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Imaginary vectors in the dual canonical basis of U q (𝔫). (English) Zbl 1044.17009
If 𝔤 is a simple Lie algebra over , 𝔫 a maximal nilpotent subalgebra of 𝔤, let 𝐁 be the canonical basis of U q (𝔫) and 𝐁 * the dual basis with respect to the natural scalar product in U q (𝔫). Berenstein and Zelevinsky had conjectured that the product b 1 b 2 is of the form q m b, for b 1 ,b 2 ,b𝐁 * 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 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 (sl ^ N ) such that VV is not irreducible.
17B37Quantum groups and related deformations
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