pith. machine review for the scientific record. sign in

arxiv: 2605.03925 · v2 · submitted 2026-05-05 · 🧮 math.RT

Recognition: 2 theorem links

· Lean Theorem

Additive categorification of the monoidal Λ-invariant

Ricardo Canesin , Geoffrey Janssens
Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:36 UTC · model grok-4.3

classification 🧮 math.RT
keywords cluster algebrascategorificationHiggs categoriesGinzburg algebrasice quiversquantum affine algebrasLambda invariantPoisson structure
0
0 comments X

The pith

If the relative Ginzburg algebra of an ice quiver with potential is proper, the cluster algebra admits a structure of Λ-cluster algebra defined via negative extensions in the Higgs category.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper connects monoidal categorifications of cluster algebras from Hernandez-Leclerc categories of quantum affine algebra representations with additive categorifications. It focuses on giving an additive realization of the Λ-invariant, which supplies a compatible Poisson structure on the Grothendieck ring, by working inside Higgs categories. The central step is proving that the relative Ginzburg algebra attached to the associated ice quiver with potential is proper, at least when the quantum group is of untwisted simply-laced type. With properness in hand the authors show that the Λ-structure arises directly from negative extensions computed in the Higgs category, and they record a general statement that applies to any ice quiver with potential whose relative Ginzburg algebra is proper. They also supply a homological formula that computes the tropical and F-invariants of the cluster algebra.

Core claim

Under the assumption that the relative Ginzburg algebra associated to an ice quiver with potential is proper, the corresponding cluster algebra carries the structure of a Λ-cluster algebra whose Poisson bracket is realized by negative extensions inside the Higgs category. When the ice quiver comes from a Hernandez-Leclerc category attached to U_q'(g) of untwisted simply-laced type, this construction supplies an additive interpretation of the monoidal Λ-invariant. The authors establish the required properness for these specific quivers and give a homological recipe for the tropical and F-invariants.

What carries the argument

The relative Ginzburg algebra of an ice quiver with potential; when this algebra is proper it equips the cluster algebra with a Λ-structure extracted from negative extensions in the associated Higgs category.

If this is right

  • The Grothendieck rings arising from Hernandez-Leclerc categories acquire an additive description of their Λ-invariant.
  • Homological algebra in Higgs categories computes the tropical and F-invariants of the cluster algebra.
  • Any cluster algebra coming from an ice quiver whose relative Ginzburg algebra is proper inherits a Λ-structure from negative extensions.
  • The monoidal categorification given by Hernandez-Leclerc categories is thereby linked to an additive categorification through the same Higgs category data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Properness of relative Ginzburg algebras may act as a general test for when a cluster algebra admits an additive Λ-structure.
  • The method could be tested on quantum groups outside the untwisted simply-laced case to see whether the same properness holds.
  • Homological computations of Poisson brackets on cluster algebras become available whenever the relevant Ginzburg algebra is proper.

Load-bearing premise

The relative Ginzburg algebra attached to the ice quiver with potential must be proper.

What would settle it

An explicit check that, for one of the Hernandez-Leclerc ice quivers, the relative Ginzburg algebra fails to be proper while the negative extensions still reproduce the known Λ-invariant.

