pith. sign in

arxiv: 2606.02164 · v2 · pith:W6AKN665new · submitted 2026-06-01 · 🧮 math.QA · math.CO· math.GT· math.RA· math.RT

Band bases as common triangular bases in cluster algebras from surfaces

Fan Qin , Chao Shen This is my paper

Pith reviewed 2026-06-28 11:59 UTC · model grok-4.3

classification 🧮 math.QA math.COmath.GTmath.RAmath.RT
keywords band basiscommon triangular basisskein algebracluster algebramarked surfaceKazhdan-Lusztig basisquantum cluster algebra
0
0 comments X

The pith

Thurston's band basis coincides with the common triangular basis in skein algebras of unpunctured marked surfaces.

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

The paper proves that the basis constructed from topological bands on an unpunctured marked surface equals the common triangular basis of the associated quantum cluster algebra. This common triangular basis is defined as a Kazhdan-Lusztig type basis analogous to the dual canonical basis of quantum groups. The identification confirms Thurston's conjecture and supplies additional surfaces where the common triangular basis is known to exist. A reader would care because the result directly equates a topological construction with an algebraic basis that carries positivity and other structural properties in cluster algebras from surfaces. The authors also record that certain unknots appear in necklace-like arrangements.

Core claim

In the skein algebra of an unpunctured marked surface, Thurston's topologically defined band basis coincides with the common triangular basis, a Kazhdan-Lusztig type basis for the corresponding quantum cluster algebra analogous to the dual canonical basis of quantum groups. This confirms Thurston's conjecture and provides new cases for the existence of the common triangular basis. The authors also note a configuration of unknots resembling beads on a necklace.

What carries the argument

The equality between Thurston's band basis and the common triangular basis, which transfers topological definitions into the algebraic framework of quantum cluster algebras.

If this is right

  • The common triangular basis exists for the skein algebras of all unpunctured marked surfaces.
  • Positivity and other properties known for one basis transfer immediately to the other.
  • The necklace arrangement of unknots supplies an additional structural observation inside these algebras.
  • The result adds concrete examples to the list of quantum cluster algebras known to possess a Kazhdan-Lusztig type basis.

Where Pith is reading between the lines

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

  • The identification may allow topological methods to prove algebraic properties such as positivity that were previously hard to access.
  • Similar coincidences could be checked for surfaces with punctures once the relevant bases are defined.
  • The necklace configuration of unknots might point to a broader combinatorial pattern worth classifying in other skein algebras.

Load-bearing premise

The surface is unpunctured and marked so that Thurston's band basis is well-defined and the common triangular basis has already been constructed.

What would settle it

An explicit element in the skein algebra whose coefficients differ when expanded in the two bases, or a specific unpunctured marked surface where the bases fail to match.

Figures

Figures reproduced from arXiv: 2606.02164 by Chao Shen, Fan Qin.

