Bases for cluster algebras

• Grant number: 18/24398-7
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: January 07, 2019
• End date: January 20, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Alexandre Grichkov
• Grantee: Alexandre Grichkov
• Visiting researcher: Alfredo Najera Chavez
• Visiting researcher institution: Universidad Nacional Autónoma de México (UNAM), Mexico
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

We would like to study the cluster algebras appearing in the context of quantum groups. For such algebras we know that the quantum cluster monomial are properly contained in the upper global basis. The elements of mid(B) and up(B) contain properly the cluster monomials. In other words, the insight of the paper [9]( M. Gross, P. Hacking, S. Keel and M. Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2) 2018, Pages 497-608. ) allows us to see beyond the cluster monomials and it is therefore natural to ask if it is possible to capture the whole structure of the upper global basis. For this one would have to extend rst the approach of [9] to the quantum world and then link this to the monoidal categorication picture. We reduce this to the following stages 1) prove that the theta basis is a Khovanskii basis. 2) Prove that a Khovanskii basis uniquely extends to a basis of a quantum cluster algebra. 3) Prove that the upper global basis is the unique extension of the theta basis when considered as a Khovanskii basis. (AU)