Completing the categorical Temperley-Lieb framework: weakened rigidity for $\mathrm{Mod}_{\mathrm{TL}}$ at roots of unity, the dilute commutor versus integrable 19-vertex weights, and fusion with braiding for the blob and affine families

The 2018 construction of Belletête and Saint-Aubin endows the category $\mathrm{Mod}_{\mathrm{TL}}$ of modules over the Temperley-Lieb (TL) algebra — functors from the TL diagram category to vector spaces, with the Read-Saleur gluing fusion $\times_f$ — with a commutor $\eta$, a twist $\theta$ and a compatible monodromy, for generic loop weight ($q$ not a root of unity). This problem asks for the completion of that framework in the regimes and for the families where it is known to fail or was never extended, as posed in the source paper's conclusions: (i) identify the appropriate weakened form of rigidity (a replacement for duals, such as duality-and-trace structures) on $\mathrm{Mod}_{\mathrm{TL}}$ at roots of unity, where genuine rigidity is obstructed because it would force $\times_f$ to be exact and the projective subcategory $\mathrm{Mod}_{\mathrm{TL}}^{\mathrm{proj}}$ is not even abelian; (ii) determine the categorical relationship between the dilute TL commutor $\eta_{1,1}$ and the Izergin-Korepin/Nienhuis 19-vertex weights that do make the dilute transfer matrices integrable — in particular whether a braided or module-category structure over a dilute analogue of $\mathrm{Mod}_{\mathrm{TL}}$ explains that integrability; and (iii) construct, or rule out, fusion-and-braiding structures for the one-boundary (blob) and affine/periodic TL families — fusion bifunctors for the blob family are entirely missing, and for the affine/periodic family, where fusion products now exist in several frameworks, no braided structure has been constructed and negative evidence exists for the natural candidates. These questions remain open: the affine-fusion existence sub-part was resolved affirmatively after 2018, but no later work supplies the weakened rigidity structure, the dilute-integrability explanation, or the braiding.

Background

The Temperley-Lieb algebra $\mathrm{TL}_N(\delta)$ is generated by $e_1,\dots,e_{N-1}$ with relations $e_i^2 = \delta e_i$, $e_i e_{i+1} e_i = e_i$ and $e_i e_j = e_j e_i$ for $|i-j|\ge 2$; diagrammatically it is spanned by crossingless link diagrams in a rectangle, with closed loops evaluating to $\delta$. Specializing $\delta = -q - q^{-1}$ connects it to the loop representation of $U_q(\mathfrak{sl}_2)$ and to the solvable O(n) and Potts loop models. Its lattice-relevant representations are the standard modules $W_{n,x}$, indexed by the number $n$ of through-lines (defects) and a defect fugacity $x$. The TL diagram category (objects natural numbers, morphisms diagrams) is monoidal and pivotal — every object carries left and right duals compatible with the monoidal structure; it is semisimple for generic $q$ and non-semisimple at roots of unity, where the Jones-Wenzl projectors (the idempotents annihilated by every generator $e_i$, projecting onto the summands without through-lines) become nontrivial idempotents.

Belletête and Saint-Aubin (2018) work in $\mathrm{Mod}_{\mathrm{TL}}$, the category whose objects are modules over the algebras $\mathrm{TL}_N$ realized as linear functors from the TL category to vector spaces; the standard modules arise as evaluation functors. Read and Saleur (2007) had defined a fusion product $\times_f$ on such modules by gluing the corresponding link diagrams through a Jones-Wenzl projector. For generic $q$, Belletête and Saint-Aubin constructed on $(\mathrm{Mod}_{\mathrm{TL}}, \times_f)$ a nontrivial commutor $\eta$ (obtained by translating one link diagram around the other and resolving through sums over shifted defect weights), a twist $\theta$ acting on $W_{n,x}$ by a power of $q$ determined by the contracted pairs of defects, and proved that $\eta$ satisfies the braid relations and is compatible with $\times_f$, giving the category a braided, ribbon-type structure (a braiding equipped with a compatible twist and duals); they computed the monodromy of two standard modules and connected it to the fusion rules and operator content of the O(n) models. This is the framework whose completion is sought.

