Two-sided invertibility of cell maps and connecting-system hypotheses in the groupoid intertwiner-twist construction for the dynamical Yang-Baxter equation

Ren (arXiv:2311.10504, 2023) recast the theory of intertwiners and twists for the dynamical Yang-Baxter equation in the language of groupoid-graded vector spaces and transfer operators: starting from a dynamical R-matrix on one groupoid and a 'connecting system' of arrows in a groupoid linking it to another, an invertible-enough map j produces a new operator, and the main theorem (Proposition 4.12) guarantees it solves the dynamical Yang-Baxter equation under quasi-uniqueness of the connecting system, right invertibility of the auxiliary maps, and the assumption that the connecting groupoid has no multi-edges. The construction is verified by hand for two examples (A_{2L-3} to D_L elliptic, and A_11 to E_6 trigonometric, arising from Ocneanu-Roche cell calculus and the Fendley-Ginsparg orbifold construction). The open problem is the verification the paper announces but does not perform: that the intertwiners of Pasquier, Roche, and Di Francesco-Zuber arise from cell twists. The unitarity conditions available in the cell-calculus literature imply only left invertibility of a cell map (Remark 7.4), so the status of these intertwiners as cell twists - and hence as sources of new dynamical R-matrices - is undetermined. No published verification of these families as cell twists is known.

Background

An $R$ -matrix is a solution of the Yang-Baxter equation (YBE), a cubic functional equation for an operator $\check{R}(z)$ depending on a spectral parameter $z$ ; such operators encode the local Boltzmann weights of exactly solvable two-dimensional lattice models and the braiding of quantum groups. A dynamical $R$ -matrix $\check{R}(z,\lambda)$ depends additionally on a dynamical parameter $\lambda$ that is shifted between tensor factors, and satisfies the dynamical Yang-Baxter equation (DYBE); the archetype is Felder's elliptic $R$ -matrix of the elliptic quantum groups, out of which grew the theory of dynamical quantum groups (Babelon-Bernard-Billey; Etingof-Varchenko; see the exposition in the book of Etingof-Latour, 2005).

Ren's formulation geometrizes the dynamical parameter. A groupoid is a small category in which every morphism (arrow) is invertible; a groupoid-graded vector space $V^{\pi}$ decomposes as a direct sum of components indexed by the arrows of $\pi$ . When the grading groupoid is finite (a 'restricted' groupoid, such as the path groupoid of an ADE Dynkin diagram, i.e. one of the simply-laced Dynkin diagrams of Lie types $A$ , $D$ , and $E$ ), the class of dynamical $R$ -matrices that can be represented is strictly larger than on ordinary or $\mathbb{Z}$ -graded spaces - this is the setting of restricted SOS (solid-on-solid) models developed by Felder and Ren (SIGMA 17 (2021) 005).

Two classical operations relate $R$ -matrices. An intertwiner is a map $C$ conjugating one dynamical $R$ -matrix into another, first arising in Baxter's 1973 proof that the zero-field eight-vertex model is equivalent to an SOS model. A twist is an invertible map $J$ that deforms a given $R$ -matrix into a new one; Drinfeld's twist of a quasitriangular (quasi-)Hopf algebra (a Hopf algebra whose coproduct is compatible with a distinguished universal $R$ -matrix) is the non-dynamical prototype, and its dynamical analogues shift the dynamical parameter. Ren re-expresses both in a transfer-operator language: $J$ is a twist exactly when $\check{R}_{J}=J^{-1}\check{R}J$ and suitable auxiliary intertwiners $Q$ exist on triple tensor products.

To twist a dynamical $R$ -matrix $\check{R}_{1}$ living on a groupoid $\pi_{1}$ into one on a different groupoid $\pi_{2}$ , one must relate the two gradings. This is done by a connecting groupoid $\pi$ and a connecting system, a coherent choice of arrows in $\pi$ realizing the paths of $\pi_{2}$ ; the system is unique when the choice is forced, and quasi-unique (Definition 4.11) when it is unique only up to arrows irrelevant to the transported operator. The main theorem, Proposition 4.12, states that if the connecting system is quasi-unique, the connecting groupoid $\pi$ has no multi-edges (no two arrows with the same source and target), and $j$ is a 'generalized twist' - right-invertible maps $q,q_{1},q_{2}$ satisfying weight-zero ('ice rule') conditions and a factorization property - then the transported operator $\check{R}_{2}=j^{-1}*_{\circ}(\check{R}_{1}j)$ (a convolution product over arrows) is a Yang-Baxter operator on $V^{\pi_{2}}$ .

The combinatorial input comes from Ocneanu's cell calculus and its lattice-model version due to Roche: a cell system is a map $c$ between groupoid-graded Hom spaces, built from local 'cells' attached to the graph, satisfying closure relations. Invertibility of such maps is defined one-sidedly with respect to the convolution products: $c$ is left invertible if a one-sided inverse exists (equations (42), (99) of the source); it is invertible only if a two-sided inverse does. Proposition 7.3 shows that an invertible cell system is a cell twist, and Proposition 4.14 then implies that $j=c^{-1}*_{\otimes}c^{-1}$ is a twist, from which a new dynamical $R$ -matrix is manufactured. Crucially, the unitarity conditions proved in the Ocneanu-Roche literature imply only left invertibility (Remark 7.4 of the source); right invertibility must be verified separately. The paper does this by hand only for its two examples (Lemmas 7.6 and 7.9: the $A_{2L-3}\to D_{L}$ elliptic case, a reformulation of Roche's example and of the Fendley-Ginsparg non-critical orbifold construction, and the $A_{11}\to E_{6}$ trigonometric case). Section 7.3 then states that for the many other intertwiners constructed by Pasquier, by Roche, and by Di Francesco-Zuber (the $SU(N)$ lattice models associated with graphs), 'we only need to check the right invertibility and also the quasi-unique connecting system condition' - a verification announced but not carried out.

Problem Statement

Within the formalism of Ren (arXiv:2311.10504); all numbered equations and definitions referenced below are to that paper. The problem is to carry out the verification announced, but not performed, in section 7.3 of the source: to determine whether the intertwiners constructed in the lattice-model literature arise from cell twists in the sense of Definitions 4.13/7.2 - Pasquier's IRF (interaction-round-a-face) intertwiners, Roche's cell systems beyond the two treated in the paper, and the Di Francesco-Zuber $SU(N)$ graph intertwiners. As the source itself states, for these families 'we only need to check the right invertiblity and also the quasi-unique connecting system condition': the unitarity conditions of the cell-calculus literature imply only left invertibility (Remark 7.4), whereas Propositions 4.14 and 7.3 require invertibility - two-sided, i.e. right invertibility in the sense of equations (44)/(99)-(100) in addition to the known left invertibility - to conclude that $j=c^{-1}*_{\otimes}c^{-1}$ is a twist yielding a new dynamical $R$ -matrix $\check{R}_{2}$ satisfying the Yang-Baxter equation. Where the cell maps fail to be two-sided invertible, or the connecting system fails to be quasi-unique, this failure should be exhibited; where they succeed, the resulting dynamical $R$ -matrices should be identified, as the paper does for its two examples (the $A_{2L-3}\to D_{L}$ elliptic case via Lemma 7.6 and the $A_{11}\to E_{6}$ trigonometric case via Lemma 7.9).

A complete answer settles, in the terminology of the source, the status of the named intertwiner families as cell twists and identifies the dynamical Yang-Baxter solutions that the construction produces from them.

References

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