Proceedings of the Japan Academy, Series A, Mathematical Sciences

Exterior differential algebras and flat connections on Weyl groups

Anatol N. Kirillov and Toshiaki Maeno

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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.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 2 (2005), 30-35.

First available in Project Euclid: 18 May 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]
Secondary: 16W30

Noncommutative differential calculus Weyl groups


Kirillov, Anatol N.; Maeno, Toshiaki. Exterior differential algebras and flat connections on Weyl groups. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 2, 30--35. doi:10.3792/pjaa.81.30.

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  • K. Bresser, F. Mueller-Hoissen, A. Dimakis and A. Sitarz, Non-commutative geometry of finite groups, J. Phys. A 29 (1996), no.,11, 2705–2735.
  • S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, in Advances in geometry, Progr. Math., 172, Birkhäuser Boston, Boston, MA, 1999, pp. 147–182.
  • S. Fomin and C. Procesi, Fibered quadratic Hopf algebras related to Schubert calculus, J. Algebra 230 (2000), no.,1, 174–183.
  • A. N. Kirillov and T. Maeno, Noncommutative algebras related with Schubert calculus on Coxeter groups, European J. Combin. 25 (2004), no.,8, 1301–1325.
  • S. Majid, Noncommutative differentials and Yang-Mills on permutation groups $S_N$, Lect. Notes Pure Appl. Math. 239 (2004), 189–214.
  • S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), no.,1, 125–170.