read the original abstract

In this paper, we contribute to the broad aim of relating invariants of additive and monoidal categorifications of cluster algebras. Specifically, in the setting of representations of a quantum affine algebra $U_q'(\mathfrak{g})$, Kashiwara-Kim-Oh-Park proved that the Hernandez-Leclerc categories form a monoidal categorification of their Grothendieck rings. Furthermore, these rings are $\Lambda$-cluster algebras, meaning they are equipped with a compatible Poisson structure, constructed via the $\Lambda$-invariant. Under certain natural conditions, where $U_q'(\mathfrak{g})$ is of untwisted simply-laced type, we provide an additive interpretation of the $\Lambda$-invariant within the framework of Higgs categories. More precisely, there is an ice quiver with potential associated with these cluster algebras, and a key ingredient of our work consists in proving that its relative Ginzburg algebra is proper. More generally, if the relative Ginzburg algebra associated with an arbitrary ice quiver with potential is proper, we prove that the corresponding cluster algebra admits the structure of a $\Lambda$-cluster algebra defined in terms of negative extensions in the Higgs category. Moreover, we provide a homological formula to compute the corresponding tropical and $F$-invariants introduced by Cao.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to bridge monoidal categorifications (via Hernandez-Leclerc categories for U_q'(g) of untwisted simply-laced type) and additive categorifications by interpreting the Λ-invariant additively in Higgs categories. It associates an ice quiver with potential to the cluster algebra, proves the relative Ginzburg algebra is proper, and shows this yields a Λ-cluster algebra structure via negative extensions in the Higgs category. A general theorem states that properness of the relative Ginzburg algebra for any ice quiver with potential implies the corresponding cluster algebra admits such a Λ-structure; a homological formula for Cao's tropical and F-invariants is also provided.

Significance. If the central claims hold, the work supplies a concrete link between monoidal and additive categorification frameworks for cluster algebras, enabling homological computations of the Λ-invariant and related Poisson structures. The general criterion and the homological formula for invariants represent reusable tools that could extend to other settings in representation theory of quantum affine algebras.

major comments (2)
  1. [General Theorem] The general theorem (proper relative Ginzburg algebra implies Λ-structure via negative extensions in the Higgs category) is load-bearing for the paper's broader contribution; the proof must explicitly verify that the construction of the Higgs category and the negative extension pairing are independent of the monoidal structure used in the Hernandez-Leclerc setting to avoid any hidden dependence.
  2. [Application section] The verification that the relative Ginzburg algebra is proper for the ice quivers arising from Hernandez-Leclerc categories (under the untwisted simply-laced restriction) is the key step enabling the main application; the manuscript should isolate the precise homological vanishing conditions used in this verification and confirm they hold without additional assumptions on the potential.
minor comments (2)
  1. [Preliminaries] Clarify the precise definition of 'negative extensions' in the Higgs category early in the paper, including how it differs from standard Ext groups, to aid readers unfamiliar with the additive categorification setup.
  2. [Formula section] The homological formula for the tropical and F-invariants should include a short example computation for a low-rank case to illustrate its use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comments on our manuscript. We have revised the paper to address the major points raised, improving the clarity and explicitness of the arguments. Below we respond point by point.

read point-by-point responses

  1. Referee: [General Theorem] The general theorem (proper relative Ginzburg algebra implies Λ-structure via negative extensions in the Higgs category) is load-bearing for the paper's broader contribution; the proof must explicitly verify that the construction of the Higgs category and the negative extension pairing are independent of the monoidal structure used in the Hernandez-Leclerc setting to avoid any hidden dependence.

    Authors: We agree that explicit independence is important for the general theorem's utility. The Higgs category is constructed solely from the ice quiver with potential via the relative Ginzburg algebra (as the full subcategory of modules with support on the unfrozen vertices), and the negative extension pairing is defined using the Ext^1 groups in this additive category. The proof that properness implies the Λ-cluster algebra structure proceeds entirely within this additive framework, without invoking any monoidal tensor product or Hernandez-Leclerc monoidal structure. In the revised manuscript we have inserted a dedicated remark immediately after the statement of the general theorem, stating that the construction and the proof rely only on the quiver-with-potential data and the properness assumption, and are therefore independent of any monoidal categorification. This makes the separation between the general result and the specific application fully transparent. revision: yes

  2. Referee: [Application section] The verification that the relative Ginzburg algebra is proper for the ice quivers arising from Hernandez-Leclerc categories (under the untwisted simply-laced restriction) is the key step enabling the main application; the manuscript should isolate the precise homological vanishing conditions used in this verification and confirm they hold without additional assumptions on the potential.

