relations between the flat connections, which conjecturally is a complete set.
Anatol N. Kirillov. Toshiaki Maeno. "Exterior differential algebras and flat connections on Weyl groups." Proc. Japan Acad. Ser. A Math. Sci. 81 (2) 30 - 35, Feb. 2005. https://doi.org/10.3792/pjaa.81.30