Two linked classification problems posed as open problems in F. F. Nichita's survey "Lie algebras and Yang–Baxter equations" (arXiv:1107.0920, 2011): (1) classify, up to the standard equivalences, all invertible constant Yang–Baxter operators R:V⊗V→V⊗VR:V\otimes V\to V\otimes V for dim⁡V=3\dim V=3 — dimension 2 was completely classified by Hietarinta in 1992, and the general classification for dim⁡V>2\dim V>2 remains open — and (2) classify all solutions (α,β,γ)(\alpha,\beta,\gamma) of the functional system (2.6)–(2.10) that governs the coloured Yang–Baxter operators produced by the associative-algebra ansatz 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.

Background

Fix a field kk and a finite-dimensional kk-vector space VV, and write τ(v⊗w)=w⊗v\tau(v\otimes w)=w\otimes v for the twist map. A Yang–Baxter operator is an invertible kk-linear map R:V⊗V→V⊗VR:V\otimes V\to V\otimes V satisfying the braid equation R12∘R23∘R12=R23∘R12∘R23R^{12}\circ R^{23}\circ R^{12}=R^{23}\circ R^{12}\circ R^{23}, where R12=R⊗IR^{12}=R\otimes I, R23=I⊗RR^{23}=I\otimes R and R13R^{13} acts on the first and third factors of a triple tensor product. Composing with the twist converts such an RR into a solution of the constant quantum Yang–Baxter equation (QYBE) R12R13R23=R23R13R12R^{12}R^{13}R^{23}=R^{23}R^{13}R^{12}, and the two forms are equivalent. These operators are the elementary building blocks of quantum group theory: every invertible solution generates a bialgebra through the FRT construction (a standard procedure, named after Faddeev, Reshetikhin and Takhtajan, that produces a matrix bialgebra whose defining relations are encoded by RR), and a suitably enhanced Yang–Baxter operator yields invariants of knots and links through its braid group representations.

Which such operators exist is a basic classification question. In dimension dim⁡V=2\dim V=2 it is completely settled: Hietarinta (1992) obtained an exhaustive list of invertible solutions of the constant QYBE, organised into eleven families (including the one given by the permutation operator, i.e. the twist map τ\tau itself) up to the standard equivalences of this literature — conjugation by S⊗SS\otimes S for an invertible linear map S:V→VS:V\to V, and rescaling by nonzero scalars. Already in dimension 3 the equation becomes a system of cubic polynomial equations in the 8181 entries of a 9×99\times 9 matrix, and no complete classification is known; Hietarinta (1993) classified only the upper-triangular solutions in the three-state case (dim⁡V=3\dim V=3). The source survey states the general problem: "Finding all Yang-Baxter operators in dimension greater than 2 is an unsolved problem" (Remark 2.3(ii)).

The same survey reviews a complementary, ansatz-based approach that manufactures solutions from algebra structures. For an associative unital kk-algebra AA and scalars α,β,γ∈k\alpha,\beta,\gamma\in k, the operator Rα,β,γA(a⊗b)=α ab⊗1+β 1⊗ab−γ a⊗bR^{A}_{\alpha,\beta,\gamma}(a\otimes b)=\alpha\,ab\otimes 1+\beta\,1\otimes ab-\gamma\,a\otimes b is a Yang–Baxter operator in exactly three explicit parameter families (Dăscălescu and Nichita, 1999), so this constant ansatz is completely classified. In the coloured (two-parameter) setting one instead seeks a function R:X×X→Endk(A⊗A)R:X\times X\to\mathrm{End}_k(A\otimes A), with XX a subset of the ground field, satisfying the two-parameter QYBE 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. Inserting the ansatz 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 with unknown kk-valued functions α,β,γ\alpha,\beta,\gamma and matching the algebra terms that appear produces the functional system (2.6)–(2.10) reproduced in the problem statement; the source records (Remark 2.7) that classifying all solutions of this system is open, the 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 being the solutions found there, while a separate one-parameter family with exponential coefficients appears as Theorem 2.10 of the source.

Two adjacent research programmes should not be confused with the problem above. The set-theoretic Yang–Baxter equation asks for bijections S:X×X→X×XS:X\times X\to X\times X on a plain set (a different equation, studied through the theory of braces — algebraic structures introduced to generate and organise such set-theoretic solutions), and the quantum-group classification of spectral-parameter RR-matrices (solutions depending additionally on a complex parameter, the spectral parameter) organises only the families arising as deformations of Lie algebras. Neither programme addresses the classification of arbitrary invertible constant operators in dimension greater than two, nor the ansatz functional system above.

