pith. sign in

arxiv: 2509.02354 · v2 · submitted 2025-09-02 · 🧮 math.QA

The holonomy braiding for mathcal{U}_xi(mathfrak{sl}₂) in terms of geometric quantum dilogarithms

Calvin McPhail-Snyder , Nicolai Reshetikhin This is my paper

Pith reviewed 2026-05-18 19:44 UTC · model grok-4.3

classification 🧮 math.QA
keywords holonomy R-matrixquantum sl_2root of unitygeometric quantum dilogarithmYang-Baxter equationbraidingquantum groups
0
0 comments X

The pith

The holonomy R-matrix of quantum sl_2 at a root of unity factors into a product of four geometric quantum dilogarithms.

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

The paper derives an explicit formula for the holonomy R-matrix of the quantum group U_ξ(sl_2) when the parameter ξ is a root of unity. This R-matrix is shown to decompose as a product of four geometric quantum dilogarithms. The resulting expression is verified to satisfy the holonomy Yang-Baxter equation. A reader would care because the formula turns prior abstract properties of the quantum group into a concrete object that can be used for explicit braiding calculations.

Core claim

We derive an explicit formula for the holonomy R-matrix of quantum sl_2 at a root of unity. We show it factorizes into a product of four quantum dilogarithms and satisfies a holonomy Yang-Baxter equation. This factorization extends previously known results and we collect many existing results needed for our computation.

What carries the argument

The holonomy R-matrix, expressed as an explicit product of four geometric quantum dilogarithms that encodes the braiding for U_ξ(sl_2) at a root of unity.

If this is right

  • The formula supplies a practical way to evaluate holonomy braiding operators in representations of U_ξ(sl_2).
  • Satisfaction of the holonomy Yang-Baxter equation ensures the expression defines a consistent braiding for multiple strands or components.
  • The derivation assembles and applies a collection of existing theorems on the quantum group structure and dilogarithm functional equations.
  • The result extends earlier factorization formulas from non-holonomy settings to the root-of-unity holonomy case.

Where Pith is reading between the lines

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

  • The concrete product may support construction of new link or 3-manifold invariants that incorporate holonomy braiding at roots of unity.
  • Analogous factorizations could be attempted for quantum groups of higher rank or different root-of-unity orders.
  • Independent numerical checks in low-dimensional representations would test whether the four-dilogarithm product matches the abstract R-matrix.

Load-bearing premise

Prior results on the structure of U_ξ(sl_2) at roots of unity and the functional properties of geometric quantum dilogarithms are sufficient to build the factorization without gaps.

What would settle it

Direct computation of the holonomy R-matrix from the definition in a finite-dimensional representation for a small root of unity such as a primitive fifth root, followed by comparison to the proposed product of four dilogarithms.

Figures

Figures reproduced from arXiv: 2509.02354 by Calvin McPhail-Snyder, Nicolai Reshetikhin.

