The noncommutative Yang–Baxter maps constructed by Konstantinou-Rizos and Nikitina over a division ring — the noncommutative KdV lift, the noncommutative Adler–Yamilov map of NLS type, and the noncommutative derivative NLS (DNLS) map — are proven to satisfy the parametric Yang–Baxter equation, but their complete integrability in the Liouville sense is undetermined: the KdV lift and the Adler–Yamilov map possess only one known first integral each, and the DNLS map possesses two functionally independent first integrals but no invariant Poisson structure has been exhibited that would complete an integrability proof. As of August 2026 the problem remains open: no follow-up work supplies the missing first integrals, an integrability proof, or a refutation.

Background

The Yang–Baxter equation is the consistency condition underlying exactly solvable lattice models in statistical mechanics and the theory of quantum groups. A set-theoretical solution of this equation — a Yang–Baxter map — is a map Ya,bY_{a,b} sending a pair of variables (x,y)(x,y) to (u,v)(u,v), with parameters a,ba,b, that satisfies the functional braid relation Ya,b12Yb,c23Ya,b12=Yb,c23Ya,b12Yb,c23Y^{12}_{a,b}Y^{23}_{b,c}Y^{12}_{a,b}=Y^{23}_{b,c}Y^{12}_{a,b}Y^{23}_{b,c} on triples of variables. A standard source of Yang–Baxter maps is the matrix refactorisation problem: given a Lax matrix La(x)L_a(x) depending on the variables and on a spectral parameter, one seeks the change of variables that makes La(u)Lb(v)=Lb(y)La(x)L_a(u)L_b(v)=L_b(y)L_a(x) hold; when this relation defines a map, that map is automatically a Yang–Baxter map. Maps of KdV, NLS and DNLS type are those arising from the Darboux matrices of the Korteweg–de Vries, nonlinear Schrödinger and derivative nonlinear Schrödinger equations respectively; a Darboux matrix is a matrix depending on a spectral parameter whose action transforms a solution of one of these differential equations into another solution of the same equation, and it is the seed from which both the maps and their Lax representations are read off.

A Yang–Baxter map is called completely integrable in the Liouville sense if the underlying space carries a Poisson bracket (an antisymmetric biderivation obeying the Jacobi identity, i.e. a Hamiltonian structure) that is invariant under the map and has constant rank 2r2r, together with k−2rk-2r functionally independent Casimir invariants (functions annihilated by the bracket and preserved by the map) and rr functionally independent first integrals in involution (pairwise Poisson-commuting preserved functions), where kk is the dimension; first integrals are functions preserved by the map. This is the standard notion of integrability for discrete dynamical systems, recalled as Definition 2.2 in the source paper.

Konstantinou-Rizos and Nikitina (Physica D 465 (2024) 134213) carried the construction to the fully noncommutative setting, where the dynamical variables belong to a division ring R\mathcal R (an associative algebra with unit in which every nonzero element is invertible, such as the quaternions) and the parameters belong to its centre Z(R)Z(\mathcal R) (the elements commuting with everything). They constructed three parametric noncommutative Yang–Baxter maps on (R×R)2(\mathcal R\times\mathcal R)^2, all with Lax representations: the noncommutative KdV lift (their map (16)) u1=y1+(a−b)(x1−y2)−1,u2=y2,v1=x1,v2=x2+(a−b)(x1−y2)−1;u_1=y_1+(a-b)(x_1-y_2)^{-1},\quad u_2=y_2,\quad v_1=x_1,\quad v_2=x_2+(a-b)(x_1-y_2)^{-1}; the noncommutative Adler–Yamilov map of NLS type (their map (38)) u1=y1−(a−b)x1(1+y2x1)−1,u2=y2,v1=x1,v2=x2+(a−b)(1+y2x1)−1y2;u_1=y_1-(a-b)x_1(1+y_2x_1)^{-1},\quad u_2=y_2,\quad v_1=x_1,\quad v_2=x_2+(a-b)(1+y_2x_1)^{-1}y_2; and the noncommutative DNLS map (their map (48)) u1=y1+(a−b)x1(a−y2x1)−1,u_1=y_1+(a-b)x_1(a-y_2x_1)^{-1}, u2=y2x1[1−(a−b)(a−y2x1)−1]−1x1−1,u_2=y_2x_1\bigl[1-(a-b)(a-y_2x_1)^{-1}\bigr]^{-1}x_1^{-1}, v1=x1[1−(a−b)(a−y2x1)−1],v2=x2+y2−y2x1[1−(a−b)(a−y2x1)−1]−1x1−1.v_1=x_1\bigl[1-(a-b)(a-y_2x_1)^{-1}\bigr],\quad v_2=x_2+y_2-y_2x_1\bigl[1-(a-b)(a-y_2x_1)^{-1}\bigr]^{-1}x_1^{-1}. For comparison, the authors prove that their commutative KdV-type map (11) is completely integrable, using three functionally independent first integrals and an invariant rank-2 Poisson bracket of which two integrals are Casimir functions.

The integrability status of the three noncommutative maps is only partially known. Map (16) has the single known first integral I1=x1+y1−x2−y2I_1=x_1+y_1-x_2-y_2; map (38) the single known first integral I=x1x2+y1y2I=x_1x_2+y_1y_2; map (48) the two functionally independent first integrals I1=x1+y1I_1=x_1+y_1 and I2=x2+y2I_2=x_2+y_2. In the commutative case the counterparts of these maps possess further first integrals — for the KdV map the integral (x2−y1)(x1−y2)(x_2-y_1)(x_1-y_2) — but the authors show (Remark 3.4) that precisely this expression fails to be invariant once the variables do not commute: the obstruction is the ordering of factors inside the noncommutative products, so the missing integrals cannot be naive transplants of the commutative ones. A second structural difficulty is that the standard definition of Liouville integrability presumes a commutative manifold, and it must first be made precise what an invariant Poisson structure, Casimir functions, involution and functional independence mean for maps whose variables lie in a division ring.

Problem Statement

Determine whether the three noncommutative Yang–Baxter maps of Konstantinou-Rizos and Nikitina over a division ring R\mathcal R — the noncommutative KdV lift (map (16)), the noncommutative Adler–Yamilov map of NLS type (map (38)), and the noncommutative DNLS map (map (48)), reproduced explicitly in the Background — are completely integrable in the Liouville sense. Determining this requires, first, a precise notion of complete Liouville integrability for maps whose variables belong to a noncommutative division ring, faithful to the commutative definition (invariant Poisson structure of constant rank, Casimir functions, involutive first integrals, functional independence). For maps (16) and (38), which currently possess a single known first integral each, it entails finding at least one further functionally independent first integral — the authors state that one more first integral must be found before an integrability proof can be completed — together with the invariant Poisson structure required to complete the proof, or otherwise establishing that the map is not completely integrable in the stated sense. For map (48), whose two known functionally independent first integrals the authors judge sufficient in number, it entails exhibiting the invariant Poisson structure of the appropriate rank under which these integrals are Casimir functions or are in involution with the remaining required integrals, thereby completing the integrability proof, or showing that the claim fails. The question concerns deciding, with proof, the complete-integrability status of each of the three maps.

References

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