Adapted algebras for the Berenstein-Zelevinsky conjecture

Let G be a simply connected semisimple complex Lie group and fix a maximal unipotent subgroup U- of G. Let q be an indeterminate and let beta* denote the dual canonical basis (cf. [19]) of the quantized algebra C-q [U-] of regular functions oil U-. Following [20], fix a Z(greater than or equal to0)(N)-parametrization of this basis, where N = dim U-. In [2], A. Berenstein and A. Zelevinsky conjecture that two elements of beta* q-commute if and only if they are multiplicative, i.e., their product is an element of beta* up to a power of q. To any reduced decomposition (w) over tilde (0) of the longest element of the Weyl group of g, we associate a subalgebra A((w) over tilde0), called adapted algebra, of Cq[U-] such that (1) A((w) over tilde0) is a q-polynomial algebra which equals C-q[U-] up to localization, (2) A((w) over tilde0) is spanned by a subset of beta*, (3) the Berenstein-Zelevinsky conjecture is true Oil A((w) over tilde0). Then we test the conjecture when one element belongs to the q-center of C-q[U-].