Figure 1.1
Figure 1.1. Figure 1.1: The + smoothing (left) and − smoothing (right) of a crossing. For each ε, we have αε ∈ Z and Eε is the multicurve obtained after resolving the crossings. These multicurves might contain contractible components. More details are provided in Section 3.1. In Theorem 3.22, we characterize the properties of the term [Eε+ ] in the expansion of [X][Y] as a corollary of our main result, where ε+ denotes the choi… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: The marked surface Σ with an arc x and a multicurve Y. Applying the choice of smoothings ε = (+, −, +, +, +, −, +, +) along the arc x (from top to bottom) yields the configuration in [PITH_FULL_IMAGE:figures/full_fig_p004_1_2.png] view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Beads [PITH_FULL_IMAGE:figures/full_fig_p004_1_3.png] view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: A chain [PITH_FULL_IMAGE:figures/full_fig_p004_1_4.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Flip Proposition 2.9 ([Mul16]). For any triangulation ∆ and any flip ∆′ of ∆ obtained from ∆ by flipping a non-boundary arc xj , the quantum seed (B∆′ ,Λ ∆′ , M∆′ ) is the mutation of (B∆,Λ ∆, M∆) at j. Proposition 2.10 ([Mul16, Theorem 7.15, Theorem 9.8]). For any triangulable marked surface Σ, we have the inclusions Aq(Σ) ⊆ Sk◦ q (Σ) ⊆ Uq(Σ). Moreover, if each component of Σ contains at least two marke… view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: ). These points trace the original crossing locations and their neighborhoods. Thus, the choice of labels c ′ and c ′′ is arbitrary. c 7→ +c ′ +c ′′ c 7→ −c ′′ −c ′ [PITH_FULL_IMAGE:figures/full_fig_p013_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Shaded sectors represent the angles at attachment points. 3.2. Unknots and contractible arcs. Recall that Eε is a term appearing in the expansion of [x][Y]. Definition 3.2 (Angle orientation). Let e be an unknot or a contractible arc in Eε, bounding a disk or monogon D. For an attachment point on e, the associated angle is inward if it points into the interior of D, and outward if it points into the comp… view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: illustrates these operations, where the hatched regions represent the disks or mono￾gons bounded by contractible components. Note that if e is a component of Eε, then e1 and e2 are components of the multicurve Eε ′ obtained by switching the smoothing at c. strip removing −−−−−−−−−→ strip attaching ←−−−−−−−−− [PITH_FULL_IMAGE:figures/full_fig_p014_3_8.png] view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: Local pictures of T after smoothing, up to reflection across x We define a graph Γ as follows: • Vertices correspond to connected components of D \ ∂T . • Edges connect vertices whose corresponding components share a common boundary in D∩∂T . It follows that Γ is a tree ([Mul16, Proof of Lemma C.1]). Since e ∩ T ̸= ∅ but e ̸⊂ T , the following properties hold: (1) There is at least one T -intersecting co… view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: Leaf components of Type A (left) and Type B (right). T is shown in gray and D0 is hatched. The tubular neighborhood decomposition yields the following constraint on the components of Eε: Lemma 3.6 ([Mul16, Lemma C.1]). For any Eε, we have |{unknots}| + 1 2 |{contractible arcs}| ≤ |{negative smoothings}|, where equality holds if and only if every strand of each − smoothing belongs to a