Zonotopes, Braids, and Quantum Groups

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 \( {\Bbb N}^r \), 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].