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Naturality of ${\rm SL}_n$ quantum trace maps for surfaces

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Pith's one-line read For every rank n, SLn quantum trace maps are natural under changes of ideal triangulation, via balanced n-th root quantum coordinate change isomorphisms that extend Fock–Goncharov quantum cluster maps.

desk verdict Completes the SL_n naturality program for all n via a clever network argument, but one asserted injectivity lemma needs proof or citation. read the letter →

arxiv 2412.16959 v1 pith:O77LTQF4 submitted 2024-12-22 math.QA

Hyun Kyu Kim , Zhihao Wang This is my paper · ORCID
classification math.QA MSC 57K3113F60
keywords SLn-skeinalgebrasquantumtracemapsclusteridealtriangulationsstatedskeinFock-Goncharovtorin-throotvariablesnetworkpathsums
open problems The Hierarchy Problem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes the naturality of the SLn-quantum trace maps of Lê and Yu under changes of ideal triangulation of a punctured bordered surface. For any two triangulations λ and λ′, it constructs a balanced n-th root quantum coordinate change isomorphism Θωλλ′ between the skew-fields of fractions of the balanced subalgebras of the n-th root Fock–Goncharov quantum tori, and proves that the quantum trace maps for λ and λ′ agree after composing with this isomorphism. The isomorphisms satisfy a consistency condition across any three triangulations and recover the usual Fock–Goncharov quantum cluster coordinate changes. The authors also show that their Θωλλ′ coincides with the coordinate change map previously constructed by Lê and Yu, thereby confirming the expectation that the SLn-quantum trace fits into the quantum cluster algebra framework. This is a step toward a quantum Fock–Goncharov duality map for SLn and PGLn.

What carries the argument

The proof is carried by three mechanisms. The first is the splitting homomorphism: cutting a surface along an ideal arc induces compatible algebra maps on both the reduced stated SLn-skein algebras and the n-th root Fock–Goncharov quantum tori, which reduce the naturality problem to the 4-gon case. The second is the extension of quantum X-mutations to the n-th root setting: each flip of triangulation is implemented by a sequence of 1/6($n^{3}$−n) mutations, promoted to isomorphisms of the mutable-balanced subalgebras using the quantum dilogarithm automorphism AdΨq(Xk). The third is the network description: the quantum trace of a stated arc is expressed as a sum over paths in the directed network dual to the n-triangulation, and Schrader–Shapiro's network mutation compatibility converts a potentially heavy algebraic identity into a combinatorial statement about these path sums.

What would settle it

Find a triangulable punctured bordered surface and a nonzero element of its reduced stated SLn-skein algebra that is mapped to zero by the splitting homomorphism for some ideal arc; such an element would disprove the injectivity assertion on which the proof of Theorem 4.1 rests. A direct attack would be to evaluate the splitting map on stated webs with two boundary points on the same side of the arc, where cancellations among the summed states could kill the image.

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Extended reading notes

Core claim

The central claim (Theorem 1.1) is that for any triangulable punctured bordered surface and any two ideal triangulations λ and λ′, there exists a skew-field isomorphism Θωλλ′ : Frac(Zω^bl(S,λ′)) → Frac(Zω^bl(S,λ)) of the balanced n-th root Fock–Goncharov algebras that extends the Fock–Goncharov quantum cluster isomorphism Φqλλ′, satisfies the consistency relation Θωλλ′Θωλ′λ′′ = Θωλλ′′, and makes the naturality equation trλ = Θωλλ′ ∘ trλ′ hold for the Lê–Yu SLn-quantum trace maps. The same map is shown to coincide with the isomorphism ΨXλλ′ that Lê and Yu had defined from the naturality equation itself. Thus the SLn-quantum trace maps for different triangulations are not merely individually defined but form one consistent system, with transition isomorphisms that are the n-th root quantum cluster coordinate changes.

Load-bearing premise

The proof's reduction from a general surface to a quadrilateral uses the assertion, made without proof or citation, that the splitting homomorphism from the reduced stated SLn-skein algebra of a surface into that of the surface cut along an ideal arc is injective; if some nonzero skein element were killed by this map, the diagram-chasing argument proving the main compatibility theorem would collapse.

