Braid Group Actions and Tensor Products

We define an action of the braid group of a simple Lie algebra on the space of imaginary roots in the corresponding quantum affine algebra. We then use this action to determine an explicit condition for a tensor product of arbitrary irreducible finite–dimensional representations is cyclic. This allows us to determine the set of points at which the corresponding R–matrix has a zero. 0. Introduction In this paper we give a sufficient condition for the tensor product of irreducible finite– dimensional representations of quantum affine algebras to be cyclic. This condition is obtained by defining a braid group action on the imaginary root vectors. We make the condition explicit in Section 5 and see that it is a natural generalization of the condition in [CP1] given in the sl2 case. This allows us for instance, to determine the finite set of points at which a tensor product of fundamental representations can fail to be cyclic. Our result proves a generalization of a recent result of Kashiwara [K], [VV], [N1]. Further, it also establishes a conjecture stated in [K], [HKOTY]. We describe our results in some detail. Let g be a complex simple finite–dimensional Lie algebra of rank n, and let Uq be the quantized untwisted affine algebra over C(q) associated to g. For every n–tuple π = (π1, · · · , πn) of polynomials with coefficients in C(q)[u] and with constant term one, there exists a unique (up to isomorphism) irreducible finite–dimensional representation V (π) of Uq. For each element w in the Weyl group W of g, let vwπ be the extremal vector defined in [K]. In this paper we compute the action of the imaginary root vectors in Uq on the elements vwπ . To do this we define in Section 2 an action of the braid group B of g on elements of (C(q)[[u]]) and prove that the eigenvalue of vwπ is the element Tw(π) where T : W → B is the canonical section defined in [Bo]. Let π(u) ∈ C(q)[u] be a polynomial that splits in C(q). Any such polynomial π can be written uniquely as a product