Every bijective non-degenerate set-theoretic solution of the Yang-Baxter (braid) equation produced by a skew brace generates, through the FRT/RTT construction, a quantum algebra whose coproduct is provably not coassociative: it is a quasi-triangular quasi-bialgebra rather than a Hopf algebra. The associator Φ\Phi that would complete this quasi-bialgebra structure has been identified only for special Lyubashenko-type solutions, and the antipode/quasi-Hopf structure has been settled only for isolated examples. The problem asks for (1) the explicit identification of the associator Φ\Phi, and of the twisted associators ΦT\Phi_{\mathcal T} obtained after the admissible Drinfeld twists, uniformly for all set-theoretic solutions, involutive and non-involutive — thereby confirming in full the conjecture of Doikou-Ghionis-Vlaar that the quantum algebra of a set-theoretic solution is a quasi-triangular quasi-bialgebra — and, as the further-study direction the source flags alongside it, (2) the determination of when and how an antipode exists, i.e. the precise quasi-Hopf algebra structure.

Background

The Yang-Baxter equation (YBE) is the consistency condition for the factorized scattering of many-body quantum systems and for transfer matrices of exactly solvable lattice models; in the braid form used here it requires a linear operator rˇ\check r on V⊗VV\otimes V to satisfy rˇ12rˇ23rˇ12=rˇ23rˇ12rˇ23\check r_{12}\check r_{23}\check r_{12}=\check r_{23}\check r_{12}\check r_{23}, where the subscripts indicate on which tensor factors the operator acts. A set-theoretic solution is a solution built from a bijection of a set rather than from linear-algebraic data: writing r(x,y)=(σx(y),τy(x))r(x,y)=(\sigma_x(y),\tau_y(x)) for maps σx,τy:X→X\sigma_x,\tau_y:X\to X, the induced operator rˇ=∑x,y∈Xeσx(y),y⊗ex,τy(x)\check r=\sum_{x,y\in X}e_{\sigma_x(y),y}\otimes e_{x,\tau_y(x)} on V=span{ex:x∈X}V=\mathrm{span}\{e_x:x\in X\} must satisfy the braid equation. The solution is non-degenerate when every σx\sigma_x and τy\tau_y is a bijection, and involutive when r2=idr^2=\mathrm{id}. The algebraic theory of such solutions rests on braces and skew braces: a skew brace is a set carrying two group operations ++ and ∘\circ satisfying the compatibility a∘(b+c)=a∘b−a+a∘ca\circ(b+c)=a\circ b-a+a\circ c, and Guarnieri and Vendramin showed that every skew brace yields a bijective non-degenerate (in general non-involutive) set-theoretic solution, while involutive non-degenerate solutions come from braces.

On the quantum side, the FRT (Faddeev-Reshetikhin-Takhtajan) or RTT construction attaches to an rr-matrix a quadratic algebra A\mathcal A generated by the entries of an LL-operator subject to relations of the form rˇ12L1L2=L2L1rˇ12\check r_{12}L_1L_2=L_2L_1\check r_{12}, and one asks for its Hopf-algebraic structure: coproduct Δ\Delta, counit ϵ\epsilon, antipode SS, and a universal RR-matrix, an element R∈A⊗A\mathcal R\in\mathcal A\otimes\mathcal A intertwining Δ\Delta and the opposite coproduct and specializing to rˇ\check r in the fundamental representation. For set-theoretic solutions the coproduct is built from the elements Vx=∑y∈Xeσx(y),yV_x=\sum_{y\in X}e_{\sigma_x(y),y} and Wy=∑x∈Xeτy(x),xW_y=\sum_{x\in X}e_{\tau_y(x),x}, and the solutions themselves arise from the permutation operator by admissible Drinfeld twists (invertible elements FF twisting the coproduct as ΔF(y)=FΔ(y)F−1\Delta^F(y)=F\Delta(y)F^{-1}). Doikou (2021) showed that all involutive set-theoretic solutions are obtained this way and observed, on special solutions, that the resulting coproduct is not coassociative. Doikou-Ghionis-Vlaar (2022) then developed the structural picture: a quasi-bialgebra is Drinfeld's relaxation of a bialgebra in which coassociativity holds only up to conjugation by an invertible associator Φ∈A⊗3\Phi\in\mathcal A^{\otimes 3}, constrained by a pentagon identity and counit conditions, and in the quasi-triangular case by compatibility identities between Φ\Phi and R\mathcal R; a quasi-Hopf algebra is a quasi-bialgebra further equipped with an antipode SS and elements α,β\alpha,\beta satisfying Drinfeld's quasi-Hopf axioms. They conjectured (Conjecture 2.8) that the quantum algebra emerging from set-theoretic solutions is a quasi-triangular quasi-bialgebra, verified the key identity only for Lyubashenko-type special solutions, and found the quasi-Hopf antipode axioms 'very limiting' for the examples they examined, leaving both the general associator and the antipode question open.