Editorial extensions

If this is right

  • For every rank n ≥ 2, the Lê–Yu SLn-quantum trace maps for different ideal triangulations are related by the balanced n-th root quantum coordinate change isomorphisms Θωλλ′, so the naturality equation holds.
  • The coordinate change isomorphisms satisfy the consistency relation Θωλλ′Θωλ′λ′′ = Θωλλ′′, making the collection of quantum trace maps a consistent system of charts for different triangulations.
  • The map Θωλλ′ coincides with Lê–Yu's map ΨXλλ′, so the quantum cluster framework and Lê–Yu's construction define the same transition isomorphisms.
  • The earlier naturality result for n = 3 is recovered by a more conceptual, non-computational proof.
  • The quantum trace maps land in the balanced subalgebra and the restricted coordinate changes (Θωλλ′)^bl map balanced fraction fields to balanced fraction fields, so the entire naturality statement holds at the level of the balanced subalgebras.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If Schrader–Shapiro's network compatibility can be extended from stated arcs to arbitrary n-webs, the same non-computational strategy would likely prove the naturality of the quantum trace on all of the skein algebra, a step the authors explicitly leave for future work.
  • The injectivity of the splitting homomorphism, asserted without proof, is the most exposed point of the argument; if it fails, the reduction to the quadrilateral would need a different justification, or the naturality theorem would require a separate proof.
  • The explicit nature of Θωλλ′ as a composition of n-th root quantum mutations gives a candidate formula for the transition maps of the sought-for quantum Fock–Goncharov duality map for SLn and PGLn, going beyond the paper's stated expectation that its methods will aid that program.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The paper proves the naturality of the SL_n quantum trace maps of Lê and Yu for triangulable punctured bordered surfaces. For any two ideal triangulations λ and λ′ of such a surface, the authors construct a balanced n-th root quantum coordinate change isomorphism Θ^ω_{λλ′} between the skew-fields of fractions of the mutable-balanced (or balanced) n-th root Fock–Goncharov algebras, extending the usual Fock–Goncharov quantum cluster isomorphism Φ^q_{λλ′}. They show that this isomorphism satisfies the consistency relation and the naturality equation tr_λ = Θ^ω_{λλ′} ∘ tr_{λ′}, and that it coincides with the coordinate change isomorphism Ψ^X_{λλ′} previously constructed by Lê and Yu. The proof avoids heavy direct computations by reducing the problem to the case of the 4-gon via splitting homomorphisms, and then using a path-sum description of quantum traces in terms of Schrader–Shapiro networks and a compatibility result of these path-sums under cluster mutations.

Significance. If the proof is correct, this is a significant and timely result. It places Lê–Yu's SL_n-quantum trace into the quantum cluster algebra framework, resolves a question explicitly raised in [LY23, §14.4], and recovers the n=3 result of [K24] by a more conceptual argument. The paper is written in a clear and organized way, and the strategy of using splitting homomorphisms together with network path-sum formulas is elegant and promising for further applications, such as a quantum Fock–Goncharov duality map for SL_n and PGL_n. The authors are also careful to record several technical debts to prior work, notably the splitting homomorphism machinery of [LS21] and the network compatibility of [SS17]; however, as detailed below, some of these debts are load-bearing and need to be addressed before the proof can be considered complete.

