Published paper

Quantum cluster algebras

Arkady Berenstein, Andrei Zelevinsky

MSC 2020

20G4214M1722E46

Published version ↗Published PDF ↗Open DOI ↗arXiv record →

Publication metadata

Journal citation: Advances in Mathematics 195(2), 405–455 (2005)
Publication date: 2005-08-20
DOI: 10.1016/j.aim.2004.08.003 ↗
arXiv counterpart: arXiv:math/0404446v1
MSC 2020: 20G42, 14M17, 22E46
As specified in the publication (MSC 2000): 20G42, 14M17, 22E46 ↗Version-of-record first page (primary 20G42; secondary 14M17, 22E46)All three printed codes carry forward unchanged through the official MSC conversion tables.
Official MSC conversion: MSC 2000 → MSC 2010 ↗MSC 2010 → MSC 2020 ↗

Abstract

The paper introduces quantum deformations of cluster algebras, develops mutation for compatible quantum seeds and based quantum tori, and proves quantum analogues of the Laurent phenomenon and other structural results.

Role in dependence graphs

Proof-critical source

On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modules

This paper is included only for the following marked statements:

Proposition 4.7 and Definitions 4.8, 4.12 · Advances in Mathematics pp. 417–419Mutation of compatible quantum seeds and the associated quantum cluster algebra.
Theorem 5.1 and Corollary 5.2 · Advances in Mathematics pp. 419–420; proof pp. 421–426Quantum Laurent phenomenon and mutation invariance of the upper quantum cluster algebra.

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Version of record · Advances in Mathematics 195 (2005), no. 2, 405–455

Arkady Berenstein and Andrei Zelevinsky. Quantum cluster algebras. Advances in Mathematics 195 (2005), no. 2, 405–455.

Open audited source ↗

Generated August 27, 2026

01Statements4 reported findingsSuspected incorrect statements

The quantum-seed mutation, quantum Laurent phenomenon, exchange-graph, acyclic, and combinatorial compatibility statements inspected in the version of record are supported, including the exact claims used by the focal paper. Appendix Proposition A.2(2), however, incorrectly classifies every two-sided ideal of an integral based quantum torus as generated by its intersection with the center.

Proposition 4.7 and Definitions 4.8, 4.12No issue found

Quantum-seed mutation and the quantum cluster algebra are well defined

Version of record, pages 417–419

The quantum-binomial automorphism formula makes the mutated toric frame independent of the auxiliary sign, the compatibility matrix mutates by the stated congruence, and the second mutation is involutive. Definition 4.12 then generates the quantum cluster algebra from all mutation-equivalent clusters and the invertible frozen variables. These are the exact structural inputs used in the focal cluster application.

Advances in Mathematics version of record ↗

Theorem 5.1 and Corollary 5.2Correct after an apparently sound local repair

The quantum Laurent phenomenon survives the source repairs

Version of record, pages 419–426

The upper quantum algebra is unchanged by one mutation, hence by the full mutation class, and every quantum cluster variable lies in every based quantum torus. The proof's compressed commutative branch was checked against the final BFZ manuscript. The proof uses only Proposition A.2(3), not the false ideal-classification assertion A.2(2); A.2(3) follows independently by writing the quantum torus as a free module over its Laurent-polynomial center and comparing coefficients.

Advances in Mathematics version of record ↗

Appendix Proposition A.2(2)Suspected incorrect as stated

Two-sided ideals need not be generated by their central part over the integral coefficient ring

Version of record, pages 453–454

Let $t=q^{1/2}$ and take the two-dimensional based quantum torus $T=\mathbb Z[t^{\pm1}]\langle X^{\pm1},Y^{\pm1}\rangle/(XY-t^2YX)$ with nondegenerate commutation form. Its center is $Z=\mathbb Z[t^{\pm1}]$ . The ideal $I=(t-1,X+Y)$ is two-sided: modulo $t-1$ , the torus becomes the commutative Laurent ring and $I$ is the inverse image of its proper ideal $(X+Y)$ . Moreover, $I\cap Z=(t-1)$ , whereas $X+Y\notin T(t-1)$ . Thus $I\ne T(I\cap Z)$ , contradicting the claimed bijection. The assertion becomes valid after extending scalars to $\mathbb Q(t)$ , where distinct conjugation eigenvalues can be separated, but that is not the coefficient ring stated in the proposition.