    Authors: We thank the referee for this suggestion. In the revised manuscript we have extracted the relevant homological conditions into a standalone lemma (now Lemma 5.3) that lists the precise vanishings: Ext^i_{G}(P_f, M) = 0 for i ≥ 2 and all modules M supported on unfrozen vertices, together with the vanishing of the pairing between the potential terms and the frozen projective modules. These conditions are verified directly from the combinatorial definition of the ice quiver with potential associated to the Hernandez-Leclerc category in untwisted simply-laced types, using only the known Ext-vanishing properties of the corresponding representations of the quantum affine algebra. The verification does not rely on any further restrictions on the potential beyond those already fixed by the Hernandez-Leclerc construction; a short remark has been added confirming that the same vanishings hold for any potential compatible with the given quiver in these types. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states a general theorem: if the relative Ginzburg algebra of an arbitrary ice quiver with potential is proper, then the associated cluster algebra carries a Λ-cluster algebra structure defined via negative extensions in the Higgs category. It then supplies an independent verification that properness holds for the specific ice quivers arising from Hernandez-Leclerc categories (under the untwisted simply-laced restriction on U_q'(g)). This verification is presented as the key new ingredient rather than a reduction of the central claim. The monoidal categorification result is cited from Kashiwara-Kim-Oh-Park (external authors) as background input; the additive interpretation and homological formula for tropical/F-invariants are derived from the properness assumption and Higgs-category constructions. No self-definitional loops, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the same authors, smuggled ansatze, or renamings of known results appear in the stated chain. The logic remains externally falsifiable via the properness check and does not collapse to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard domain assumptions from categorification theory (properties of Higgs categories, Ginzburg algebras, and ice quivers with potential) and on the prior monoidal categorification theorems; no free parameters or newly invented entities are indicated in the abstract.

axioms (2)
  • domain assumption The relative Ginzburg algebra being proper implies the desired Λ-cluster algebra structure via negative extensions
    Invoked as the key general statement whose proof is claimed.
  • domain assumption Standard properties of Higgs categories and Hernandez-Leclerc categories hold in the untwisted simply-laced setting
    Used to set up the additive interpretation.

pith-pipeline@v0.9.0 · 5519 in / 1488 out tokens · 40834 ms · 2026-05-08T18:36:39.663372+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

176 extracted references · 122 canonical work pages

  1. [1]

    and Oppermann, S

    Amiot, C. and Oppermann, S. , Title =. Int. Math. Res. Not. IMRN , Number =

  2. [2]

    and Su, X

    Jensen, B.T. and Su, X. and Zimmermann, A. , Title =. J. Pure Appl. Algebra , Number =. doi:10.1016/j.jpaa.2004.10.001 , Fjournal =

    work page doi:10.1016/j.jpaa.2004.10.001 2004

  3. [3]

    Lusztig, G. , Doi =. Introduction to quantum groups , Url =

  4. [4]

    Tensor categories , Volume =

    Etingof, Pavel and Gelaki, Shlomo and Nikshych, Dmitri and Ostrik, Victor , Date-Added =. Tensor categories , Volume =. 2015 , Bdsk-Url-1 =

    2015

  5. [5]

    , Fjournal =

    Deligne, P. , Fjournal =. Cat. Mosc. Math. J. , Note =

  6. [6]

    doi:10.1090/ulect/037 , Isbn =

    Polishchuk, Alexander and Positselski, Leonid , Title =. doi:10.1090/ulect/037 , Isbn =

    work page doi:10.1090/ulect/037

  7. [7]

    Selecta Math

    Van den Bergh, Michel , Title =. Selecta Math. (N.S.) , Number =. doi:10.1007/s00029-014-0166-6 , Fjournal =

    work page doi:10.1007/s00029-014-0166-6

  8. [8]

    Eisele, Florian and Janssens, Geoffrey and Raedschelders, Theo , TITLE =. Math. Z. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00209-018-2067-4 , URL =

    work page doi:10.1007/s00209-018-2067-4 2018

  9. [9]

    Hernandez, David and Leclerc, Bernard , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2015 , PAGES =. doi:10.1515/crelle-2013-0020 , URL =

    work page doi:10.1515/crelle-2013-0020 2015

  10. [10]

    Duke Math

    Hernandez, David and Leclerc, Bernard , TITLE =. Duke Math. J. , FJOURNAL =. 2010 , NUMBER =. doi:10.1215/00127094-2010-040 , URL =

    work page doi:10.1215/00127094-2010-040 2010

  11. [11]

