Vertex-type quantum R-matrices (solutions of the quantum Yang-Baxter equation) frequently satisfy, in addition, the associative Yang-Baxter equation (AYBE), a quadratic functional equation of matrix multiplication type with two spectral parameters and a Planck constant. For dynamical (IRF/face-type) quantum R-matrices, which depend on extra shift parameters and satisfy the Gervais-Neveu-Felder (dynamical) Yang-Baxter equation, no analogue of the AYBE is known: the problem is to find a quadratic associative-type relation, with appropriately shifted spectral and dynamical arguments, that is satisfied by the fully dynamical R-matrices (in particular the rational and elliptic GL_N ones), or to prove that no such relation of the expected form exists. A concrete companion target (required for a full resolution) is to prove or disprove explicitly the gauge equivalence between the difference-form rational GL_N vertex R-matrix of Atalikov-Zotov (2023) and the non-difference rational R-matrices obtained from vector bundles on the cuspidal cubic curve by Burban-Kreussler and Burban-Henrich.

Background

A quantum R-matrix in this context is a matrix-valued function R~12(z)\tilde R_{12}(z) of a spectral parameter zz and a Planck constant ℏ\hbar, acting on the tensor product of two fundamental GLNGL_N representations and satisfying the quantum Yang-Baxter equation (QYBE). The associative Yang-Baxter equation (AYBE), introduced in this form by Fomin-Kirillov and Polishchuk, is a quadratic relation for R-matrices of vertex type depending on a second parameter η\eta: schematically R12ηR23η=R13ηR12η−ℏ+R23η−ℏR13ηR^{\eta}_{12}R^{\eta}_{23}=R^{\eta}_{13}R^{\eta-\hbar}_{12}+R^{\eta-\hbar}_{23}R^{\eta}_{13}, where the superscript denotes the value of the second parameter. It generalizes the scalar addition formula for the elliptic Kronecker function (the doubly periodic function underlying elliptic solutions of the QYBE), and a unitary skew-symmetric solution of the AYBE automatically satisfies the QYBE, so the AYBE packages a hierarchy of R-matrix identities (classical limits, unitarity, higher-order relations) into one equation. Elliptic, trigonometric and rational solutions of the AYBE, and their geometric classification via vector bundles on elliptic curves and their degenerations, have been studied extensively (Polishchuk; Schedler; Burban-Kreussler; Burban-Henrich; Lekili-Polishchuk). Dynamical or IRF-type (face-type) R-matrices R~12(ℏ,z1−z2∣q)\tilde R_{12}(\hbar, z_1-z_2 \mid q) arise in face (interaction-round-a-face) lattice models and in Felder's elliptic quantum groups (dynamical generalizations of quantum groups built from such R-matrices): besides the spectral parameters they depend on dynamical parameters q1,…,qNq_1,\dots,q_N, and they satisfy the quantum dynamical (Gervais-Neveu-Felder) Yang-Baxter equation, in which consecutive factors act on the dynamical arguments by shifts along the tensor factors. Vertex and dynamical R-matrices are related by the IRF-Vertex (intertwining) transformation, a gauge conjugation by a matrix g(z,q)g(z,q). Semi-dynamical R-matrices (Arutyunov-Chekhov-Frolov type) depend separately on z1,z2z_1,z_2 and satisfy yet another Yang-Baxter-type equation. Sechin and Zotov (2016) proved a version of the AYBE with shifted spectral parameters for the semi-dynamical elliptic R-matrix, and showed that the plain AYBE fails for Felder's fully dynamical R-matrix (in the rational case the residual is an explicit spectral-parameter-independent tensor expression), leaving only a dressed cubic identity with non-cancelling gauge factors. Atalikov and Zotov (2023) constructed the higher-rank (GLNGL_N) rational vertex R-matrix degenerating for N=2N=2 to the 11-vertex R-matrix (a distinguished rational non-difference GL2GL_2 R-matrix with eleven nonzero matrix entries), proved its AYBE, and derived the rational IRF-Vertex relations to the dynamical and semi-dynamical R-matrices; in that context they state that, to their best knowledge, an analogue of the AYBE for dynamical R-matrices is unknown. They also note that the non-difference rational GLNGL_N R-matrix obtained geometrically from vector bundles on the cuspidal cubic curve (a singular rational degeneration of the elliptic curve whose vector bundles nevertheless yield R-matrices) is presumably gauge equivalent to their difference-form R-matrix, without proof. The AYBE is the algebraic engine behind constructions of Lax pairs for higher-rank Landau-Lifshitz field theories, integrable deformations of the principal chiral model, and long-range spin chains, so an associative equation for dynamical R-matrices would extend this machinery to the dynamical/IRF class of models.

Problem Statement

Find an associative Yang-Baxter-type equation for fully dynamical (IRF-type) quantum R-matrices, i.e. a quadratic relation of AYBE type for R-matrices R~12(ℏ,z1−z2∣q)\tilde R_{12}(\hbar, z_1-z_2 \mid q) satisfying the Gervais-Neveu-Felder (quantum dynamical) Yang-Baxter equation, at natural generality — the rational, trigonometric and elliptic classes alike, and with no restriction on the rank NN — or prove that none exists within the class of quadratic relations built from the R-matrix entries with shifted spectral and dynamical arguments. The relation should reduce, in the vertex limit (when the dynamical parameters decouple and the R-matrix becomes of vertex type), to the standard associative Yang-Baxter equation, and should reproduce, on semi-dynamical R-matrices, the known shifted AYBE of Sechin-Zotov; a resolution must either exhibit the equation and prove it for the dynamical R-matrices, or prove an obstruction (the plain AYBE is already known to fail, with an explicit rational-case residual). As a concrete companion target within the same circle of ideas — required for a full resolution alongside the main problem — prove or disprove explicitly the gauge equivalence between the difference-form rational GLNGL_N vertex R-matrix of Atalikov-Zotov (2023) and the non-difference rational R-matrix solutions constructed geometrically from vector bundles on the cuspidal cubic curve by Burban-Kreussler and Burban-Henrich.

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