If \(\mathfrak{g}\) is a simple Lie algebra over \(\mathbb C\), \(\mathfrak{n}\) a maximal nilpotent subalgebra of \(\mathfrak{g}\), let \(\mathbf B\) be the canonical basis of \(U_q(\mathfrak{n})\) and \({\mathbf B}^{\ast}\) the dual basis with respect to the natural scalar product in \(U_q(\mathfrak{n})\). Berenstein and Zelevinsky had conjectured that the product \(b_1b_2\) is of the form \(q^mb\), for \(b_1,b_2,b\in{\mathbf B}^{\ast}\) if and only if \(b_1\) and \(b_2\) \(q\)-commute. This would imply that \(b_1^2\) is always of the form \(q^mb\), for \(b_1 \in {\mathbf B}^{\ast}\). Such vectors are called real, otherwise they are called imaginary. The paper shows that there are imaginary vectors except when \(\mathfrak{g}\) if of type \(A_1,A_2,A_3,A_4,B_2\). The author uses this to exhibit an explicit irreducible representation \(V\) for \(U_q(\hat{sl}_N)\) such that \(V\otimes V\) is not irreducible.