The 2015 survey "Yang–Baxter Equations, Computational Methods and Applications" records, among its open problems, a long-standing system of five functional equations (its equations (15)–(19)), originating from Nichita–Parashar (2006), that governs when operators of the form R(u,v)(a⊗b)=α(u,v)1⊗ab+β(u,v)ab⊗1−γ(u,v)b⊗aR(u,v)(a\otimes b)=\alpha(u,v)1\otimes ab+\beta(u,v)ab\otimes 1-\gamma(u,v)b\otimes a built from a unital associative algebra satisfy the two-parameter (coloured) quantum Yang–Baxter equation. Only scattered solution families are known (constant triples, a linear two-parameter family, and an exponential one-parameter specialization), and the classification of all solutions has been explicitly stated as open in 2006, 2011 and 2015; later work by the same school (2018, 2020) still treats the surrounding Yang–Baxter classification problems as unresolved. No classification or refutation appears in the literature to date: the problem is open.

Background

The Yang–Baxter equation (YBE) is an algebraic consistency condition that first arose in exactly solved lattice models of statistical mechanics and in the quantum many-body problem, and is now central to quantum groups, knot theory, and integrable systems. In its operator form one seeks an invertible linear map R:V⊗V→V⊗VR:V\otimes V\to V\otimes V on the tensor square of a vector space VV over a field kk. Using leg notation R12=R⊗IR^{12}=R\otimes I, R23=I⊗RR^{23}=I\otimes R, R13=(I⊗τ)(R⊗I)(I⊗τ)R^{13}=(I\otimes\tau)(R\otimes I)(I\otimes\tau), where τ(v⊗w)=w⊗v\tau(v\otimes w)=w\otimes v is the twist map and II the identity, the braid form of the equation reads R12R23R12=R23R12R23R^{12}R^{23}R^{12}=R^{23}R^{12}R^{23}; composing with the twist gives the equivalent quantum Yang–Baxter equation (QYBE) R12R13R23=R23R13R12R^{12}R^{13}R^{23}=R^{23}R^{13}R^{12}. Despite decades of work, the classification of all solutions is a recognized open problem; complete classifications exist only in restricted regimes such as small matrix size with constant entries.

In most physical applications the RR-matrix depends on a spectral parameter. The coloured, or two-parameter, form makes such dependence explicit: a map R:X×X→End⁡k(V⊗V)R:X\times X\to\operatorname{End}_k(V\otimes V), where XX is a parameter set (typically a subset of kk), must satisfy

R12(u,v) R13(u,w) R23(v,w)=R23(v,w) R13(u,w) R12(u,v)for all u,v,w∈X.R^{12}(u,v)\,R^{13}(u,w)\,R^{23}(v,w)=R^{23}(v,w)\,R^{13}(u,w)\,R^{12}(u,v)\qquad\text{for all }u,v,w\in X.

A productive source of solutions is algebra structure. Dăscălescu and Nichita (1999) proved that for a unital associative kk-algebra AA with dim⁡A≥2\dim A\geq2 and constants α,β,γ∈k\alpha,\beta,\gamma\in k, the map a⊗b↦α ab⊗1+β 1⊗ab−γ a⊗ba\otimes b\mapsto\alpha\,ab\otimes 1+\beta\,1\otimes ab-\gamma\,a\otimes b is a Yang–Baxter operator exactly in three cases: α=γ≠0\alpha=\gamma\neq0 with β≠0\beta\neq0; β=γ≠0\beta=\gamma\neq0 with α≠0\alpha\neq0; and α=β=0\alpha=\beta=0 with γ≠0\gamma\neq0. Up to the twist map τ(a⊗b)=b⊗a\tau(a\otimes b)=b\otimes a this constant-coefficient operator coincides with the coloured ansatz R(u,v)R(u,v) of the Problem Statement: the coefficients (α,β,γ)(\alpha,\beta,\gamma) sit here on (ab⊗1, 1⊗ab, a⊗b)(ab\otimes1,\,1\otimes ab,\,a\otimes b), the τ\tau-images of (1⊗ab, ab⊗1, b⊗a)(1\otimes ab,\,ab\otimes1,\,b\otimes a) there. Operators of this shape carry knot-theoretic weight: the link invariant they induce is the Alexander polynomial (a classical two-variable knot invariant), and they seed bialgebras of FRT type (bialgebras universally attached to a matrix solution of the YBE via the Faddeev–Reshetikhin–Takhtajan construction).

