Annals of Combinatorics 4 (2000) 433-468

Zonotopes, Braids, and Quantum Groups

Marcelo Aguiar

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 [2] 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 $ {\Bbb N}\hspace{1pt}^r$, 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 $ U_q^0(A)$ 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 $ U_q^0(A)$ 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 [2].

Keywords: Braids, Cartan matrices, zonotopes, quantum groups


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