Exterior differential algebras and flat connections on Weyl groups

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. Introduction The study of higher order differential structures on Hopf algebras was initiated by S. L. Woronowicz [6], and further developed by K. Bresser et al.[1] and S. Majid [5] for algebras of functions on finite groups. In particular, S. Majid has introduced and studied flat connections on the symmetric group SN . In our paper, we study the algebra generated by flat connections in a sense of Majid on a finite Weyl group. This is an interesting problem which is not treated in [5]. We consider the differential structure with respect to the set of reflections. Since the complete set of the defining relations of the left-invariant exterior differential algebra has not yet been determined in general, we will work on its quadratic version Λquad for the root system of type A or D, and on its quartic version Λquar for the root system of type B. Our main result describes a set of relations among flat connections on Weyl groups of type A and D. Conjecturally, these relations are complete set of relations among flat connections in Λquad. We expect some connections of our construction with Schubert calculus on flag varieties [4]. 1 Woronowicz exterior algebra Woronowicz exterior algebra was introduced in [6] for the study of higher order differential structure on the quantum groups. In the category of modules over a commutative algebra, the exterior products of a module are constructed by using the canonical action Both of the authors were supported by Grant-in-Aid for Scientific Research.