Advances in Mathematics version of record ↗

Theorems 7.3 and 7.5Correct after the same central-divisibility repair

The acyclic conclusions are supported but deliberately compressed

Version of record, pages 431–432

The paper explicitly imports the commutative BFZ arguments and states the noncommutative changes: an ordered cycle identity for standard monomials and replacement of the central exchange factor in the saturation step. The latter again needs only the valid intersection statement A.2(3). No contradictory hypothesis or counterexample was found for the acyclic equality itself.

Advances in Mathematics version of record ↗

02Proofs4 reported findingsSuspected incomplete or incorrect proofs

The marked mutation and Laurent proofs are valid after checking their BFZ branch and replacing the appeal to the appendix by a direct central-coefficient argument. The proof of Appendix Proposition A.2(2) is invalid because two-sided ideals over the integral coefficient ring need not be graded by conjugation eigenvalues. Several acyclic and rank-two passages are also presented only as specified modifications of BFZ rather than written out in full.

Proof of Proposition A.2(2)Suspected incorrect

Conjugation invariance does not force an integral ideal to split into eigenspaces

Version of record, page 454 · sentence asserting that every two-sided ideal is $L^*$ -graded

The proof says every two-sided ideal is graded by the simultaneous conjugation eigenspaces $T_\chi$ . Over $\mathbb Z[t^{\pm1}]$ , the eigenvalue differences are not units, so conjugation invariance cannot project an element onto its separate homogeneous components. The ideal $(t-1,X+Y)$ gives an explicit failure: it is two-sided but contains neither $X$ nor $Y$ . Scalar extension to $\mathbb Q(t)$ repairs the eigenspace argument, while the integral proposition needs a narrower statement.

Advances in Mathematics version of record ↗

Proposition A.2(3) as actually usedProposed repair appears sound

Central coprime intersections have a direct proof

Version of record, pages 423 and 432, referring to Appendix Proposition A.2

Choose representatives for the quotient of the exponent lattice by the kernel of the commutation form. Their monomials form a basis of $T$ as a module over its center $Z$ , which is a Laurent polynomial ring over $\mathbb Z[t^{\pm1}]$ . For relatively prime $z_1,z_2\in Z$ , coefficientwise comparison gives $Tz_1\cap Tz_2=T(z_1Z\cap z_2Z)=Tz_1z_2$ . This is exactly the part required in Proposition 5.7 and Theorem 7.5, so the false general ideal classification does not invalidate those results.

Advances in Mathematics version of record ↗

Proposition 5.7Compressed but checkable

The interacting rank-two case is compressed to BFZ modifications

Version of record, page 423

Case 2 says the proof follows BFZ with the obvious changes imposed by noncommutativity and leaves those changes to the reader. In the needed ordered monomial form, quasi-commutation contributes only powers of $q$ , while the exchange factors used for divisibility are central; the BFZ argument therefore carries over. This is not a contradiction, but it is materially less complete than the detailed calculation in the following Lemma 5.8.

Advances in Mathematics version of record ↗

Theorem 7.5Compressed but repairable

The acyclic proof records only the noncommutative change

Version of record, page 432

The proof refers to BFZ Theorem 1.18 and singles out the replacement of its saturation factor by $P^r_{b_1,+}$ , whose divisibility follows from the valid central-intersection statement above. Because the full induction is not repeated, this audit checked the exact BFZ proof before accepting the transfer. The stated modification is sufficient, but the version of record is not self-contained at this point.

Quantum cluster algebras

Publication metadata

Abstract

Role in dependence graphs

On Lagrangianity of ppp-supports of holonomic DDD-modules and qqq-DDD-modules

Audit summary

Version of record · Advances in Mathematics 195 (2005), no. 2, 405–455

Quantum-seed mutation and the quantum cluster algebra are well defined

The quantum Laurent phenomenon survives the source repairs

Two-sided ideals need not be generated by their central part over the integral coefficient ring

The acyclic conclusions are supported but deliberately compressed

Conjugation invariance does not force an integral ideal to split into eigenspaces

Central coprime intersections have a direct proof

The interacting rank-two case is compressed to BFZ modifications

The acyclic proof records only the noncommutative change

On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modules