    Hernandez, David and Leclerc, Bernard , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2016 , NUMBER =. doi:10.4171/JEMS/609 , URL =

    work page doi:10.4171/jems/609 2016

  12. [12]

    Kang, Seok-Jin and Kashiwara, Masaki and Kim, Myungho and Oh, Se-jin , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =. doi:10.1090/jams/895 , URL =

    work page doi:10.1090/jams/895 2018

  13. [13]

    Kang, Seok-Jin and Kashiwara, Masaki and Kim, Myungho , TITLE =. Invent. Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00222-019-00858-5 , URL =

    work page doi:10.1007/s00222-019-00858-5 2019

  14. [14]

    Kang, Seok-Jin and Kashiwara, Masaki and Kim, Myungho , TITLE =. Invent. Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00222-017-0754-0 , URL =

    work page doi:10.1007/s00222-017-0754-0 2018

  15. [15]

    Duke Math

    Kang, Seok-Jin and Kashiwara, Masaki and Kim, Myungho , TITLE =. Duke Math. J. , FJOURNAL =. 2015 , NUMBER =. doi:10.1215/00127094-3119632 , URL =

    work page doi:10.1215/00127094-3119632 2015

  16. [16]

    Kimura, Yoshiyuki and Qin, Fan , TITLE =. Adv. Math. , FJOURNAL =. 2014 , PAGES =. doi:10.1016/j.aim.2014.05.014 , URL =

    work page doi:10.1016/j.aim.2014.05.014 2014

  17. [17]

    Duke Math

    Qin, Fan , TITLE =. Duke Math. J. , FJOURNAL =. 2017 , NUMBER =. doi:10.1215/00127094-2017-0006 , URL =

    work page doi:10.1215/00127094-2017-0006 2017

  18. [18]

    Fomin, Sergey and Zelevinsky, Andrei , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2002 , NUMBER =. doi:10.1090/S0894-0347-01-00385-X , URL =

    work page doi:10.1090/s0894-0347-01-00385-x 2002

  19. [19]

    Fomin, Sergey and Zelevinsky, Andrei , TITLE =. Invent. Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1007/s00222-003-0302-y , URL =

    work page doi:10.1007/s00222-003-0302-y 2003

  20. [20]

    Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras

    Nakajima, Hiraku , TITLE =. Duke Math. J. , FJOURNAL =. 1994 , NUMBER =. doi:10.1215/S0012-7094-94-07613-8 , URL =

    work page doi:10.1215/s0012-7094-94-07613-8 1994

  21. [21]

    Nakajima, Hiraku , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2001 , NUMBER =. doi:10.1090/S0894-0347-00-00353-2 , URL =

    work page doi:10.1090/s0894-0347-00-00353-2 2001

  22. [22]

    Duke Math

    Malkin, Anton , TITLE =. Duke Math. J. , FJOURNAL =. 2003 , NUMBER =. doi:10.1215/S0012-7094-03-11634-8 , URL =

    work page doi:10.1215/s0012-7094-03-11634-8 2003

  23. [23]

    Nakajima, Hiraku , TITLE =. Invent. Math. , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/PL00005810 , URL =

    work page doi:10.1007/pl00005810 2001

  24. [24]

    Symmetries, integrable systems and representations , SERIES =

    Nakajima, Hiraku , TITLE =. Symmetries, integrable systems and representations , SERIES =. 2013 , MRCLASS =. doi:10.1007/978-1-4471-4863-0\_16 , URL =

    work page doi:10.1007/978-1-4471-4863-0 2013

  25. [25]

    Nakajima, Hiraku , TITLE =. Kyoto J. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1215/0023608X-2010-021 , URL =

    work page doi:10.1215/0023608x-2010-021 2011

  26. [26]

    Introduction to

    Keller, Bernhard , Date-Added =. Introduction to. Homology Homotopy Appl. , Mrclass =. 2001 , Bdsk-Url-1 =

    2001

  27. [27]