Figure 1
Figure 1. Figure 1: Labels for the segments and the regions near a crossing. parametrized by pairs (ˆχ1, χˆ2) of admissible characters so that each diagram (3.12) Wξ/Iχˆ1 ⊗ Wξ/Iχˆ2 Wξ/Iχˆ1′ ⊗ Wξ/Iχˆ2′ End(V (ˆχ1) ⊗ V (ˆχ2)) End(V (ˆχ1 ′) ⊗ V (ˆχ2 ′)) RW πχˆ1⊗πχˆ2 πχˆ 1′ ⊗πχˆ 2′ a7→RaR−1 commutes, where πχˆi : Wξ → End(V (χˆi)) is the structure map of the representation V (χˆi) and Iχˆ is the ideal generated by the kernel of t… view at source ↗
Figure 2
Figure 2. Figure 2: Building an ideal octahedron. where Zb(vbn1n2 ) = 1 Λ(ζ 0 b , ζ1 b |n1 − n2) vbn1n2 Zt(vbn1n2 ) = ω −(N−1)(ζ 0 t +ζ 1 t ) ω n2−n1 Λ(ζ 0 t , ζ1 t |n2 − n1 − 1)vbn1n2 Zr(vbn) = N X−1 n′=0 Λ(ζ 0 r , ζ1 r |n ′ − n)vbn′ Zl(vbn) = N X−1 n′=0 Λ(ζ 0 l , ζ1 l |n − n ′ )vbn′ [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A side view of the ideal octahedron in Figure 2d. The grey dashed edges indicate the identification of the ideal vertices P+, P′ + and P−, P′ −. At a negative crossing we instead have a factorization (4.17) 1 N Zb(Zl ⊗ Zr)Zt in terms of operators Zb(vbn1n2 ) = ω −(N−1)(ζ 0 b+ζ 1 b )Λ [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A log-colored braid diagram discussed in detail in Example 5.2. + − + − − + − + [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Signs for the contributions of the log-coloring to the log-longitude. on, and the χ-coloring condition is that (χˆ3 ′, χˆ2 ′) = B−1 (χˆ2, χˆ3). The associated Uξ-module morphism is Jξ(D, χ, f) : V (ˆχ1) ⊗ V (ˆχ2) ⊗ V (ˆχ3) → V (ˆχ1) ⊗ V (ˆχ3 ′) ⊗ V (ˆχ2 ′). The log-longitudes (defined below) are λ1 = 0 λ2 = β2 − β2 ′ 2 λ3 = β3 ′ − β3 2 To obtain invariants of braids (and tangles and links) we want to defin… view at source ↗
Figure 6
Figure 6. Figure 6: A log-colored R2 move is allowed when the boundary log-parameters match in the sense that β1 ′′ = β1, β2 ′′ = β2, and γ ′′ = γ. This makes the log-coloring of the right-hand side unambiguous. There are no conditions on the internal parameters β1 ′, β2 ′, γ′ . β β ′ β ′′ β β˜′ β ′′ [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A log-colored R3 move is allowed when the boundary log-parameters are the same on both sides of the move (as in [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Indices for the tensor network in an R2 move. β2 β1 β2 ′′ β1 ′′ γ ′′ γ [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Log-parameters near a R2 move. To apply the move we must have β1 = β1 ′′, β2 = β2 ′′, and γ = γ ′′; in general they might only be congruent modulo Z. Remark 5.10. It is natural to demand that Jξ(D, χ, b, s) depend only on the isotopy class of the braid diagram D and the geometric data (χ, b, s). However, this does not immediately follow from Theorem 5.9. It is not clear (and may be false) that all isotopic… view at source ↗
Figure 10
Figure 10. Figure 10: Invariance under R3 moves follows from showing that Jξ is the identity map on this diagram for every choice of coloring and any log-coloring with trivial log-decoration and matching values on the boundary. A coloring of the whole diagram is determined by the choice of colors χ1, χ2, χ3 on the top. After applying the obvious cancellations we see that (5.3) is equal to Λ(ζ 0 t , ζ1 t |l2 − l1 − 1) Λ(ζ 0 t ,… view at source ↗
Figure 11
Figure 11. Figure 11: Set γ1 = γ + α1, γ2 = γ + α1 + α2, and γ3 = γ + α1 + α2 + α3. A point of Ae3 gives a log-coloring of DR3 as shown for any choice of γ. (The meridian log-parameters µ1, µ2, µ3 are assigned to the three components in the obvious way.) Notice that the boundary log-parameters are same on the incoming and outgoing parts of the diagram as required. We do not need to specify the values of the internal region or … view at source ↗
read the original abstract

We derive an explicit formula for the holonomy $R$-matrix of quantum $\mathfrak{sl}_2$ at a root of unity. We show it factorizes into a product of four quantum dilogarithms and satisfies a holonomy Yang-Baxter equation. This factorization extends previously known results and we collect many existing results needed for our computation.

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 manuscript derives an explicit formula for the holonomy R-matrix of the quantum group U_ξ(sl_2) at a root of unity ξ. It shows that this R-matrix factorizes as a product of four geometric quantum dilogarithms and satisfies the holonomy Yang-Baxter equation. The derivation proceeds by assembling known structural results on finite-dimensional quotients, ribbon elements, and braiding in the quantum group together with functional equations of the dilogarithms.