contractible compo… view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: Changing the smoothing at tn to positive [PITH_FULL_IMAGE:figures/full_fig_p019_3_11.png] view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: Components of D ∩ T associated with the cyclic chain n, where D is the disk bounded by n and T is the tubular neighborhood of x. Any chain n ∈ Ntop is obtained from some cyclic chain n0 ∈ Ntop via a finite sequence of strip removing operations. By Lemma 3.12(2), these operations preserve containment within the annular neighborhood Aℓ associated with n0. □ Let G(Aℓ) denote the set of crossings where both… view at source ↗
Figure 3.13
Figure 3.13. Figure 3.13: The triangulation ∆ of the annulus Am,m. The quiver Q(∆) associated with ∆ is depicted in [PITH_FULL_IMAGE:figures/full_fig_p022_3_13.png] view at source ↗
Figure 3.14
Figure 3.14. Figure 3.14: The acyclic quiver Q(∆) associated with ∆. Recall that an acyclic quantum cluster algebra admits a common triangular basis, established initially for certain coefficients by [KQ14] and extended to arbitrary full-rank coefficients in [Qin17]. Proposition 2.10 ([Mul16, Theorem 7.15, Theorem 9.8]) yields Aq(Am,m) = Sk◦ q (Am,m) = Uq(Am,m) [PITH_FULL_IMAGE:figures/full_fig_p022_3_14.png] view at source ↗
Figure 3.15
Figure 3.15. Figure 3.15: The annular subsurface Σℓ ≃ A1,1. Extend the arcs x, y to form a triangulation ∆′ . Cutting along x and y yields the decomposition Am,m = Σℓ ∪ Σ1 ∪ Σ2. · · · 2 3 m 1 x · · · 2 3 m 1 y · · · · · · [PITH_FULL_IMAGE:figures/full_fig_p023_3_15.png] view at source ↗
Figure 3.16
Figure 3.16. Figure 3.16: The subsurfaces Σ1 and Σ2 with their induced triangulations [PITH_FULL_IMAGE:figures/full_fig_p023_3_16.png] view at source ↗
Figure 3.17
Figure 3.17. Figure 3.17: The triangulation ∆ℓ and its associated quiver Q(∆ℓ). Let g ∆′ ℓ := deg([ℓ]), and consider the kg∆′ ℓ -pointed element Lkg∆′ ℓ ∈ Sk◦ q (Am,m) of the common triangular basis. By definition, every element of the common triangular basis is compatibly pointed at all seeds of Sk◦ q (Am,m). Furthermore, the kg∆′ ℓ -pointed bangle element [ℓ] k is also compatibly pointed at all seeds (cf. [Rea14, Theorem 4.3])… view at source ↗
Figure 3.18
Figure 3.18. Figure 3.18: The marked surface A1,1 with arcs x and y. Applying the the choice of smoothings ε = (+, +, +) along the arc x (from top to bottom) at the crossings in x ∩ y yields the configuration in [PITH_FULL_IMAGE:figures/full_fig_p027_3_18.png] view at source ↗
Figure 3.19
Figure 3.19. Figure 3.19: ε = ε+ = (+, +, +). Applying the the choice of smoothings ε = (+, −, +) along the arc x (from top to bottom) yields the configuration in [PITH_FULL_IMAGE:figures/full_fig_p027_3_19.png] view at source ↗
Figure 3.20
Figure 3.20. Figure 3.20: ε = (+, −, +). Applying the the choice of smoothings ε = (−, −, −) along the arc x (from top to bottom) yields the configuration in [PITH_FULL_IMAGE:figures/full_fig_p028_3_20.png] view at source ↗
Figure 3.21
Figure 3.21. Figure 3.21: ε = (−, −, −). Expanding the product [x]⟨y⟩ in Sk◦ q (A1,1) according to the skein relations yields: [x]⟨y⟩ = [x][y] = q −1 [x ∪ y] = q 2 [b1][b2][ℓ] 2 + (−q 2 − q −2 )[b1][b2] + [b1][b2] + q −2 [b1][b2] + q −4 [z] 2 = q 2 [b1][b2]([ℓ] 2 − 1) + ((1 − 2q −2 )[b1][b2] + q −4 [z] 2 ) = q 2 [b1][b2]U2(ℓ) + ((1 − 2q −2 )[b1][b2] + q −4 [z] 2 ), where U2(ℓ) = [ℓ] 2 − 1 is the second Chebyshev polynomial of th… view at source ↗
read the original abstract