Nichita and Parashar (2006) let the three coefficients become functions of the colour parameters, considering

R(u,v)(a⊗b)=α(u,v) 1⊗ab+β(u,v) ab⊗1−γ(u,v) b⊗a,u,v∈X,R(u,v)(a\otimes b)=\alpha(u,v)\,1\otimes ab+\beta(u,v)\,ab\otimes 1-\gamma(u,v)\,b\otimes a,\qquad u,v\in X,

with α,β,γ\alpha,\beta,\gamma arbitrary kk-valued functions on X×XX\times X. Expanding both sides of the coloured QYBE using only the associativity of the product in AA and equating coefficients yields a closed system of five functional equations in the three unknown functions (displayed in the problem statement below, as equations (15)–(19) of the 2015 survey). Every solution of the system produces a coloured Yang–Baxter operator on A⊗AA\otimes A for every unital associative kk-algebra AA. Only scattered solution families are known: the constant triples just described; the linear family α(u,v)=p(u−v)\alpha(u,v)=p(u-v), β(u,v)=q(u−v)\beta(u,v)=q(u-v), γ(u,v)=pu−qv\gamma(u,v)=pu-qv for constants p,q∈kp,q\in k, which gives invertible operators whenever pu≠qvpu\neq qv and qu≠pvqu\neq pv; and the exponential family R(u,v)=S(u−v)R(u,v)=S(u-v), i.e. α(u,v)=eu−v−1\alpha(u,v)=e^{u-v}-1, β(u,v)=q(eu−v−1)\beta(u,v)=q(e^{u-v}-1), γ(u,v)=eu−v−q\gamma(u,v)=e^{u-v}-q, where S(λ)(a⊗b)=(eλ−1) 1⊗ab+q(eλ−1) ab⊗1−(eλ−q) b⊗aS(\lambda)(a\otimes b)=(e^{\lambda}-1)\,1\otimes ab+q(e^{\lambda}-1)\,ab\otimes 1-(e^{\lambda}-q)\,b\otimes a. Whether these exhaust the system is unknown: the originating 2006 paper, a 2011 survey (Remark 2.7: "It is an open problem to classify its solutions"), and the 2015 survey all state the classification as open, and later work (2018, 2020) by the same school still treats the surrounding classification problems as unresolved.

Problem Statement

Let kk be a field, X⊆kX\subseteq k a parameter set, and let α,β,γ:X×X→k\alpha,\beta,\gamma:X\times X\to k be kk-valued functions. Following Nichita–Parashar (2006), associate to them the operators

R(u,v)(a⊗b)=α(u,v) 1⊗ab+β(u,v) ab⊗1−γ(u,v) b⊗a,u,v∈X,R(u,v)(a\otimes b)=\alpha(u,v)\,1\otimes ab+\beta(u,v)\,ab\otimes 1-\gamma(u,v)\,b\otimes a,\qquad u,v\in X,

defined on A⊗AA\otimes A for any unital associative kk-algebra AA. On A⊗A⊗AA\otimes A\otimes A, write II for the identity and set R12(u,v)=R(u,v)⊗IR^{12}(u,v)=R(u,v)\otimes I, R23(v,w)=I⊗R(v,w)R^{23}(v,w)=I\otimes R(v,w), and R13(u,w)=(I⊗τ)(R(u,w)⊗I)(I⊗τ)R^{13}(u,w)=(I\otimes\tau)(R(u,w)\otimes I)(I\otimes\tau), where τ(x⊗y)=y⊗x\tau(x\otimes y)=y\otimes x is the twist map. Requiring RR to satisfy the two-parameter (coloured) quantum Yang–Baxter equation R12(u,v)R13(u,w)R23(v,w)=R23(v,w)R13(u,w)R12(u,v)R^{12}(u,v)R^{13}(u,w)R^{23}(v,w)=R^{23}(v,w)R^{13}(u,w)R^{12}(u,v) for all u,v,w∈Xu,v,w\in X — expanding both sides using only the associativity of the product in AA — yields the following system of functional equations, displayed as (15)–(19) in the 2015 survey and as (2.6)–(2.10) in arXiv:1107.0920:

(β(v,w)−γ(v,w))(α(u,v)β(u,w)−α(u,w)β(u,v))+(α(u,v)−γ(u,v))(α(v,w)β(u,w)−α(u,w)β(v,w))=0,(\beta(v,w)-\gamma(v,w))\big(\alpha(u,v)\beta(u,w)-\alpha(u,w)\beta(u,v)\big)+(\alpha(u,v)-\gamma(u,v))\big(\alpha(v,w)\beta(u,w)-\alpha(u,w)\beta(v,w)\big)=0,

β(v,w)(β(u,v)−γ(u,v))(α(u,w)−γ(u,w))+(α(v,w)−γ(v,w))(β(u,w)γ(u,v)−β(u,v)γ(u,w))=0,\beta(v,w)\big(\beta(u,v)-\gamma(u,v)\big)\big(\alpha(u,w)-\gamma(u,w)\big)+\big(\alpha(v,w)-\gamma(v,w)\big)\big(\beta(u,w)\gamma(u,v)-\beta(u,v)\gamma(u,w)\big)=0,