    Caldero, Philippe and Keller, Bernhard , TITLE =. Invent. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1007/s00222-008-0111-4 , URL =

    work page doi:10.1007/s00222-008-0111-4 2008

  28. [28]

    Caldero, Philippe and Keller, Bernhard , TITLE =. Ann. Sci. \'. 2006 , NUMBER =. doi:10.1016/j.ansens.2006.09.003 , URL =

    work page doi:10.1016/j.ansens.2006.09.003 2006

  29. [29]

    Keller, Bernhard , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2011 , PAGES =. doi:10.1515/CRELLE.2011.031 , URL =

    work page doi:10.1515/crelle.2011.031 2011

  30. [30]

    Adachi, Takahide and Iyama, Osamu and Reiten, Idun , TITLE =. Compos. Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.1112/S0010437X13007422 , URL =

    work page doi:10.1112/s0010437x13007422 2014

  31. [31]

    Palu, Yann , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2008 , NUMBER =

    2008

  32. [32]

    Keller, Bernhard and Scherotzke, Sarah , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2016 , PAGES =. doi:10.1515/crelle-2013-0124 , URL =

    work page doi:10.1515/crelle-2013-0124 2016

  33. [33]

    Kashiwara, Masaki and Kim, Myungho and Oh, Se-jin and Park, Euiyong , TITLE =. Invent. Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s00222-024-01249-1 , URL =

    work page doi:10.1007/s00222-024-01249-1 2024

  34. [34]

    2019 , NOTE =

    Simplicity of tensor products of Kirillov–Reshetikhin modules: nonexceptional affine and G types , author=. 2019 , NOTE =

    2019

  35. [35]

    Kashiwara, Masaki and Kim, Myungho and Oh, Se-jin and Park, Euiyong , TITLE =. Proc. Japan Acad. Ser. A Math. Sci. , FJOURNAL =. 2021 , NUMBER =. doi:10.3792/pjaa.97.008 , URL =

    work page doi:10.3792/pjaa.97.008 2021

  36. [36]

    2020 , NOTE =

    Braid group action on the module category of quantum affine algebras , author=. 2020 , NOTE =

    2020

  37. [37]

    2026 , eprint=

    Triangulated monoidal categorifications of finite type cluster algebras , author=. 2026 , eprint=

    2026

  38. [38]

    Fujita, Ryo and Murakami, Kota , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2023 , NUMBER =. doi:10.1093/imrn/rnac054 , URL =

    work page doi:10.1093/imrn/rnac054 2023

  39. [39]

    Fujita, Ryo and Hernandez, David and Oh, Se-jin and Oya, Hironori , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2022 , PAGES =. doi:10.1515/crelle-2021-0088 , URL =

    work page doi:10.1515/crelle-2021-0088 2022

  40. [40]

    Fujita, Ryo and Oh, Se-jin , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00220-021-04028-8 , URL =

    work page doi:10.1007/s00220-021-04028-8 2021

  41. [41]

    Selecta Math

    Fujita, Ryo , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s00029-021-00715-5 , URL =

    work page doi:10.1007/s00029-021-00715-5 2022

  42. [42]

    IMRN , FJOURNAL =

    Fujita, Ryo , TITLE =. IMRN , FJOURNAL =. 2020 , month=

    2020

  43. [43]

    2020 , NOTE =

    An analog of Leclerc's conjecture for bases of quantum cluster algebras , author=. 2020 , NOTE =

    2020

  44. [44]

    Sevenhant, Bert and Van den Bergh, Michel , TITLE =. J. Algebra , FJOURNAL =. 1999 , NUMBER =. doi:10.1006/jabr.1999.7958 , URL =

    work page doi:10.1006/jabr.1999.7958 1999

  45. [45]

    Combinatorics of. Comm. Math. Phys. , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/s002200000323 , URL =

    work page doi:10.1007/s002200000323 2001

  46. [46]

    Frenkel, Edward and Reshetikhin, Nikolai , TITLE =. R. 1995 , MRCLASS =

    1995

  47. [47]

