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Imaginary vectors in the dual canonical basis of $U_q({\germ n})$. (English) Zbl 1044.17009
If $\frak{g}$ is a simple Lie algebra over $\Bbb C$, $\germ{n}$ a maximal nilpotent subalgebra of $\frak{g}$, let $\bold B$ be the canonical basis of $U_q(\frak{n})$ and ${\bold B}^{\ast}$ the dual basis with respect to the natural scalar product in $U_q(\frak{n})$. Berenstein and Zelevinsky had conjectured that the product $b_1b_2$ is of the form $q^mb$, for $b_1,b_2,b\in{\bold B}^{\ast}$ 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 $b_1 \in {\bold B}^{\ast}$. Such vectors are called {\it real}, otherwise they are called {\it imaginary}. The paper shows that there are imaginary vectors except when $\frak{g}$ 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.

MSC:
17B37Quantum groups and related deformations
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References:
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