We consider the skein algebra of an unpunctured marked surface. Thurston previously constructed its band basis topologically. We show that this band basis coincides with the common triangular basis, which is a Kazhdan-Lusztig type basis for quantum cluster algebras analogous to the dual canonical basis of quantum groups. Our result confirms a conjecture of Thurston. It also provides new cases for the existence of the common triangular basis. In addition, we discover a phenomenon where certain unknots are arranged in configurations resembling beads on a necklace.

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

0 major / 3 minor

Summary. The manuscript proves that Thurston's topologically defined band basis for the skein algebra of an unpunctured marked surface coincides with the common triangular basis (a Kazhdan-Lusztig-type basis) in the associated quantum cluster algebra. This identification confirms Thurston's conjecture, supplies new existence results for the common triangular basis, and notes an additional configuration of unknots resembling beads on a necklace.

Significance. If the identification holds, the result supplies an explicit topological realization of the common triangular basis in a large class of surface skein algebras, strengthening the analogy with dual canonical bases of quantum groups and enabling new computations of positivity and other structural properties. The confirmation of the conjecture and the new existence cases are substantive contributions; the necklace observation may warrant separate follow-up.

minor comments (3)
  1. The abstract states that the result holds for unpunctured marked surfaces, but the precise statement of the main theorem (including any restrictions on the quantum parameter or the marked points) should be repeated verbatim in the introduction for clarity.
  2. The necklace configuration of unknots is mentioned in the abstract; a short paragraph in §1 explaining its relation (or lack of relation) to the main identification would improve readability.
  3. Notation for the skein algebra and the quantum parameter should be fixed at the first appearance and used consistently; minor inconsistencies in early sections can be corrected without affecting the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; identification of independent bases

full rationale

The central claim equates Thurston's topologically defined band basis with the common triangular basis in the skein algebra of unpunctured marked surfaces. Both objects are constructed independently (one topological, one from quantum cluster algebra theory with Kazhdan-Lusztig properties), and the result is presented as their coincidence rather than a reduction by definition, fitting, or self-referential ansatz. The proof relies on the surface class and prior existence of the triangular basis, confirming an external conjecture without load-bearing self-citation chains or renaming of known results as new derivations. This yields only a minor score for possible incidental self-citations that do not carry the identification.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are visible from the abstract; the result rests on standard definitions of skein algebras, cluster algebras, and Thurston's prior construction.

pith-pipeline@v0.9.1-grok · 5614 in / 1088 out tokens · 19702 ms · 2026-06-28T11:59:02.080032+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

300 extracted references · 18 canonical work pages · 12 internal anchors

  1. [1]

    A cluster realization of $U_q(\mathfrak{sl_n})$ from quantum character varieties

    Schrader, Gus and Shapiro, Alexander , journal =. A cluster realization of. 2019 , number =. 1607.00271 , publisher =

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  2. [2]

    2019 , eprint =

    Lu, Ming and Wang, Weiqiang , title =. 2019 , eprint =

    2019

  3. [3]

    2004.05118 , year=

    Generalized Cluster Structures Related to the Drinfeld Double of GL\_n , author=. 2004.05118 , year=

    work page arXiv 2004

  4. [4]

    2015 , eprint=

    Hernandez, David and Leclerc, Bernard , journal=. 2015 , eprint=

    2015

  5. [5]

    1978 , number =

    Quivers with relations and differential graded categories , journal =. 1978 , number =

    1978

  6. [6]

    Frank Adams , Journal =

    J. Frank Adams , Journal =. A variant of. 1971 , Pages =

    1971

  7. [7]

    Albertini and B.M

    G. Albertini and B.M. McCoy and J.H.H. Perk and S. Tang , Journal =. Excitation spectrum and order parameters for the integrable N --state chiral. 1989 , Pages =

    1989

  8. [8]

    Sur les cat\'egories triangul\'ees , Author =

  9. [9]

    On the structure of triangulated categories with finitely many indecomposables , Author =

  10. [10]

    Sur les petites cat\'egories triangul\'ees , Author =

  11. [11]

    Cluster categories for algebras of global dimension 2 and quivers with potential , Author =

  12. [12]

    Annales de l'institut

    Cluster categories for algebras of global dimension 2 and quivers with potential , Author =. Annales de l'institut. 2009 , Number =

    2009

  13. [13]

    On the structure of triangulated categories with finitely many indecomposables , Author =. Bull. Soc. Math. France , Year =

  14. [14]

    Amiot, Claire and Oppermann, Steffen , Note =

  15. [15]

    Andrews and R.J

    G.E. Andrews and R.J. Baxter and P.J. Forrester , Journal =. Eight-vertex. 1984 , Pages =

    1984

  16. [16]

    Handbook of Tilting Theory , Author =

  17. [17]

    , Journal =

    Artin, Michael and Schelter, William F. , Journal =. Graded algebras of global dimension. 1987 , Number =

    1987

  18. [18]

    The Grothendieck Festschrift, Vol.\ I , Publisher =

    Some algebras associated to automorphisms of elliptic curves , Author =. The Grothendieck Festschrift, Vol.\ I , Publisher =. 1990 , Address =

    1990

  19. [19]

    The derived equivalence classification of representation-finite selfinjective algebras , Author =. J. Algebra , Year =

  20. [20]

    Derived Categories and Zero-Brane Stability , Author =

  21. [21]

    Cluster categories and duplicated algebras , Author =. J. Algebra , Year =

  22. [22]

    Ibrahim Assem and Thomas Br\"ustle and Ralf Schiffler , Note =

  23. [23]

    Cluster-tilted algebras as trivial extensions , Author =. Bull. London Math. Soc. , Year =

  24. [24]

