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Partially compactified quantum cluster structures on simple algebraic groups and the full Berenstein--Zelevinsky conjecture

Fan Qin, Milen Yakimov

TL;DR

This work provides a general method to lift quantum cluster structures from localizations to partially compactified quantum cluster algebras, enabling constructions on quantized coordinate rings $R_q[G]$ of complex simple groups. By analyzing quantum double Bruhat cells through $S_w^+$ and $S_u^-$ and using Berenstein--Zelevinsky seeds, the authors prove the full Berenstein--Zelevinsky conjecture: all seeds associated to signed words for $(u,w)$ are mutation-connected and yield the same quantum cluster structure. They furthermore develop a principled approach (Theorem C) to recover a noncommutative algebra from a localization, securing a bridge between localized and globally defined cluster structures. The combination of codimension-2 arguments, canonical and triangular bases, and mutation connectivity yields a robust framework for constructing partially compactified quantum cluster algebras on quantized coordinate rings, with direct implications for canonical basis theory and the geometry of double Bruhat cells.

Abstract

The construction of partially compactified cluster algebras on coordinate rings is handled by using codimension 2 arguments on cluster covers. An analog of this in the quantum situation is highly desirable but has not been found yet. In this paper, we present a general method for the construction of partially compactified quantum cluster algebra structures on quantized coordinate rings from that of quantum cluster algebra structures on localizations. As an application, we construct a partially compactified quantum cluster algebra structure on the quantized coordinate ring of every connected, simply connected complex simple algebraic group. Along the way, we settle in full the Berenstein--Zelevinsky conjecture that all quantum double Bruhat cells have quantum cluster algebra structures associated to seeds indexed by arbitrary signed words, and prove that all such seeds are linked to each by mutations.

Partially compactified quantum cluster structures on simple algebraic groups and the full Berenstein--Zelevinsky conjecture

TL;DR

This work provides a general method to lift quantum cluster structures from localizations to partially compactified quantum cluster algebras, enabling constructions on quantized coordinate rings of complex simple groups. By analyzing quantum double Bruhat cells through and and using Berenstein--Zelevinsky seeds, the authors prove the full Berenstein--Zelevinsky conjecture: all seeds associated to signed words for are mutation-connected and yield the same quantum cluster structure. They furthermore develop a principled approach (Theorem C) to recover a noncommutative algebra from a localization, securing a bridge between localized and globally defined cluster structures. The combination of codimension-2 arguments, canonical and triangular bases, and mutation connectivity yields a robust framework for constructing partially compactified quantum cluster algebras on quantized coordinate rings, with direct implications for canonical basis theory and the geometry of double Bruhat cells.

Abstract

The construction of partially compactified cluster algebras on coordinate rings is handled by using codimension 2 arguments on cluster covers. An analog of this in the quantum situation is highly desirable but has not been found yet. In this paper, we present a general method for the construction of partially compactified quantum cluster algebra structures on quantized coordinate rings from that of quantum cluster algebra structures on localizations. As an application, we construct a partially compactified quantum cluster algebra structure on the quantized coordinate ring of every connected, simply connected complex simple algebraic group. Along the way, we settle in full the Berenstein--Zelevinsky conjecture that all quantum double Bruhat cells have quantum cluster algebra structures associated to seeds indexed by arbitrary signed words, and prove that all such seeds are linked to each by mutations.

Paper Structure

This paper contains 24 sections, 35 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Partially compactified quantum cluster structures on quantized coordinate rings
  3. Statements of main results
  4. Classical and quantum cluster algebras
  5. Basics of cluster algebras
  6. Quantum cluster algebras associated with signed words
  7. Quantum groups
  8. Semisimple Lie algebras
  9. Quantum groups and quantum function algebras
  10. Quantum double Bruhat cells
  11. Quantum Schubert cells
  12. Dual canonical basis
  13. Cluster algebra structures on quantum double Bruhat cells
  14. Quantum seeds on reduced double Bruhat cells
  15. Quantum seeds on double Bruhat cells
  16. ...and 9 more sections

Key Result

Lemma 2.4

For any $k\in I_{\operatorname{uf}}$ and $z\in \mathcal{LP}({\bf s})\cap \mathcal{LP}(\mu_k {\bf s})$, we have $\nu_j^{\bf s}(z)=\nu_j^{\mu_k {\bf s}}(z)$ whenever $j\neq k$.

Theorems & Definitions (78)

  • Definition 2.1: fomin2002cluster
  • Definition 2.2: BerensteinZelevinsky05
  • Definition 2.3
  • Lemma 2.4: qin2023analogs
  • Definition 2.5
  • Definition 2.6: Dominance order qin2017triangularcerulli2015caldero
  • Definition 2.7: qin2017triangular
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10: FominZelevinsky07Tran09DerksenWeymanZelevinsky09gross2018canonical
  • ...and 68 more