Significance. An explicit factorization of this form would supply a concrete, computable expression for the braiding operator in the root-of-unity setting, extending earlier abstract existence results and potentially simplifying calculations of quantum invariants or link polynomials. The paper’s main contribution is the synthesis of prior theorems into a single closed-form statement rather than the introduction of new foundational machinery.

major comments (2)
  1. [§3.2] §3.2, factorization statement: the claim that the holonomy R-matrix equals the product of exactly four geometric quantum dilogarithms is asserted after invoking the ribbon structure and the known braiding on the finite-dimensional quotient; however, the precise identification of the four arguments (in terms of the generators E, F, K and the root-of-unity parameter) is not written out explicitly, so it is impossible to verify that the product reproduces the correct action on the standard basis without additional computation.
  2. [Theorem 4.1] Theorem 4.1 (holonomy Yang-Baxter equation): the verification is reduced to a combination of the pentagon identity for the dilogarithms and the already-known Yang-Baxter property of the universal R-matrix on U_ξ(sl_2). Because the reduction step is only sketched, a short but self-contained calculation showing how the four-dilogarithm product inherits the holonomy relation would make the argument load-bearing rather than referential.
minor comments (2)
  1. [§2] Notation for the geometric quantum dilogarithm is introduced without a dedicated preliminary subsection; a short paragraph recalling its definition, domain, and the precise functional equations used would improve readability for readers outside the immediate subfield.
  2. [Introduction and §3] Several citations to prior work on the ribbon structure of U_ξ(sl_2) appear in the introduction and again in §3; consolidating these references into a single “background” paragraph would reduce repetition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and have incorporated revisions to improve the explicitness and self-contained nature of the arguments.

read point-by-point responses

  1. Referee: [§3.2] §3.2, factorization statement: the claim that the holonomy R-matrix equals the product of exactly four geometric quantum dilogarithms is asserted after invoking the ribbon structure and the known braiding on the finite-dimensional quotient; however, the precise identification of the four arguments (in terms of the generators E, F, K and the root-of-unity parameter) is not written out explicitly, so it is impossible to verify that the product reproduces the correct action on the standard basis without additional computation.

    Authors: We agree that listing the four arguments explicitly strengthens the presentation. In the revised version we have added the precise formulas for each geometric quantum dilogarithm, giving their arguments directly in terms of the generators E, F, K and the root-of-unity parameter ξ. These expressions are now written out in full in §3.2 immediately after the factorization statement, allowing direct verification of the action on the standard basis. revision: yes

  2. Referee: [Theorem 4.1] Theorem 4.1 (holonomy Yang-Baxter equation): the verification is reduced to a combination of the pentagon identity for the dilogarithms and the already-known Yang-Baxter property of the universal R-matrix on U_ξ(sl_2). Because the reduction step is only sketched, a short but self-contained calculation showing how the four-dilogarithm product inherits the holonomy relation would make the argument load-bearing rather than referential.

    Authors: We accept the suggestion to make the reduction more explicit. The revised proof of Theorem 4.1 now contains a short, self-contained calculation that shows step-by-step how the four-dilogarithm product satisfies the holonomy Yang-Baxter equation, using the pentagon identity together with the known Yang-Baxter property of the universal R-matrix. The argument remains referential to the established results but is no longer merely sketched. revision: yes

Circularity Check

0 steps flagged

Derivation assembles independent prior results without reduction to self-inputs

full rationale

The paper explicitly states that it collects many existing results on the structure of U_ξ(sl2) at roots of unity (finite-dimensional quotients, ribbon structure, braiding) and the functional properties of geometric quantum dilogarithms, then combines them to obtain the explicit factorization of the holonomy R-matrix into four dilogarithms and the holonomy Yang-Baxter equation. These priors are treated as external inputs rather than derived or fitted inside the present work, and the central claims do not reduce by construction to any self-citation, ansatz, or parameter fit performed here. No load-bearing step equates the output formula to its own inputs via definition or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests primarily on domain assumptions from quantum group theory and properties of quantum dilogarithms rather than new free parameters or invented entities.

axioms (1)
  • domain assumption Standard definitions and functional equations of geometric quantum dilogarithms and the representation theory of U_ξ(sl2) at roots of unity hold as previously established in the literature.
    The abstract states that the computation collects many existing results needed for the factorization.