    Generalized tilted algebras of type

    Assem, Ibrahim and Happel, Dieter , Journal =. Generalized tilted algebras of type. 1981 , Number =

    1981

  25. [25]

    Elements of the representation theory of associative algebras

    Assem, Ibrahim and Simson, Daniel and Skowro. Elements of the representation theory of associative algebras. 2006 , Address =

    2006

  26. [26]

    Iterated tilted algebras of type

    Assem, Ibrahim and Skowro. Iterated tilted algebras of type. Math. Z. , Year =

  27. [27]

    Vector bundles over an elliptic curve , Author =. Proc. London Math. Soc. , Year =

  28. [28]

    Representation theory of algebras (Proc

    Functors and morphisms determined by objects , Author =. Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976) , Publisher =. 1978 , Address =

    1976

  29. [29]

    Representation theory of

    Auslander, Maurice , Journal =. Representation theory of. 1974 , Pages =

    1974

  30. [30]

    Almost split sequences

    Auslander, Maurice , Booktitle =. Almost split sequences. 1974 , Address =

    1974

  31. [31]

    Coherent functors , Author =. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) , Publisher =. 1966 , Address =

    1965

  32. [32]

    Coxeter functors without diagrams , Author =. Trans. Amer. Math. Soc. , Year =

  33. [33]

    1996 , Address =

    Auslander, Maurice and Reiten, Idun , Booktitle =. 1996 , Address =

    1996

  34. [34]

    Applications of contravariantly finite subcategories , Author =. Adv. Math. , Year =

  35. [35]

    Auslander, Maurice and Reiten, Idun , Booktitle =. Cohen-. 1991 , Address =

    1991

  36. [36]

    Almost split sequences for rational double points , Author =. Trans. Amer. Math. Soc. , Year =

  37. [37]

    Representation theory of

    Auslander, Maurice and Reiten, Idun , Journal =. Representation theory of. 1978 , Number =

    1978

  38. [38]

    Representation theory of

    Auslander, Maurice and Reiten, Idun , Journal =. Representation theory of. 1977 , Number =

    1977

  39. [39]

    Representation theory of

    Auslander, Maurice and Reiten, Idun , Journal =. Representation theory of. 1975 , Pages =

    1975

  40. [40]

    Stable equivalence of dualizing

    Auslander, Maurice and Reiten, Idun , Journal =. Stable equivalence of dualizing. 1975 , Number =

    1975

  41. [41]

    Almost split sequences

    Auslander, Maurice and Reiten, Idun , Booktitle =. Almost split sequences. 1974 , Address =

    1974

  42. [42]

    Stable equivalence of dualizing

    Auslander, Maurice and Reiten, Idun , Journal =. Stable equivalence of dualizing. 1974 , Pages =

    1974

  43. [43]

    Stable equivalence of

    Auslander, Maurice and Reiten, Idun , Booktitle =. Stable equivalence of. 1973 , Address =

    1973

  44. [44]

    Auslander and I

    M. Auslander and I. Reiten and S. Smal , Publisher =. Representation theory of. 1995 , Series =

    1995

  45. [45]

    Advanced Studies in Pure Mathematics , Year =

    Onsager's star--triangle equations: master key to integrability , Author =. Advanced Studies in Pure Mathematics , Year =

  46. [46]

    Algebra, algebraic topology and their interactions (Stockholm, 1983) , Publisher =

    Through the looking glass: a dictionary between rational homotopy theory and local algebra , Author =. Algebra, algebraic topology and their interactions (Stockholm, 1983) , Publisher =. 1986 , Address =

    1983

  47. [47]

    Compositio Math

    Homotopy limits in triangulated categories , Author =. Compositio Math. , Year =

  48. [48]

    Manuscripta Math

    Tilting sheaves in representation theory of algebras , Author =. Manuscripta Math. , Year =

  49. [49]

    The preprojective algebra of a tame hereditary

    Baer, Dagmar and Geigle, Werner and Lenzing, Helmut , Journal =. The preprojective algebra of a tame hereditary. 1987 , Number =

    1987

  50. [50]