At roots of unity the construction breaks in structural ways. First, rigidity (the existence of left and right duals for all objects, with evaluation and coevaluation maps satisfying the triangle identities) cannot be implemented: a rigid braided structure would force the fusion bifunctor $\times_f$ to be exact in each variable, but at roots of unity kernels and cokernels of morphisms between projective modules need not be projective. Second, the subcategory $\mathrm{Mod}_{\mathrm{TL}}^{\mathrm{proj}}$ of projective modules is stable under $\times_f$ but is not abelian (kernels and cokernels need not stay within it), so it cannot directly host the standard machinery. The neighboring literature provides candidate structures — traces on projective representations of generically semisimple pivotal categories (Geer and Patureau-Mirand); the unique modified trace (a linear functional defined only on a suitable ideal of morphisms and cyclic there, replacing the trace that duals would provide) on the projective ideal of the idempotent-completed TL diagram category at odd roots of unity, together with non-semisimple 3-manifold invariants (De Renzi and Murakami); Frobenius-type weakened duality structures on the Jones-Wenzl algebra object (Grant and Pugh); extended traces along pivotal monoidal functors, with module categories over TL at even roots of unity as an application (Hardiman); and limits of traces of TL (Moore) — but none of these has been adapted to the functor module category $\mathrm{Mod}_{\mathrm{TL}}$ with the gluing fusion $\times_f$.

The dilute Temperley-Lieb algebra adds generators allowing lattice sites to be empty, and underlies the dilute O(n) loop models whose critical behavior is described by the Izergin-Korepin (dilute $A_2^{(2)}$) 19-vertex model; the integrable Boltzmann weights (the local face-operator entries of the transfer matrix) are those of Izergin and Korepin, in the form used by Nienhuis. Belletête (2015) computed the fusion rules for dilute TL modules. In the 2018 paper the authors evaluate the elementary braiding $\eta_{1,1}$ — the commutor on the fusion of two one-defect standard modules — in both the ordinary and the dilute settings: in the dilute case the naive diagrammatic operator $x_i$ fails the inversion relation required of local face operators, while the Izergin-Korepin/Nienhuis weights $\hat{x}_i$ satisfy all three integrability conditions (the Yang-Baxter-type, inversion and boundary relations). Whether a categorical commutor on a dilute analogue of $\mathrm{Mod}_{\mathrm{TL}}$ underlies the integrability of these weights is not known.

Finally, the one-boundary TL (blob) algebra introduces a boundary generator $b$, and the affine/periodic TL algebra adds translation generators and underlies loop models on the annulus and torus. For these families the 2018 framework was never extended: the source states it is not known whether one can define a fusion product between their modules, or make a braided category out of their link diagrams. Subsequent work has constructed fusion in the affine/periodic case in several inequivalent frameworks (topological gluing of standard modules; two commuting fusion products; two-marked-point fusion, culminating in a general bifunctor on enlarged periodic TL modules), while Gainutdinov and Saleur showed that the natural affine fusion is non-commutative and admits no braiding (only a semi-braiding between two chiral products); for the blob family only an ad hoc lattice-fusion prescription from the literature on logarithmic minimal models (non-semisimple lattice models whose continuum limits are logarithmic, non-diagonalizable conformal field theories) exists. What is missing everywhere is the categorical completion: coherent fusion together with braiding, twists, monodromy and duality/traces, or a rigorous identification of the obstruction.

Problem Statement

Overarching objective: complete — or rigorously delimit — the categorical framework (fusion, braiding, twist, duality/traces) for module categories over Temperley-Lieb-type algebras in the regimes and families where the Belletête-Saint-Aubin construction on $(\mathrm{Mod}_{\mathrm{TL}}, \times_f)$ is absent or fails. Concretely, with $\mathrm{TL}$ the Temperley-Lieb category at loop weight $\delta = -q - q^{-1}$, $\mathrm{Mod}_{\mathrm{TL}}$ the category of modules realized as functors $\mathrm{TL} \to \mathrm{Vect}_{\mathbb{C}}$, and $\times_f$ the Read-Saleur fusion through a Jones-Wenzl projector:

(1) Weakened rigidity at roots of unity. For $q$ a root of unity, answer the source's question of what the appropriate weaker forms of rigidity for $(\mathrm{Mod}_{\mathrm{TL}}, \times_f)$ are: identify a structure that replaces rigidity (a replacement for duals) — for example a system of possibly non-invertible duality morphisms, or a modified or relative trace on a suitable ideal such as the projective subcategory $\mathrm{Mod}_{\mathrm{TL}}^{\mathrm{proj}}$. A characterization theorem (uniqueness or classification of such structures) is an equally valid answer.