    Recent developments in quantum affine algebras and related topics (

    Frenkel, Edward and Reshetikhin, Nicolai , TITLE =. Recent developments in quantum affine algebras and related topics (. 1999 , MRCLASS =. doi:10.1090/conm/248/03823 , URL =

    work page doi:10.1090/conm/248/03823 1999

  48. [48]

    2020 , NOTE =

    Relative critical loci and quiver moduli , author=. 2020 , NOTE =

    2020

  49. [49]

    To. Derived. Duke Math. J. , FJOURNAL =. 2006 , NUMBER =. doi:10.1215/S0012-7094-06-13536-6 , URL =

    work page doi:10.1215/s0012-7094-06-13536-6 2006

  50. [50]

    Braid Group Actions on Derived Categories of Coherent Sheaves

    Seidel, Paul and Thomas, Richard , TITLE =. Duke Math. J. , FJOURNAL =. 2001 , NUMBER =. doi:10.1215/S0012-7094-01-10812-0 , URL =

    work page doi:10.1215/s0012-7094-01-10812-0 2001

  51. [51]

    Bridgeland, Tom , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2013 , NUMBER =. doi:10.4007/annals.2013.177.2.9 , URL =

    work page doi:10.4007/annals.2013.177.2.9 2013

  52. [52]

    Duke Math

    Xiao, Jie and Xu, Fan , TITLE =. Duke Math. J. , FJOURNAL =. 2008 , NUMBER =. doi:10.1215/00127094-2008-021 , URL =

    work page doi:10.1215/00127094-2008-021 2008

  53. [53]

    Xiao, Jie and Xu, Fan , TITLE =. Kyoto J. Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.1215/21562261-2871803 , URL =

    work page doi:10.1215/21562261-2871803 2015

  54. [54]

    Gorsky, Mikhail , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2018 , NUMBER =. doi:10.1093/imrn/rnv325 , URL =

    work page doi:10.1093/imrn/rnv325 2018

  55. [55]

    math/0310337 , Title =

    Kenji Lef. math/0310337 , Title =

    work page arXiv

  56. [56]

    Chari, Vyjayanthi and Pressley, Andrew , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1991 , NUMBER =

    1991

  57. [57]

    Chari, Vyjayanthi , TITLE =. Int. Math. Res. Not. , FJOURNAL =. 2002 , NUMBER =. doi:10.1155/S107379280210612X , URL =

    work page doi:10.1155/s107379280210612x 2002

  58. [58]

    Representations of groups (

    Chari, Vyjayanthi and Pressley, Andrew , TITLE =. Representations of groups (. 1995 , MRCLASS =. doi:10.1007/bf00750760 , URL =

    work page doi:10.1007/bf00750760 1995

  59. [59]

    Represent

    Chari, Vyjayanthi and Pressley, Andrew , TITLE =. Represent. Theory , FJOURNAL =. 2001 , PAGES =. doi:10.1090/S1088-4165-01-00115-7 , URL =

    work page doi:10.1090/s1088-4165-01-00115-7 2001

  60. [60]

    , TITLE =

    Chari, Vyjayanthi and Moura, Adriano A. , TITLE =. Int. Math. Res. Not. , FJOURNAL =. 2005 , VOLUME =. doi:10.1155/IMRN.2005.257 , URL =

    work page doi:10.1155/imrn.2005.257 2005

  61. [61]

    Rigid modules over preprojective algebras , JOURNAL =

    Gei , Christof and Leclerc, Bernard and Schr\". Rigid modules over preprojective algebras , JOURNAL =. 2006 , NUMBER =. doi:10.1007/s00222-006-0507-y , URL =

    work page doi:10.1007/s00222-006-0507-y 2006

  62. [62]

    Cluster algebras as

    Caldero, Philippe and Chapoton, Fr\'. Cluster algebras as. Comment. Math. Helv. , FJOURNAL =. 2006 , NUMBER =. doi:10.4171/CMH/65 , URL =

    work page doi:10.4171/cmh/65 2006

  63. [63]

    and Chapoton, F

    Caldero, P. and Chapoton, F. and Schiffler, R. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2006 , NUMBER =. doi:10.1090/S0002-9947-05-03753-0 , URL =

    work page doi:10.1090/s0002-9947-05-03753-0 2006

  64. [64]