major comments (4)
  1. [§4.1, diagram (50); §4.2, Lemma 4.5(2)] The proof uses the injectivity of the splitting homomorphism S_e on the reduced stated SL_n-skein algebra, but §2.2 only records that S_e is an algebra homomorphism and does not prove or cite injectivity. In the proof of Lemma 4.5(2), the conclusion 'Since S_{e'_1} is injective, it follows that ...' is unsupported, and in the proof of Theorem 4.1 the sentence 'Observe that the two long vertical arrows in the above diagram are injective' refers to these splitting maps. Without injectivity of the right vertical arrow, the chain of equalities in the proof of Theorem 4.1 cannot be used to deduce the upper triangle from the outer square, and the path-sum formula for tr_{λ'}(a_{ij}) in Lemma 4.5(2) may fail. The authors should either prove this injectivity or cite a precise theorem (e.g., from [LS21]) and state the theorem.
  2. [§4.2, Lemma 4.7] Lemma 4.7 is stated without proof and is entirely delegated to [SS17, Prop. 4.2], which is not stated in the paper. Since Lemma 4.7 is the key step that bypasses the heavy computation in the quadrilateral case, the authors should state the relevant proposition from [SS17] and verify that its hypotheses are satisfied in their network mutation sequence. As written, the reader cannot check the correctness of Lemma 4.7 without consulting an arXiv preprint, and the application of [SS17, Prop. 4.2] is asserted rather than demonstrated.
  3. [§3.2, Lemma 3.4] Lemma 3.4 is used in the proof of Lemma 4.8 and Proposition 4.9 to propagate the vanishing conditions on Q(u,v)t_v under the monomial transformation ν'_k, but the proof is omitted with only a remark that it is a 'straightforward computation.' Given that this lemma is load-bearing for the construction of the n-th root quantum mutation and for the proof of balancedness, a full proof should be included, or a precise statement of the cited [FG09b, Lem.2.7] / [K21b, Lem.3.7(1)] should be provided so that the reader can verify the implication.
  4. [§4.3, proof of Proposition 4.9] In the proof of Proposition 4.9, after applying S_E to the equation, the paper asserts that 'the balancedness of Z^{t''''} = S_E(Z^{t^{(0)}})$ implies that of Z^{t^{(0)}}$'. Lemma 4.10 only proves the forward implication (balanced elements map to balanced elements); the reverse implication is not self-evident and requires justification, e.g., using injectivity of S_E and the triangle-wise structure of the balancedness condition. Without this, the conclusion t^{(0)} ∈ B_λ is not established.
minor comments (3)
  1. [§4.2, equation (53)] The surjectivity of the concatenation map in (53) relies on the claim that every path in N'(ij) meets the diagonal ideal arc e'_1 exactly once. This claim should be justified explicitly, since it is essential for the bijection between paths in the cut surface and paths in the original network.
  2. [References] The reference [SS17] is listed as an arXiv preprint; if it has since been published or updated, the citation should be amended. More importantly, the specific proposition used (Prop. 4.2 of [SS17]) is not quoted in the text; readers would benefit from a precise statement.
  3. [§2.1, equation (7)] Equation (7) appears to state q^{1/n} - q^{-1/n} = q - q^{-1}, which is not dimensionally consistent with the usual SL_n skein relations; please check this relation against [LY23] and clarify the intended identity.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the naturality is derived from Lê–Yu's quantum trace and Schrader–Shapiro's independent network mutation compatibility; the identification with Ψ^X is a consequence, not an input. A separate non-circular gap is an unproved injectivity of the splitting homomorphism.

full rationale

The derivation chain is not circular. The balanced n-th root quantum coordinate change Θ is constructed in §3 from quantum cluster mutations (Definitions 3.3, 3.6, 3.7, and the composition in §3.4), independently of the naturality equation. The compatibility trλ = Θ ∘ trλ′ is then proved in §4 by cutting to the 4-gon and using Schrader–Shapiro's network mutation compatibility [SS17, Prop. 4.2] (quoted as Lemma 4.7) together with the triangle path-sum formula [LY23, Thm. 10.5] (Lemma 4.5(1)). The Lê–Yu map Ψ^X is not assumed; it is only shown to coincide with (Θ)^bl afterward by a uniqueness argument in Theorem 4.16, so Theorem 1.1(4) is a derived consequence. The self-citations to the first author's [K24] are technical, parameter-free lemmas (Weyl-ordered Laurent monomials under ν′, ∗-structure preservation) used as tools; they are not equivalent to the target naturality statement. I do flag one genuine non-circular gap: the paper asserts without proof or citation that the splitting homomorphism on reduced stated SL_n skein algebras is injective — "Observe that the two long vertical arrows in the above diagram are injective" (§4.1, after diagram (50)) and "Since S_{e'_1} is injective, it follows that ..." (§4.2, Lemma 4.5(2)). This injectivity is load-bearing for the reduction to P4 and for the path-sum formula in Lemma 4.5(2). It is a missing structural fact, not a circular step: supplying a proof or citation would fix the gap without changing the argument's independence from its conclusions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities; the only inputs are the definitions of the stated skein algebra and quantum torus algebras, plus a set of prior theorems. The most fragile external input is the unproved injectivity of the splitting homomorphism, followed by the black-box [SS17, Prop. 4.2].