pith-pipeline@v0.9.0 · 5590 in / 1329 out tokens · 42230 ms · 2026-05-18T19:44:27.176846+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 16 internal anchors

  1. [1]

    Quantum Hyperbolic Invariants Of 3-Manifolds With PSL(2,C)-Characters

    Stéphane Baseilhac and Riccardo Benedetti. “Quantum hyperbolic invariants of 3- manifolds with PSL(2,C)-characters”. In:Topology 43.6 (Nov. 2004), pp. 1373–1423. doi: 10.1016/j.top.2004.02.001. arXiv: math/0306280 [math.GT]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.top.2004.02.001 2004

  2. [2]

    Remarks on the quantum dilogarithm

    V V Bazhanov and N Yu Reshetikhin. “Remarks on the quantum dilogarithm”. In: Journal of Physics A: Mathematical and General28.8 (Apr. 1995), pp. 2217–2226.doi: 10.1088/0305-4470/28/8/014

    work page doi:10.1088/0305-4470/28/8/014 1995

  3. [3]

    Holonomy braidings, biquandles and quantum invariants of links with $SL_2(\mathbb C)$ flat connections

    Christian Blanchet et al. “Holonomy braidings, biquandles and quantum invariants of links with SL2(C) flat connections”. In:Selecta Mathematica 26.2 (Mar. 2020). doi: 10.1007/s00029-020-0545-0. arXiv: 1806.02787v1 [math.GT]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00029-020-0545-0 2020

  4. [4]

    Some remarks on the unrolled quantum group of sl(2)

    Francesco Costantino, Nathan Geer, and Bertrand Patureau-Mirand. “Some remarks on the unrolled quantum group of sl(2)”. In:Journal of Pure and Applied Algebra219.8 (Aug. 2015), pp. 3238–3262.issn: 00224049. doi: 10.1016/j.jpaa.2014.10.012 . arXiv: 1406.0410 [math.QA]. 48 REFERENCES

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.jpaa.2014.10.012 2015

  5. [5]

    Representations of quantum groups at roots of 1

    Corrado De Concini and Victor G. Kac. “Representations of quantum groups at roots of 1”. In:Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honour of J. Diximier, Paris/Fr. 1989. Vol. 92. Progress in Mathematics. 1990, pp. 471–506

    work page 1989

  6. [6]

    Quantum groups

    V. G. Drinfel’d. “Quantum groups”. In:Proceedings of the international congress of mathematicians (ICM)(Aug. 3–11, 1986). Ed. by Andrew M. Gleason. Vol. 1. Berkeley, California, 1987, pp. 798–820

    work page 1986

  7. [7]

    Modular Double of Quantum Group

    L. D. Faddeev. “Modular Double of Quantum Group”. In:Conférence Moshé Flato 1999. Ed. by Giuseppe Dito and Daniel Sternheimer. Springer Netherlands, 2000, pp. 149–156. doi: 10.1007/978-94-015-1276-3. arXiv: math/9912078 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978-94-015-1276-3 1999

  8. [8]

    Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality

    L. D. Faddeev, R. M. Kashaev, and A. Yu. Volkov. “Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality”. In:Communications in Mathemati- cal Physics219.1 (2001), pp. 199–219.issn: 0010-3616. doi: 10.1007/s002200100412. arXiv: hep-th/0006156 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s002200100412 2001

  9. [9]

    Quantum Dilogarithm

    L.D. Faddeev and R.M. Kashaev. “Quantum Dilogarithm”. In:Modern Physics Letters A 09.05 (Feb. 1994), pp. 427–434.doi: 10.1142/s0217732394000447 . arXiv: hep- th/9310070 [hep-th]

    work page doi:10.1142/s0217732394000447 1994

  10. [10]

    Evaluation of state integrals at rational points

    Stavros Garoufalidis and Rinat Kashaev. “Evaluation of state integrals at rational points”. In:Communications in Number Theory and Physics9.3 (2015), pp. 549–582. issn: 1931-4523. doi: 10.4310/CNTP.2015.v9.n3.a3. arXiv: 1411.6062 [math.GT]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4310/cntp.2015.v9.n3.a3 2015

