Multiplicative properties of dual canonical bases of quantum groups
Let \(C=(a_{i,j})_{i,j \in I}\) be a Cartan matrix of finite simply laced type, let \(\mathcal U\) be the quantized enveloping algebra over \(\mathbb Q(v)\) associated with \(C\), and let \(\mathcal U^+\) be its positive part. Let \(\mathcal B\) be the canonical basis of \(\mathcal U^+\), and \(\mathcal B^*\) be the dual of \(\mathcal B\) with respect to the standard inner product of \(\mathcal U^+\). Two elements of \(\mathcal B^*\) are said to quasi-commute if they commute up to a twist (i.e. a power of \(v\)) and to be multiplicative if their product equals an element of \(\mathcal B^*\) up to a twist. In [Adv. Sov. Math. 16, 51-89 (1993; Zbl 0794.17007)], \textit{A. Berenstein} and \textit{A. Zelevinsky} studied multiplicative properties of \(\mathcal B^*\) for type \(A\), and conjectured that two elements of \(\mathcal B^*\) quasi-commute if and only if they are multiplicative. This conjecture was verified in that paper for types \(A_2\) and \(A_3\). The aim of this paper is to develop some further tools for the study of \(\mathcal B^*\) and to prove parts of the conjecture. The author proves that two quasi-commuting elements of \(\mathcal B^*\) are multiplicative provided one of the elements is a ``small'' quantum minor. The main working tool is Ringel's Hall algebra approach.
quasi-commuting multiplicative representations of quivers
scientific article; zbMATH DE number 1251298
scientific article