assumptions (7)
  • domain assumption The reduced stated SLn-skein algebra Sω(S) admits an algebraic quantum trace homomorphism trλ to Zω(S,λ) with image in the balanced subalgebra (Theorem 2.4).
    Invoked throughout §2-4; imported from [LY23] and [K20].
  • domain assumption The splitting homomorphism Se on reduced stated skein algebras is injective.
    Asserted in the proof of Theorem 4.1 after diagram (50) without proof or citation; load-bearing for the reduction to the quadrilateral.
  • domain assumption The quantum trace of a stated arc in a triangle is a sum over paths in the dual network, [LY23, Theorem 10.5].
    Basis for Lemma 4.5(1) and (2).
  • domain assumption Path sums in networks are compatible with cluster mutations, [SS17, Proposition 4.2].
    Yields Lemma 4.7, the key computational reduction of Proposition 4.2.
  • domain assumption For polygon surfaces, trλ induces an isomorphism of fraction fields, [LY23, Theorem 11.7 and §14.2].
    Used in Lemma 4.11 and Proposition 4.12.
  • standard math Any two ideal triangulations are connected by a flip sequence, and the associated cluster seeds are connected by the specified mutation sequence.
    Used throughout; standard result cited to [L09] and [FG06].
  • standard math Skew-fields of fractions exist for the quantum torus algebras used.
    Cohn's theorem [C95]; needed for the fraction field formulations.

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Pith. "Pith review of Naturality of ${\rm SL}_n$ quantum trace maps for surfaces." pith.science (2026). https://pith.science/paper/O77LTQF4

@misc{pith2026241216959,
  author       = {Pith},
  title        = {Pith review of: Naturality of $\rm SL_n$ quantum trace maps for surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O77LTQF4}},
  note         = {Machine review of arXiv:2412.16959}
}
abstract

The ${\rm SL}_n$-skein algebra of a punctured surface $\mathfrak{S}$, studied by Sikora, is an algebra generated by isotopy classes of $n$-webs living in the thickened surface $\mathfrak{S} \times (-1,1)$, where an $n$-web is a union of framed links and framed oriented $n$-valent graphs satisfying certain conditions. For each ideal triangulation $\lambda$ of $\mathfrak{S}$, L\^e and Yu constructed an algebra homomorphism, called the ${\rm SL}_n$-quantum trace, from the ${\rm SL}_n$-skein algebra of $\mathfrak{S}$ to a so-called balanced subalgebra of the $n$-root version of Fock and Goncharov's quantum torus algebra associated to $\lambda$. We show that the ${\rm SL}_n$-quantum trace maps for different ideal triangulations are related to each other via a balanced $n$-th root version of the quantum coordinate change isomorphism, which extends Fock and Goncharov's isomorphism for quantum cluster varieties. We avoid heavy computations in the proof, by using the splitting homomorphisms of L\^e and Sikora, and a network dual to the $n$-triangulation of $\lambda$ studied by Schrader and Shapiro.

Figures

Figures reproduced from arXiv: 2412.16959 by the authors.

Figure 1
Figure 1. The left is C(p)ij and the right is C⃗(p)ij . For a pb surface S, S ω(S) = Sω(S)/Ibad is called the reduced stated SLn-skein algebra, defined in [LY23], where I bad is the two-sided ideal of Sω(S) generated by all bad arcs [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Barycentric coordinates ijk and a 4-triangulation with its quiver [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Vertices of the n-triangulation quiver involved in the flip of triangulations; the above example picture is for the case n = 4 At the quantum level, we first define the quantum coordinate change isomorphism for the skew-fields of fractions of the usual Fock-Goncharov algebras Φ q λλ′ := µ q v1 ◦ · · · ◦ µ q vr : Frac(Xq(S, λ′ )) → Frac(Xq(S, λ)). (32) Note that the order of the mutation sequence seems to be reversed… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Two triangulations λ and λ ′ of P4 In fact, the core computational part of the proof of the entire Theorem 4.1, hence the main difficulty thereof, lies in this case of P4, i.e. in the proof of Proposition 4.2, which we postpone until the next subsection. At the moment,…
Figure 5
Figure 5. Figure 5: Three corner arcs a, b, c in P4. Four edges of P4 are labeled by e2, e3, e4, e5 Lemma 4.4. The reduced stated SLn-skein algebra S ω(P4) is generated by aij , bij , cij for i, j ∈ {1, . . . , n} with i ≥ j. Thus it suffices to check (47) when applied to these generators…
Figure 6
Figure 6. Figure 6: (left-turn) Networks for triangulations of P4; when n = 4 three oriented edges. As can be verified by inspection in [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]