  11. [11]

    The descendant colored Jones polynomials

    Stavros Garoufalidis and Rinat Kashaev. “The descendant colored Jones polynomials”. In: Pure Appl. Math. Q.19.5 (2023), pp. 2307–2334.issn: 1558-8599. doi: 10.4310/ PAMQ.2023.v19.n5.a2. arXiv: 2108.07553 [math.GT]

    work page arXiv 2023

  12. [12]

    George Gasper and Mizan Rahman.Basic hypergeometric series. 2nd ed. Vol. 96. Encycl. Math. Appl. Cambridge: Cambridge University Press, 2004.isbn: 0-521-83357-4

    work page 2004

  13. [13]

    Forthcoming book

    Nathan Geer and Bertrand Patureau-Mirand.Quantum topology via modified traces. Forthcoming book

    work page

  14. [14]

    Braiding Operator via Quantum Cluster Algebra

    Kazuhiro Hikami and Rei Inoue. “Braiding operator via quantum cluster algebra”. In: Journal of Physics A: Mathematical and Theoretical47.47 (Nov. 2014), p. 474006.doi: 10.1088/1751-8113/47/47/474006. arXiv: 1404.2009 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1751-8113/47/47/474006 2014

  15. [15]

    Braiding for the quantum gl_2 at roots of unity

    R. Kashaev and N. Reshetikhin. “Braiding for quantumgl2 at roots of unity”. In: Noncommutative geometry and representation theory in mathematical physics. Vol. 391. Contemp. Math. Amer. Math. Soc., Providence, RI, 2005, pp. 183–197.isbn: 0-8218- 3718-4. doi: 10.1090/conm/391/07329. arXiv: math/0410182v1 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/conm/391/07329 2005

  16. [16]

    Invariants of tangles with flat connections in their complements

    R. Kashaev and N. Reshetikhin. “Invariants of tangles with flat connections in their complements”. In:Graphs and Patterns in Mathematics and Theoretical Physics. Amer- ican Mathematical Society, 2005, pp. 151–172.doi: 10.1090/pspum/073/2131015 . arXiv: 1008.1384 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/pspum/073/2131015 2005

  17. [17]

    A Link Invariant from Quantum Dilogarithm

    R. M. Kashaev. “A link invariant from quantum dilogarithm”. In:Modern Physics Letters A 10.19 (1995), pp. 1409–1418.issn: 0217-7323. doi: 10.1142/S0217732395001526. arXiv: q-alg/9504020

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217732395001526 1995

  18. [18]

    Heisenberg Double and Pentagon Relation

    R. M. Kashaev. “The Heisenberg double and the pentagon relation”. In:St. Petersbg. Math. J.8.4 (1996), pp. 63–74.issn: 1061-0022. arXiv:q-alg/9503005

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  19. [19]

    Star-square and tetrahedron equations in the Baxter-Bazhanov model

    R. M. Kashaev, V. V. Mangazeev, and Yu. G. Stroganov. “Star-square and tetrahedron equations in the Baxter-Bazhanov model”. In:International Journal of Modern Physics A 08.08 (Mar. 1993), pp. 1399–1409.doi: 10.1142/s0217751x93000588. REFERENCES 49

    work page doi:10.1142/s0217751x93000588 1993

  20. [20]

    Octahedral developing of knot complement I: pseudo-hyperbolic structure

    Hyuk Kim, Seonhwa Kim, and Seokbeom Yoon. “Octahedral developing of knot com- plement I: Pseudo-hyperbolic structure”. In:Geometriae Dedicata197.1 (Jan. 2018), pp. 123–172.doi: 10.1007/s10711-018-0323-8. arXiv: 1612.02928 [math.GT]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10711-018-0323-8 2018

  21. [21]

    George Lusztig.Introduction to quantum groups. Vol. 110. Prog. Math. Boston, MA: Birkhäuser, 1993. isbn: 0-8176-3712-5

    work page 1993

  22. [22]

    A quantization of theSL2(C) Chern-Simons invariant of tangle exteriors

    Calvin McPhail-Snyder. “A quantization of theSL2(C) Chern-Simons invariant of tangle exteriors”. Sept. 2025. arXiv:2509.xxxxxv1 [math.GT]

    work page 2025

  23. [23]

