Analogs of the dual canonical bases for cluster algebras from Lie theory

TL;DR

This paper constructs common triangular bases for quantum cluster algebras from Lie theory, providing analogs of dual canonical bases, and establishes their quasi-categorification in symmetric cases, with applications to quantum unipotent subgroups.

Contribution

It introduces new cluster operations to extend known results, constructs analogs of dual canonical bases, and proves A=U for these quantum cluster algebras, advancing the understanding of their structure.

Findings

Construction of common triangular bases for most quantum cluster algebras from Lie theory

Proof of A=U for these quantum cluster algebras

Monoidal categorifications in type ADE via positive braids

Abstract

We construct common triangular bases for almost all the known (quantum) cluster algebras from Lie theory. These bases provide analogs of the dual canonical bases, long anticipated in cluster theory. In cases where the generalized Cartan matrices are symmetric, we show that these cluster algebras and their bases are quasi-categorified. We base our approach on the combinatorial similarities among cluster algebras from Lie theory. For this purpose, we introduce new cluster operations to propagate structures across different cases, which allow us to extend established results on quantum unipotent subgroups to other such algebras. We also obtain fruitful byproducts. First, we prove A=U for these quantum cluster algebras. Additionally, we discover rich structures of the locally compactified quantum cluster algebras arising from double Bott-Samelson cells, including T-systems, standard bases, and Kazhdan-Lusztig type algorithms. Notably, in type ADE, we obtain their monoidal categorifications via monoidal categories associated with positive braids. As a special case, these categories provide monoidal categorifications of the quantum function algebras in type ADE.