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

TL;DR

This work constructs common triangular bases for almost all Lie-theoretic cluster algebras (classical and quantum), yielding natural analogs of dual canonical bases and establishing quasi-categorification in the symmetric Cartan setting. It develops a unifying strategy based on cluster operations (freezing, base changes) and a correction technique to propagate bases across related seeds, proving A=U and extending results from quantum unipotent subgroups to broader Lie-theoretic varieties. The authors apply these bases to a wide range of Lie-theoretic objects, including double Bott-Samelson cells, braid varieties, double Bruhat cells, and coordinate rings of groups, Grassmannians, and related varieties, obtaining standard bases, $T$-systems, and monoidal categorifications in ADE. The framework yields a comprehensive, structurally coherent picture linking cluster algebras from Lie theory with monoidal categorifications, geometric interpretations, and explicit bases with positivity properties, significantly broadening the scope of dual-canonical-type phenomena in cluster theory.

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.

Figures (6)

• Figure 6.1: The quiver for $\ddot{B}(\ddot{{\bf t}}(1,2,1,-1,-2,-1))$.
• Figure 6.2: The quiver for a seed of $\Bbbk[G^{w_{0},s_{1}w_{0}}]$, $w_{0}=s_{1}s_{2}s_{1}$
• Figure 6.3: A quiver and an extension
• Figure 7.1: The quiver for $\dot{{\bf t}}(1,2,1,2,1,2)$
• Figure 8.1: Quivers with weights associated with $\ddot{B}$.

Theorems & Definitions (211)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4: Quantized coordinate rings and integral forms
• Remark 1.5: Deformation quantization
• Conjecture 1.6
• Theorem 1.7
• Theorem 1.8: Proposition \ref{['prop:T-systems']} Theorem \ref{['thm:dBS_PBW']}
• Conjecture 1.9
• Theorem 1.10: Theorem \ref{['thm:ADE-braid-categorification']}, Theorem \ref{['thm:ADE-braid-categorification-ddBS']}