Notes: from dual canonical bases to triangular bases of quantum cluster algebras

TL;DR

This paper demonstrates that the dual canonical basis of quantum unipotent subgroups associated with symmetrizable Kac-Moody algebras forms the common triangular basis of the related quantum cluster algebra, encompassing all quantum cluster monomials.

Contribution

It establishes that the dual canonical basis is the triangular basis of quantum cluster algebras for symmetrizable Kac-Moody cases, extending prior results.

Findings

Dual canonical basis equals the triangular basis in quantum cluster algebras.

The basis includes all quantum cluster monomials.

Extension of previous results to broader algebra classes.

Abstract

These notes are mainly based on arXiv:2003.13674 and a series of talks given in the workshop CARTEA. For any symmetrizable Kac-Moody algebra $\mathfrak{g}$ and any Weyl group element $w$, the corresponding quantum unipotent subgroup $A_{q}[N_{-}(w)]$ possesses the dual canonical basis $\mathbf{B}^*$. We show that the dual canonical basis is the (common) triangular basis of the quantum cluster algebra. Consequently, we deduce that the basis contains all quantum cluster monomials, extending previous results by the author and Kang-Kashiwara-Kim-Oh.