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Imaginary vectors in the dual canonical basis of ${U}_{q}\left(𝔫\right)$. (English) Zbl 1044.17009
If $𝔤$ is a simple Lie algebra over $ℂ$, $𝔫$ a maximal nilpotent subalgebra of $𝔤$, let $𝐁$ be the canonical basis of ${U}_{q}\left(𝔫\right)$ and ${𝐁}^{*}$ the dual basis with respect to the natural scalar product in ${U}_{q}\left(𝔫\right)$. Berenstein and Zelevinsky had conjectured that the product ${b}_{1}{b}_{2}$ is of the form ${q}^{m}b$, for ${b}_{1},{b}_{2},b\in {𝐁}^{*}$ if and only if ${b}_{1}$ and ${b}_{2}$ $q$-commute. This would imply that ${b}_{1}^{2}$ is always of the form ${q}^{m}b$, for ${b}_{1}\in {𝐁}^{*}$. Such vectors are called real, otherwise they are called imaginary. The paper shows that there are imaginary vectors except when $𝔤$ 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 ${U}_{q}\left({\stackrel{^}{sl}}_{N}\right)$ such that $V\otimes V$ is not irreducible.
##### MSC:
 17B37 Quantum groups and related deformations
canonical basis
##### References:
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