(2) Dilute commutor and integrability. Determine the exact relationship between the dilute elementary commutor $\eta_{1,1}$ (the source's Section 5 computation) and the Izergin-Korepin/Nienhuis 19-vertex weights $\hat{x}_i$: either construct a possibly weakened braided structure, or a module-category structure, over a dilute analogue of $\mathrm{Mod}_{\mathrm{TL}}$ whose categorical commutor reproduces the integrable weights — thereby explaining why $\hat{x}_i$ satisfies all three integrability conditions (the Yang-Baxter-type, inversion and boundary relations) while the naive diagrammatic operator $x_i$ fails the inversion relation — or prove that no such structure exists, with precisely stated hypotheses.

(3) Fusion and braiding for the blob and affine/periodic families. For the one-boundary (blob) algebra: construct a fusion bifunctor on its module families, or prove impossibility, with commutativity and associativity statements comparable to those now available for the periodic algebra. For the affine/periodic family, where fusion products now exist in several frameworks: construct a commutor/braided structure compatible with one of the existing fusion products (together with the associated twist and monodromy), or prove a no-go theorem clarifying whether the known obstructions — the non-commutativity of the affine fusion rules and the absence of braiding in the Gainutdinov-Saleur framework — are essential or framework-dependent.

Complete resolution requires each of (1)-(3) to be settled, either by an explicit construction with proofs of the required coherence and compatibility axioms, or by a rigorous impossibility theorem with explicitly stated, physically relevant hypotheses; contributions resolving individual parts constitute meaningful partial progress. Out of scope: purely formal restatements; constructions on different categories that do not restrict to the objects above (for example only on the TL diagram category rather than $\mathrm{Mod}_{\mathrm{TL}}$, or only at generic $q$); and no-go results that silently strengthen the hypotheses beyond those of the source.