Problem Statement

The task is the pair of classification questions that the source survey (Nichita, arXiv:1107.0920) states as open problems, each asking for a complete, effectively checkable description of the solutions in one of the two settings the source leaves open.

(1) Constant Yang–Baxter operators beyond dimension two. Over a field kk, let VV be a finite-dimensional kk-vector space with dim⁡V=3\dim V=3. Classify all Yang–Baxter operators on VV: invertible kk-linear maps R:V⊗V→V⊗VR:V\otimes V\to V\otimes V satisfying

R12∘R23∘R12=R23∘R12∘R23R^{12}\circ R^{23}\circ R^{12}=R^{23}\circ R^{12}\circ R^{23}

up to the standard equivalences (conjugation by S⊗SS\otimes S for an invertible linear map S:V→VS:V\to V, and multiplication by nonzero scalars); equivalently, via R∘τR\circ\tau, classify all invertible solutions of the constant quantum Yang–Baxter equation. Dimension 2 is completely classified (Hietarinta, 1992); the requested result is the analogous complete classification for dim⁡V=3\dim V=3; the classification for all dimensions greater than two is a wider open problem, of which the dim⁡V=3\dim V=3 case is the first step, and is not part of the required deliverable.

(2) The coloured associative-algebra ansatz system. Let X⊆kX\subseteq k and let α,β,γ:X×X→k\alpha,\beta,\gamma:X\times X\to k be functions. Classify all triples (α,β,γ)(\alpha,\beta,\gamma) satisfying, for all u,v,w∈Xu,v,w\in X, the functional system

(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))(\alpha(u,v)\beta(u,w)-\alpha(u,w)\beta(u,v))+(\alpha(u,v)-\gamma(u,v))(\alpha(v,w)\beta(u,w)-\alpha(u,w)\beta(v,w))=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)(\beta(u,v)-\gamma(u,v))(\alpha(u,w)-\gamma(u,w))+(\alpha(v,w)-\gamma(v,w))(\beta(u,w)\gamma(u,v)-\beta(u,v)\gamma(u,w))=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)(\alpha(u,w)-\gamma(u,w))+\alpha(v,w)\gamma(u,w)(\gamma(u,v)-\alpha(u,v))+\gamma(v,w)(\alpha(u,v)\gamma(u,w)-\alpha(u,w)\gamma(u,v))=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)(\beta(u,w)-\gamma(u,w))+\beta(v,w)\gamma(u,w)(\gamma(u,v)-\beta(u,v))+\gamma(v,w)(\beta(u,v)\gamma(u,w)-\beta(u,w)\gamma(u,v))=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)(\alpha(v,w)-\gamma(v,w))(\beta(u,w)-\gamma(u,w))+(\beta(u,v)-\gamma(u,v))(\alpha(u,w)\gamma(v,w)-\alpha(v,w)\gamma(u,w))=0

(equations (2.6)–(2.10) of the source). Every such triple yields, for every associative unital kk-algebra AA, a coloured Yang–Baxter operator 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 satisfying the two-parameter quantum Yang–Baxter equation, and the system is obtained precisely by inserting this ansatz into that equation; the known 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 does not exhaust the system.

For (1) this is an explicit list or parametrization of the equivalence classes of invertible Yang–Baxter operators on a three-dimensional space (dim⁡V=3\dim V=3), each listed representative a Yang–Baxter operator and every Yang–Baxter operator on a three-dimensional space equivalent to one listed; for (2) it is an explicit parametrization of all triples (α,β,γ)(\alpha,\beta,\gamma) solving the functional system. The two questions are scientifically linked: the constant coefficient analogue of the ansatz in (2) is completely classified (Dăscălescu and Nichita, 1999), and that solved template is exactly what both open questions ask to extend — to all operators regardless of origin in (1), and to the coloured, functional-equation setting in (2).

References

  1. Florin F. Nichita, Lie algebras and Yang-Baxter equations, arXiv:1107.0920 (2011), DOI 10.48550/arXiv.1107.0920, https://arxiv.org/abs/1107.0920 (source paper)
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