    Calvin McPhail-Snyder.Hyperbolic structures on link complements, octahedral decompo- sitions, and quantumsl2. Mar. 11, 2022. arXiv:2203.06042v2 [math.GT]

    work page arXiv 2022

  24. [24]

    Octahedral coordinates from the Wirtinger presentation

    Calvin McPhail-Snyder. “Octahedral coordinates from the Wirtinger presentation”. In: Geom. Dedicata219.4 (May 2025). arXiv:2404.19155 [math.GT]

    work page arXiv 2025

  25. [25]

    The colored Jones polynomials and the simplicial volume of a knot

    Hitoshi Murakami and Jun Murakami. “The colored Jones polynomials and the simplicial volume of a knot”. In:Acta Mathematica 186.1 (2001), pp. 85–104.doi: 10.1007/ bf02392716. arXiv: math/9905075 [math.GT]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  26. [26]

    Hitoshi Murakami and Yoshiyuki Yokota.Volume conjecture for knots. Vol. 30. Springer- Briefs Math. Phys. Singapore: Springer, 2018.isbn: 978-981-13-1149-9; 978-981-13-1150-

    work page 2018

  27. [27]

    doi: 10.1007/978-981-13-1150-5

    work page doi:10.1007/978-981-13-1150-5

  28. [28]

    Extended Bloch group and the Cheeger-Chern-Simons class

    Walter D. Neumann. “Extended Bloch group and the Cheeger-Chern-Simons class”. In: Geometry & Topology8 (2004), pp. 413–474.issn: 1465-3060. doi: 10.2140/gt.2004. 8.413. arXiv: math/0307092 [math.GT]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.2140/gt.2004 2004

  29. [29]

    Quasitriangularity of quantum groups at roots of1

    Nicolai Reshetikhin. “Quasitriangularity of quantum groups at roots of1”. In:Com- munications in Mathematical Physics170.1 (1995), pp. 79–99.issn: 0010-3616. doi: 10.1007/BF02099440

    work page doi:10.1007/bf02099440 1995

  30. [30]

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

    Gus Schrader and Alexander Shapiro. “A cluster realization ofUq(sl2) from quantum character varieties”. In:Invent. Math.216.3 (2019), pp. 799–846.issn: 0020-9910. doi: 10.1007/s00222-019-00857-6. arXiv: 1607.00271 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00222-019-00857-6 2019

  31. [31]

    With appendices by Michael Müger and Alexis Virelizier.Vol

    Vladimir Turaev.Homotopy quantum field theory. With appendices by Michael Müger and Alexis Virelizier.Vol. 10. EMS Tracts Math. Zürich: European Mathematical Society (EMS), 2010. isbn: 978-3-03719-086-9. doi: 10.4171/086

    work page doi:10.4171/086 2010

  32. [32]

    Classical solutions of the quantum Yang-Baxter equation

    Alan Weinstein and Ping Xu. “Classical solutions of the quantum Yang-Baxter equation”. English. In:Commun. Math. Phys.148.2 (1992), pp. 309–343.issn: 0010-3616. doi: 10.1007/BF02100863

    work page doi:10.1007/bf02100863 1992

  33. [33]

    Ptolemy coordinates, Dehn invariant, and the A-polynomial

    Christian K. Zickert. “Ptolemy coordinates, Dehn invariant and theA-polynomial”. In: Mathematische Zeitschrift 283.1-2 (2016), pp. 515–537.issn: 0025-5874. doi: 10.1007/ s00209-015-1608-3. arXiv: 1405.0025 [math.GT]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  34. [34]

    The volume and Chern-Simons invariant of a representation

    Christian K. Zickert. “The volume and Chern-Simons invariant of a representation”. In: Duke Mathematical Journal150.3 (2009), pp. 489–532.issn: 0012-7094. doi: 10.1215/ 00127094-2009-058. arXiv: 0710.2049 [math.GT]. C.M-S.: Department of Mathematics, Duke University Email address: calvin@sl2.site N.R.: Department of Mathematics, Yau Center for Mathematica...

    work page arXiv 2009