References

1. Jonathan Belletête, Yvan Saint-Aubin, 'Fusion and monodromy in the Temperley-Lieb category', SciPost Physics 5, 041 (2018), DOI 10.21468/SciPostPhys.5.4.041, https://doi.org/10.21468/SciPostPhys.5.4.041 (arXiv:1802.09203, https://arxiv.org/abs/1802.09203)
2. Nathan Read, Hubert Saleur, 'Associative-algebraic approach to logarithmic conformal field theories', Nuclear Physics B 777, 316-351 (2007), DOI 10.1016/j.nuclphysb.2007.03.033, https://doi.org/10.1016/j.nuclphysb.2007.03.033 (arXiv:hep-th/0701117, https://arxiv.org/abs/hep-th/0701117)
3. Jonathan Belletête, 'The fusion rules for the Temperley-Lieb algebra and its dilute generalization', Journal of Physics A 48, 395205 (2015), DOI 10.1088/1751-8113/48/39/395205, https://doi.org/10.1088/1751-8113/48/39/395205 (arXiv:1505.02112, https://arxiv.org/abs/1505.02112)
4. Azat M. Gainutdinov, Hubert Saleur, 'Fusion and braiding in finite and affine Temperley-Lieb categories', preprint (2016), arXiv:1606.04530, https://arxiv.org/abs/1606.04530
5. Azat M. Gainutdinov, Jesper Lykke Jacobsen, Hubert Saleur, 'A fusion for the periodic Temperley-Lieb algebra and its continuum limit', Journal of High Energy Physics 2018(11), 117 (2018), DOI 10.1007/JHEP11(2018)117, https://doi.org/10.1007/JHEP11(2018)117 (arXiv:1712.07076, https://arxiv.org/abs/1712.07076)
6. Jonathan Belletête, Yvan Saint-Aubin, 'On the computation of fusion over the affine Temperley-Lieb algebra', Nuclear Physics B 937, 333-370 (2018), DOI 10.1016/j.nuclphysb.2018.10.016, https://doi.org/10.1016/j.nuclphysb.2018.10.016 (arXiv:1802.03575, https://arxiv.org/abs/1802.03575)
7. Jonathan Belletête, Azat M. Gainutdinov, Jesper Lykke Jacobsen, Hubert Saleur, Tiago S. Tavares, 'Topological defects in lattice models and affine Temperley-Lieb algebra', Communications in Mathematical Physics 400, 1203-1254 (2023), DOI 10.1007/s00220-022-04618-0, https://doi.org/10.1007/s00220-022-04618-0 (arXiv:1811.02551, https://arxiv.org/abs/1811.02551)
8. Yacine Ikhlef, Alexi Morin-Duchesne, 'Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models', SciPost Physics 12, 030 (2022), DOI 10.21468/SciPostPhys.12.1.030, https://doi.org/10.21468/SciPostPhys.12.1.030 (arXiv:2105.10240, https://arxiv.org/abs/2105.10240)
9. Yacine Ikhlef, Alexi Morin-Duchesne, 'Fusion of irreducible modules in the periodic Temperley-Lieb algebra', SciPost Physics 17, 132 (2024), DOI 10.21468/SciPostPhys.17.5.132, https://doi.org/10.21468/SciPostPhys.17.5.132 (arXiv:2312.14837, https://arxiv.org/abs/2312.14837)
10. Yacine Ikhlef, Alexi Morin-Duchesne, 'Fusion in the periodic Temperley-Lieb algebra: general definition of a bifunctor', SciPost Physics Core 9, 053 (2026), DOI 10.21468/SciPostPhysCore.9.3.053, https://doi.org/10.21468/SciPostPhysCore.9.3.053 (arXiv:2509.11756, https://arxiv.org/abs/2509.11756)
11. Youssef Mousaaid, Alistair Savage, 'Affinization of monoidal categories', Journal de l'École polytechnique — Mathématiques 8, 791-829 (2021), DOI 10.5802/jep.158, https://doi.org/10.5802/jep.158 (arXiv:2010.13598, https://arxiv.org/abs/2010.13598)
12. Marco De Renzi, Jun Murakami, 'Non-semisimple 3-manifold invariants derived from the Kauffman bracket', Quantum Topology 13, 255-333 (2022), DOI 10.4171/QT/164, https://doi.org/10.4171/QT/164 (arXiv:2007.10831, https://arxiv.org/abs/2007.10831)
13. Nathan Geer, Bertrand Patureau-Mirand, 'The trace on projective representations of quantum groups', Letters in Mathematical Physics 108, 117-140 (2018), DOI 10.1007/s11005-017-0993-4, https://doi.org/10.1007/s11005-017-0993-4 (arXiv:1610.09129, https://arxiv.org/abs/1610.09129)
14. Joseph Grant, Mathew Pugh, 'Frobenius algebra objects in Temperley-Lieb categories at roots of unity', Glasgow Mathematical Journal, 1-20 (2026), first-view article, DOI 10.1017/S0017089525100876, https://doi.org/10.1017/S0017089525100876 (arXiv:2409.12920, https://arxiv.org/abs/2409.12920)
15. Stephen T. Moore, 'Limits of traces of Temperley-Lieb algebras', Journal of Algebra 642, 95-147 (2024), DOI 10.1016/j.jalgebra.2023.12.018, https://doi.org/10.1016/j.jalgebra.2023.12.018 (arXiv:2212.11592, https://arxiv.org/abs/2212.11592)
16. Leonard Hardiman, 'Extending the trace of a pivotal monoidal functor', Communications in Mathematical Physics 381, 1055-1090 (2021), DOI 10.1007/s00220-020-03909-8, https://doi.org/10.1007/s00220-020-03909-8 (arXiv:1911.09024, https://arxiv.org/abs/1911.09024)
17. Florence Boileau, Alexi Morin-Duchesne, Yvan Saint-Aubin, 'Fusion hierarchies, T-systems and Y-systems for the dilute $A_2^{(2)}$ loop models on a strip', Journal of Statistical Mechanics: Theory and Experiment 2023, 033102 (2023), DOI 10.1088/1742-5468/acb7d7, https://doi.org/10.1088/1742-5468/acb7d7 (arXiv:2211.09017, https://arxiv.org/abs/2211.09017)
18. A. G. Izergin, V. E. Korepin, 'The inverse scattering method approach to the quantum Shabat-Mikhailov model', Communications in Mathematical Physics 79, 303-316 (1981), DOI 10.1007/BF01208496, https://doi.org/10.1007/BF01208496
19. Bernard Nienhuis, 'Critical and multicritical O(n) models', Physica A 163, 152-157 (1990), DOI 10.1016/0378-4371(90)90325-M, https://doi.org/10.1016/0378-4371(90)90325-M
20. Alexi Morin-Duchesne, Jørgen Rasmussen, David Ridout, 'Boundary algebras and Kac modules for logarithmic minimal models', Nuclear Physics B 899, 677-769 (2015), DOI 10.1016/j.nuclphysb.2015.08.017, https://doi.org/10.1016/j.nuclphysb.2015.08.017 (arXiv:1503.07584, https://arxiv.org/abs/1503.07584)