α(u,v)β(v,w)(α(u,w)−γ(u,w))+α(v,w)γ(u,w)(γ(u,v)−α(u,v))+γ(v,w)(α(u,v)γ(u,w)−α(u,w)γ(u,v))=0,\alpha(u,v)\beta(v,w)\big(\alpha(u,w)-\gamma(u,w)\big)+\alpha(v,w)\gamma(u,w)\big(\gamma(u,v)-\alpha(u,v)\big)+\gamma(v,w)\big(\alpha(u,v)\gamma(u,w)-\alpha(u,w)\gamma(u,v)\big)=0,

α(u,v)β(v,w)(β(u,w)−γ(u,w))+β(v,w)γ(u,w)(γ(u,v)−β(u,v))+γ(v,w)(β(u,v)γ(u,w)−β(u,w)γ(u,v))=0,\alpha(u,v)\beta(v,w)\big(\beta(u,w)-\gamma(u,w)\big)+\beta(v,w)\gamma(u,w)\big(\gamma(u,v)-\beta(u,v)\big)+\gamma(v,w)\big(\beta(u,v)\gamma(u,w)-\beta(u,w)\gamma(u,v)\big)=0,

α(u,v)(α(v,w)−γ(v,w))(β(u,w)−γ(u,w))+(β(u,v)−γ(u,v))(α(u,w)γ(v,w)−α(v,w)γ(u,w))=0,\alpha(u,v)\big(\alpha(v,w)-\gamma(v,w)\big)\big(\beta(u,w)-\gamma(u,w)\big)+\big(\beta(u,v)-\gamma(u,v)\big)\big(\alpha(u,w)\gamma(v,w)-\alpha(v,w)\gamma(u,w)\big)=0,

all to hold for every u,v,w∈Xu,v,w\in X. Every solution triple (α,β,γ)(\alpha,\beta,\gamma) of this system yields a coloured Yang–Baxter operator on A⊗AA\otimes A for every unital associative kk-algebra AA. The problem, stated as open in the source survey and in the 2006 and 2011 papers it builds on, is: classify all solutions (α,β,γ)(\alpha,\beta,\gamma) of this system — that is, give a provably complete description of the solution space, showing where the known families sit inside it (the constant triples classified in the constant-coefficient case; the linear family α(u,v)=p(u−v)\alpha(u,v)=p(u-v), β(u,v)=q(u−v)\beta(u,v)=q(u-v), γ(u,v)=pu−qv\gamma(u,v)=pu-qv; and the exponential one-parameter family) and which solutions give invertible operators. The classification is sought up to reparametrization of XX and the α↔β\alpha\leftrightarrow\beta leg-swap symmetry, over all kk-valued functions on X×XX\times X, and without imposing regularity hypotheses — continuity, measurability, invertibility, or non-vanishing of coefficients — that the sources do not impose.

References

  1. Dăscălescu, S.; Nichita, F. — Yang-Baxter operators arising from (co)algebra structures. Communications in Algebra 27(12):5833–5845, 1999. DOI: 10.1080/00927879908826793. https://doi.org/10.1080/00927879908826793
  2. Nichita, F. F.; Parashar, D. — Spectral-parameter dependent Yang-Baxter operators and Yang-Baxter systems from algebra structures. Communications in Algebra 34(8):2713–2726, 2006. DOI: 10.1080/00927870600651661; arXiv:math/0607710. https://arxiv.org/abs/math/0607710
  3. Nichita, F. F.; Popovici, B. P. — Yang-Baxter operators from (G, θ)-Lie algebras. arXiv:1011.2072, 2010. https://arxiv.org/abs/1011.2072
  4. Nichita, F. F. — Lie algebras and Yang-Baxter equations. arXiv:1107.0920, 2011. https://arxiv.org/abs/1107.0920
  5. Nichita, F. F. — Yang–Baxter Equations, Computational Methods and Applications. Axioms 4(4):423–435, 2015. DOI: 10.3390/axioms4040423; arXiv:1506.03610. https://arxiv.org/abs/1506.03610
  6. Nichita, F. F. — The Unification of Nonassociative Structures. arXiv:1807.04264, 2018. https://arxiv.org/abs/1807.04264
  7. Nichita, F. F. — On Jordan algebras and some unification results. arXiv:2002.05345, 2020. https://arxiv.org/abs/2002.05345
  8. Wang, S.; Zhang, X.; Guo, S. — Hom-Yang-Baxter equations and Hom-Yang-Baxter systems. Communications in Algebra 51(4):1462–1478, 2022. DOI: 10.1080/00927872.2022.2137518. https://doi.org/10.1080/00927872.2022.2137518
  9. Van Caudenberg, D.; Bogaerts, B.; Vendramin, L. — Incremental SAT-Based Enumeration of Solutions to the Yang-Baxter Equation. arXiv:2501.14363, 2025. https://arxiv.org/abs/2501.14363
  10. Maity, S.; Padmanabhan, P.; Korepin, V. — Non-hermitian integrable systems from constant non-invertible solutions of the Yang-Baxter equation. arXiv:2503.08109, 2025. https://arxiv.org/abs/2503.08109