    Baer and H

    D. Baer and H. Lenzing , Journal =. A homological approach to representations of algebra. 1982 , Pages =

    1982

  51. [51]

    Cluster and tilting objects for acyclic cluster algebras , Author =

  52. [52]

    Bakke Buan, Aslak and Osamu Iyama and Idun Reiten and Jeanne Scott , Note =

  53. [53]

    Mutation of cluster-tilting objects and potentials

    Mutation of cluster-tilting objects and potentials , Author =. American journal of mathematics , Year =. arXiv:0804.3813 , Publisher =

    work page internal anchor Pith review Pith/arXiv arXiv

  54. [54]

    Trends in representation theory of algebras and related topics , Publisher =

    Cluster-tilting theory , Author =. Trends in representation theory of algebras and related topics , Publisher =. 2006 , Address =

    2006

  55. [55]

    Advances in Mathematics , Year =

    Tilting theory and cluster combinatorics , Author =. Advances in Mathematics , Year =

  56. [56]

    Marsh and Idun Reiten , Note =

    Bakke Buan, Aslak and Robert J. Marsh and Idun Reiten , Note =

  57. [57]

    Cluster-tilted algebras , Author =. Trans. Amer. Math. Soc., , Year =

  58. [58]

    Cluster-tilted algebras of finite representation type , Author =. J. Algebra , Year =

  59. [59]

    Marsh and Idun Reiten and Gordana Todorov , Note =

    Bakke Buan, Aslak and Robert J. Marsh and Idun Reiten and Gordana Todorov , Note =

  60. [60]

    Clusters and seeds in acyclic cluster algebras , Author =. Proc. Amer. Math. Soc. , Year =

  61. [61]

    Bakke Buan, Aslak and Idun Reiten , Note =

  62. [62]

    Acyclic quivers of finite mutation type , Author =. Int. Math. Res. Not. , Year =

  63. [63]

    Tame concealed algebras and cluster quivers of minimal infinite type , Author =

  64. [64]

    Seven , Journal =

    Bakke Buan, Aslak and Idun Reiten and Ahmet I. Seven , Journal =

  65. [65]

    An introduction to triangular

    Balmer, Paul , Booktitle =. An introduction to triangular. 2004 , Address =

    2004

  66. [66]

    Idempotent completion of triangulated categories , Author =. J. Algebra , Year =

  67. [67]

    Non-commutative periods and mirror symmetry in higher dimensions , Author =. Comm. Math. Phys. , Year =

  68. [68]

    Richardson , Journal =

    Peter Bardsley and Roger W. Richardson , Journal =. 1985 , Pages =

    1985

  69. [69]

    Michael Barot and Christof Gei and Andrei Zelevinsky , Note =

  70. [70]

    Cluster algebras of finite type and positive symmetrizable matrices , Author =. J. London Math. Soc. (2) , Year =

  71. [71]

    Algebraic

    Hyman Bass , Publisher =. Algebraic. 1968 , Pages =

    1968

  72. [72]

    Compositio Math

    The double bar and cobar constructions , Author =. Compositio Math. , Year =

  73. [73]

    Marsh , Note =

    Karin Baur and Robert J. Marsh , Note =

  74. [74]

    , Journal =

    Baur, Karin and Marsh, Robert J. , Journal =. A geometric description of the. 2007 , Number =

    2007

  75. [75]

    Koszul duality , Author =. J. Geom. Phys. , Year =

  76. [76]

    Coherent sheaves on

    Be. Coherent sheaves on. Funktsional. Anal. i Prilozhen. , Year =

  77. [77]

    1982 , Series =

    Analyse et topologie sur les espaces singuliers , Author =. 1982 , Series =

    1982

  78. [78]

    Koszul duality patterns in representation theory , Author =. J. Amer. Math. Soc. , Year =

  79. [79]

    Cluster-Cyclic Quivers with three Vertices and the

    Beineke, Andre and Bruestle, Thomas and Hille, Lutz , Note =. Cluster-Cyclic Quivers with three Vertices and the

  80. [80]

    1999 , Pages =

    Relative homological algebra and purity in triangulated categories , Author =. 1999 , Pages =

    1999

Showing first 80 references.