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Works this paper leans on

38 extracted references · 32 canonical work pages

  1. [1]

    D. G. L. Allegretti and H. Kim, A duality map for quantum cluster varieties from surfaces , Adv. Math. 306 (2017), 1164–1208

    work page 2017

  2. [2]

    A.Berenstein and A.Zelevinsky, Quantum cluster algebras, Adv. Math. 195 (2) (2005), 405--455

    work page 2005

  3. [3]

    Bonahon and H

    F. Bonahon and H. Wong, Quantum traces for representations of surface groups in SL_2( C) , Geom. Topol. 15 (3) (2011), 1569--1615

    work page 2011

  4. [4]

    Bonahon and H

    F. Bonahon and H. Wong, Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations , Invent. Math. 204 (2016), 195–243

    work page 2016

  5. [5]

    Cautis, J

    S. Cautis, J. Kamnitzer, and S. Morrison, Webs and quantum skew Howe duality , Math. Ann. 360 (2014), 351--390

    work page 2014

  6. [6]

    Chekhov and V

    L. Chekhov and V. V. Fock, A quantum Teichm\"uller space , Theor. Math. Phys. 120 (1999) 511–528

    work page 1999

  7. [7]

    L. O. Chekhov and M. Shapiro, Log-canonical coordinates for symplectic groupoid and cluster algebras, Int. Math. Res. Notices 2023 no. 11 (2023), 9565--9652

    work page 2023

  8. [8]

    P. M. Cohn, Skew fields: theory of general division rings , Encyclopedia of Mathematics and its Applications 57 , Cambridge Univ. Press, 1995

    work page 1995

Show all 38 references
  1. [9]

    Costantino and T

    F. Costantino and T. T. Q. L\^e, Stated skein algebras of surfaces , J. Eur. Math. Soc. 24 (12) (2022), 4063--4142

    2022

  2. [10]

    D. C. Douglas, Quantum traces for SL _n( C ) : The case n = 3 . J. Pure Appl. Algebra 228 (7) (2024), article no. 107652

    2024

  3. [11]

    L. D. Faddeev, Discrete Heisenberg-Weyl group and modular group , Lett. Math. Phys. 34 (1995), 249--254

    1995

  4. [12]

    L. D. Faddeev and R. M. Kashaev, Quantum dilogarithm , Mod. Phys. Lett. A9 (1994) 427--434

    1994

  5. [13]

    V. V. Fock, Dual Teichm\"uller spaces , preprint (1997). arXiv:hep-th/9702018

    1997 arXiv

  6. [14]

    V. V. Fock and A. B. Goncharov, Moduli spaces of local systems and higher Teichm\"uller theory , Publ. Math. Inst. Hautes \'Etudes Sci. 103 (2006), 1--211

    2006

  7. [15]

    V. V. Fock and A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. \' E c. Norm. Sup\' e r. (4) 42 (2009), no. 6, 865--930

    2009

  8. [16]

    V. V. Fock and A. B. Goncharov, The quantum dilogarithm and representations of the quantum cluster varieties , Invent. Math. 175 (2) (2009), 223--286

    2009

  9. [17]

    W. M. Goldman, The symplectic nature of fundamental groups of surfaces , Adv. Math. 54 (1984), 200–225

    1984

  10. [18]

    A. B. Goncharov and L. Shen, Quantum geometry of moduli spaces of local systems and representation theory , arXiv:1904.10491

    1904 arXiv

  11. [19]

    Hiatt, Quantum traces in quantum Teichm\"uller theory , Algebr

    C. Hiatt, Quantum traces in quantum Teichm\"uller theory , Algebr. Geom. Topol. 10 (3) (2010), 1245--1283

    2010

  12. [20]

