A solution to the quantum Yang--Baxter equation associated with a universal two-valued algebraic group
Abstract
We construct spectral-parameter solutions of the quantum Yang--Baxter equation from the associativity condition of the universal symmetric $2$-algebraic two-valued group, whose multiplication law depends on four coefficients k2,k4,k6,k8 subject to the single relation 4k8−k42+k2k6=0 -- which coincides with the three-dimensional WDVV equation for Dubrovin's normalised potential. Every three-dimensional Frobenius manifold with a cyclic vector field carries such a spectral two-valued group, whose finite branch points are the eigenvalues of the multiplication operator. For an arbitrary unital algebra we show that a two-parameter spectral R-operator built from multiplication and unit has Yang--Baxter defect proportional to an explicit tensor built from the associator; for tangent Frobenius algebras this yields a regular, unitary 9×9 R-matrix satisfying the quantum Yang--Baxter equation exactly at WDVV points, with spectral parameter the Abel coordinate of a bielliptic curve whose elliptic quotient is the spectral curve, together with an associated integrable chain. The A3 Frobenius manifold is worked out explicitly.
Related
- Global Strong Solution to Multi-Dimensional Compressible Quantum Navier-Stokes Equation in Bounded Domains with Large Initial Data
- A multiplicative universal property for real algebraic K-theory
- On algebraic dynamics of birational automorphisms of Calabi-Yau manifolds and the universal Coxeter groups
- A Low-rank ADI Algorithm for the Numerical Solution of Large Discrete-time Non-symmetric Algebraic Riccati Equations
- On convergence of formal Dulac series, satisfying an algebraic q-difference equation
BibTeX
@misc{buchstaber2026solution,
title = {{A solution to the quantum Yang--Baxter equation associated with a universal two-valued algebraic group}},
author = {Victor M. Buchstaber},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02354},
eprint = {2610.02354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02354},
}
RIS
TY - UNPB
TI - A solution to the quantum Yang--Baxter equation associated with a universal two-valued algebraic group
AU - Victor M. Buchstaber
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02354
UR - https://arxiv.org/abs/2610.02354
AB - We construct spectral-parameter solutions of the quantum Yang--Baxter equation from the associativity condition of the universal symmetric $2$-algebraic two-valued group, whose multiplication law depends on four coefficients k_2,k_4,k_6,k_8 subject to the single relation 4k_8−k_4^2+k_2k_6=0 -- which coincides with the three-dimensional WDVV equation for Dubrovin's normalised potential. Every three-dimensional Frobenius manifold with a cyclic vector field carries such a spectral two-valued group, whose finite branch points are the eigenvalues of the multiplication operator. For an arbitrary unital algebra we show that a two-parameter spectral R-operator built from multiplication and unit has Yang--Baxter defect proportional to an explicit tensor built from the associator; for tangent Frobenius algebras this yields a regular, unitary 9×9 R-matrix satisfying the quantum Yang--Baxter equation exactly at WDVV points, with spectral parameter the Abel coordinate of a bielliptic curve whose elliptic quotient is the spectral curve, together with an associated integrable chain. The A_3 Frobenius manifold is worked out explicitly.
ER -