Annals of Combinatorics 4 (2000) 433-468
Zonotopes, Braids, and Quantum Groups
Department of Mathematics, Northwestern University, Evanston,
IL 60208, USA
Received September 16, 1998
AMS Subject Classification: 05A30, 17B37
Abstract. Various results about the action of the binomial braids and other braid analogs  on some particular higher dimensional representations of the braid groups are presented. These representations are constructed from a fixed integer square matrix A. The common nullspace of the binomial braids is studied in some detail. This space is graded over , where r is the size of A. Our main results state that the non-trivial components occur only on the lattice points of a certain hypersurface in r-space that is canonically associated to A, and that when A is the symmetrization of a Cartan matrix C of finite type, these lattice points are closely related to the vertices of the zonotope associated to C (the precise relationship is given in Theorem 5.3). The same action is used to construct a quantum group from an arbitrary integer square matrix A. The simplest choices of A yield the usual polynomial and Eulerian Hopf algebras of Joni and Rota (in the corresponding representations, the binomial braids become the usual binomial or q-binomial coefficients). The other choice we consider is that when A is the symmetrization of a symmetrizable Cartan matrix C. Some of the previous results are used to prove that in this case coincides with the usual quantum group of Drinfeld and Jimbo. The quantum group is actually defined in a more general setting involving Hopf algebras and crossed bimodules. This paper is a continuation of .
Keywords: Braids, Cartan matrices, zonotopes, quantum groups
1. M. Aguiar, Internal categories and quantum groups, Ph.D. Thesis, Cornell University, Ithaca, 1997.
2. M. Aguiar, Braids, q-binomials and quantum groups, Adv. Pure and Appl. Math. 20 (1998) 323–365 .
3. V.G. Drinfeld, Quantum Groups, in: Proc. Int. Cong. Math. Berkeley, 1986, pp. 798-820.
4. J. Goldman and G.C. Rota, On the foundations of combinatorial theory IV. Finite vector spaces and eulerian generating functions, Stud. in Appl. Math. vXLIX (3) (1970) 239–258.
5. J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, 1972.
6. P. Hanlon and R. Stanley, A q-deformation of a trivial symmetric group action, Trans. Amer. Math. Soc. 350 (1998) 4445–4459.
7. J.C. Jantzen, Lectures on Quantum Groups, Graduate Texts in Mathematics, Vol. 6, American Mathematical Society, 1996.
8. M. Jimbo, A q-difference analogue of U(g) and the Yang–Baxter equation, Lett. Math. Phys. 10 (1985) 63–69.
9. S.A. Joni and G.C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61 (1979) 93–139; Joseph P.S. Kung, Ed., Gian-Carlo Rota on Combinatorics: Introductory Papers and Commentaries, Birkhäuser, Boston, 1995 (reprint).
10. A. Joyal and R. Street, Braided tensor categories, Adv. Math. 102 (1993), 20–78.
11. V.G. Kac, Infinite dimensional Lie Algebras, Cambridge University Press, 1990.
12. C. Kassel, Quantum Groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, 1995.
13. G. Lusztig, Introduction to quantum groups, Progr. Math. 110, Birkh¨auser, 1993.
14. S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS 82, 1993.
15. P. Schauenburg, A characterization of the Borel-like subalgebras of quantum enveloping algebras, Comm. in Algebra 24 (1996) 2811–2823.
16. H. Samelson, Notes on Lie Algebras, Second Ed., Universitext, Springer-Verlag, 1990.
17. E.J. Taft, The order of the antipode of a finite-dimensional Hopf algebra, Proc. Nat. Acad. Sci. USA 68 (1971) 2631–2633.
18. A. Varchenko, Bilinear form of a real configuration of hyperplanes, Adv. Math. 97 (1993) 110–144.
19. G.M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, Vol. 152, Spriger- Verlag, 1995.