  Let $n$ be the maximal nilpotent subalgebra of a simple complex Lie algebra
$g$. We introduce the notion of imaginary vector in the dual canonical basis of
$U_q(n)$, and we give examples of such vectors for types $A_n (n\ge 5)$, $B_n
(n\ge 3)$, $C_n (n\ge 3)$, $D_n (n\ge 4)$, and all exceptional types. This
disproves a conjecture of Berenstein and Zelevinsky about $q$-commuting
products of vectors of the dual canonical basis. It also shows the existence of
finite-dimensional irreducible representations of quantum affine algebras whose
tensor square is not irreducible.