    R. M. Kashaev, Quantization of Teichm\"uller spaces and the quantum dilogarithm , Lett. Math. Phys. 43 (1998) 105–115

    1998

  13. [21]

    R. M. Kashaev and T. Nakanishi, Classical and quantum dilogarithm identities, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), article no. 102

    2011

  14. [22]

    Kim, SL _3 -laminations as bases for PGL _3 cluster varieties for surfaces , arXiv:2011.14765 (2020)

    H. Kim, SL _3 -laminations as bases for PGL _3 cluster varieties for surfaces , arXiv:2011.14765 (2020)

    2020 arXiv

  15. [23]

    Kim, Phase constants in the Fock-Goncharov quantum cluster varieties, Anal.Math

    H. Kim, Phase constants in the Fock-Goncharov quantum cluster varieties, Anal.Math. Phys. 11 (2021), no. 1, article no. 2

    2021

  16. [24]

    Kim, Irreducible self-adjoint representations of quantum Teichm\"uller space and the phase constants , J

    H. Kim, Irreducible self-adjoint representations of quantum Teichm\"uller space and the phase constants , J. Geom. Phys. 162 (2021), 104103

    2021

  17. [25]

    Kim, Naturality of SL _3 quantum trace maps for surfaces , Quantum Topol., online first (2024), arXiv:2104.06286

    H. Kim, Naturality of SL _3 quantum trace maps for surfaces , Quantum Topol., online first (2024), arXiv:2104.06286

    2024 arXiv

  18. [26]

    H. Kim, T. T. Q. L\^e, and M. Son, SL _2 quantum trace in quantum Teichm\"uller theory via writhe , Algebr. Geom. Topol. 23 (1) (2023), 339-418

    2023

  19. [27]

    Kim and L

    H. Kim and L. Shen, On the canonicity of SL3-PGL3 duality maps for cluster varieties for surfaces (tentative title), in preparation

  20. [28]

    Kuperberg, Spiders for Rank 2 Lie Algebras , Commun

    G. Kuperberg, Spiders for Rank 2 Lie Algebras , Commun. Math. Phys. 180 (1996), 109–151

    1996

  21. [29]

    Labardini-Fragoso, Quivers with potentials associated to triangulated surfaces, Proceedings of the London Mathematical Society 98(3) (2009): 797-839

    D. Labardini-Fragoso, Quivers with potentials associated to triangulated surfaces, Proceedings of the London Mathematical Society 98(3) (2009): 797-839

    2009

  22. [30]

    T. T. Q. L\^e, Triangular decomposition of skein algebras , Quantum Topol. 9 (3) (2018), 591--632

    2018

  23. [32]

    T. T. Q. L \^e , T. Yu, Quantum traces for SL _n -skein algebras, arXiv:2201.00045 (2023)

    2023 arXiv

  24. [33]

    T. T. Q. L\^ e , T. Yu, Quantum traces and embeddings of stated skein algebras into quantum tori, Selecta Math. (N.S.) 28 (2022), no. 4, Paper No. 66, 48 pp

    2022

  25. [34]

    Poudel, A comparison between SL_n spider categories , arXiv:2210.09289

    A. Poudel, A comparison between SL_n spider categories , arXiv:2210.09289

    arXiv

  26. [35]

    Przytycki, Fundamentals of Kauffman bracket skein modules , Kobe J

    J. Przytycki, Fundamentals of Kauffman bracket skein modules , Kobe J. Math. 16 (1999) 45–66

    1999

  27. [36]

    Schrader, and A

    G. Schrader, and A. Shapiro Continuous tensor categories from quantum groups I: algebraic aspects, arXiv:1708.08107 (2017)

    2017 arXiv

  28. [37]

    A. S. Sikora, SL _n -character varieties as spaces of graphs , T. Am. Math. Soc. 35 no.7 (2001), 2773–2804

    2001

  29. [38]

    A. S. Sikora, Skein theory for SU(n) -quantum invariants , Alg. Geom. Topol. 5 (2005), 865– 897

    2005

  30. [39]

    V. G. Turaev, ``Algebra of loops on surfaces, algebra of knots, and quantization" in Braid group, knot theory and statistical mechanics , Adv. Ser. Math. Phys. 9 , World Sci. Publ., Teaneck, NJ, 1989. pp.59–95

    1989

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