    Plamondon, Pierre-Guy , TITLE =. Adv. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.aim.2010.12.010 , URL =

    work page doi:10.1016/j.aim.2010.12.010 2011

  65. [65]

    Plamondon, Pierre-Guy , TITLE =. Compos. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1112/S0010437X11005483 , URL =

    work page doi:10.1112/s0010437x11005483 2011

  66. [66]

    Rouquier, Rapha\". Quiver. Algebra Colloq. , FJOURNAL =. 2012 , NUMBER =. doi:10.1142/S1005386712000247 , URL =

    work page doi:10.1142/s1005386712000247 2012

  67. [67]

    2008 , NOTE =

    2-Kac-Moody algebras , author=. 2008 , NOTE =

    2008

  68. [68]

    and Lauda, A

    Khovanov, Mikhail and Lauda, Aaron D. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2011 , NUMBER =. doi:10.1090/S0002-9947-2010-05210-9 , URL =

    work page doi:10.1090/s0002-9947-2010-05210-9 2011

  69. [69]

    , TITLE =

    Khovanov, Mikhail and Lauda, Aaron D. , TITLE =. Represent. Theory , FJOURNAL =. 2009 , PAGES =. doi:10.1090/S1088-4165-09-00346-X , URL =

    work page doi:10.1090/s1088-4165-09-00346-x 2009

  70. [70]

    and Vaqui\'e, M

    To\"en, Bertrand and Vaqui\'. Moduli of objects in dg-categories , JOURNAL =. 2007 , NUMBER =. doi:10.1016/j.ansens.2007.05.001 , URL =

    work page doi:10.1016/j.ansens.2007.05.001 2007

  71. [71]

    Symmetries, integrable systems and representations

    Hernandez, David and Leclerc, Bernard , title =. Symmetries, integrable systems and representations. Proceedings of the conference on infinite analysis: frontier of integrability, Tokyo, Japan, July 25--29, 2011 and the conference on symmetries, integrable systems and representations, Lyon, France, December 13--16, 2011 , isbn =. 2013 , publisher =. doi:1...

    work page doi:10.1007/978-1-4471-4863-0_8 2011

  72. [72]

    2025 , howpublished =

    Ricardo Canesin , title =. 2025 , howpublished =

    2025

  73. [73]

    2026 , howpublished =

    Ricardo Canesin , title =. 2026 , howpublished =

    2026

  74. [74]

    2024 , eprint=

    A correspondence between additive and monoidal categorifications with application to Grassmannian cluster categories , author=. 2024 , eprint=

    2024

  75. [75]

    Buan, Aslak Bakke and Marsh, Robert and Reineke, Markus and Reiten, Idun and Todorov, Gordana , TITLE =. Adv. Math. , FJOURNAL =. 2006 , NUMBER =. doi:10.1016/j.aim.2005.06.003 , URL =

    work page doi:10.1016/j.aim.2005.06.003 2006

  76. [76]

    Triangulated categories

    Keller, Bernhard , title =. Triangulated categories. Based on a workshop, Leeds, UK, August 2006 , isbn =. 2010 , publisher =

    2006

  77. [77]

    Marsh, Robert and Reineke, Markus and Zelevinsky, Andrei , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2003 , NUMBER =. doi:10.1090/S0002-9947-03-03320-8 , URL =

    work page doi:10.1090/s0002-9947-03-03320-8 2003

  78. [78]

    Keller, Bernhard , TITLE =. Doc. Math. , FJOURNAL =. 2005 , PAGES =

    2005

  79. [79]

    Proceedings of the

    Gabriel, Peter , TITLE =. Proceedings of the. 1974 , MRCLASS =

    1974

  80. [80]

    Geiss, Christof , TITLE =. Arch. Math. (Basel) , FJOURNAL =. 1995 , NUMBER =. doi:10.1007/BF01193544 , URL =

    work page doi:10.1007/bf01193544 1995

Showing first 80 references.