Positivity in cluster algebras and its relations with categorifications of cluster algebras and total positivity in semisimple algebraic groups

点击翻译按钮获取中文摘要

In every cluster algebra there is a natural notion of positivity, first noticed by Sherman and Zelevinsky. This project has two main objectives: on one hand investigate this notion in relation with the well–known notion of total positivity in semisimple algebraic groups, in the case when the cluster algebra is the coordinate ring of the corresponding group. On the other hand, our aim is to interpret the positive elements in the language of categorification of the cluster algebra theory. The final goal is to be able to give an explicit, satisfactory answer to the following question: are cluster algebras and their categorifications the right framework to study total positivity?