The source paper, Doikou-Rybolowicz (2022/2024), extends the story to the non-involutive case: it produces new families of non-involutive solutions from (skew) braces — including zz-deformed families and, for two-sided skew braces, a solution attached to every element of the set — proves that the known twists remain admissible in the non-involutive frame, identifies the twisted rr-matrices and twisted coproducts of the VxV_x and WyW_y, and concludes that the underlying quantum algebra is again a quasi-triangular quasi-bialgebra: the coproducts are not coassociative and the quasi-triangularity identities fail by the nontrivial counit action. What is missing is the associator itself — in the involutive case the twisted associator is trivial, ΦT=1⊗1⊗1\Phi_{\mathcal T}=1\otimes 1\otimes 1, so even there the plain associator Φ\Phi remains to be identified. The paper closes by identifying as 'the next important step in this frame' the associators Φ\Phi and ΦT\Phi_{\mathcal T} and the full quasi-bialgebra structure for all set-theoretic solutions; by noting that the antipode and quasi-Hopf structure were discussed only briefly, for some special set-theoretic solutions, and that 'further study in this direction is required'; and (as separate further directions) by listing Baxterization — the construction of spectral-parameter solutions R(λ)R(\lambda) from the parameter-free set-theoretic ones — together with the derivation of a universal RR-matrix among 'the most challenging questions in this context', as well as the RTT/reflection-algebra representation theory of the new solution families.

Problem Statement

Let XX be a finite set and let r(x,y)=(σx(y),τy(x))r(x,y)=(\sigma_x(y),\tau_y(x)) be a bijective, non-degenerate set-theoretic solution of the braid equation — for instance the solution produced by a finite skew brace — acting on V=span{ex:x∈X}V=\mathrm{span}\{e_x:x\in X\} through rˇ=∑x,y∈Xeσx(y),y⊗ex,τy(x)\check r=\sum_{x,y\in X}e_{\sigma_x(y),y}\otimes e_{x,\tau_y(x)}. Let A\mathcal A be the associated quantum algebra (generated by the entries of the LL-operator with the quadratic relations rˇ12L1L2=L2L1rˇ12\check r_{12}L_1L_2=L_2L_1\check r_{12}, equivalently described together with the elements Vx=∑yeσx(y),yV_x=\sum_{y}e_{\sigma_x(y),y} and Wy=∑xeτy(x),xW_y=\sum_{x}e_{\tau_y(x),x}), equipped with the coproduct Δ\Delta and counit ϵ\epsilon constructed in Doikou-Rybolowicz, for which quasi-triangularity and coassociativity provably fail. The problem, as posed at the end of that paper, has as its stated next important step part (1), together with the further-study direction recorded in part (2):

(1) Identify explicitly the associator Φ∈A⊗3\Phi\in\mathcal A^{\otimes 3} — and, for each admissible Drinfeld twist T\mathcal T of the set-theoretic solution (the twists F=∑x∈Xex,x⊗VxF=\sum_{x\in X}e_{x,x}\otimes V_x and F^=∑y∈XWy⊗ey,y\hat F=\sum_{y\in X}W_y\otimes e_{y,y} constructed in Doikou (2021), which Propositions 4.3–4.4 of Doikou-Rybolowicz show to remain admissible in the non-involutive case), the twisted associators ΦT\Phi_{\mathcal T} — such that (A,Δ,ϵ,Φ)(\mathcal A,\Delta,\epsilon,\Phi) is a quasi-bialgebra in Drinfeld's sense (coassociativity up to conjugation by Φ\Phi, the pentagon identity, and the counit identities for Φ\Phi) and Φ\Phi satisfies the compatibility identities with the set-theoretic rr-matrix underlying quasi-triangularity — that is, the quasi-triangularity identities are required in their fundamental-representation form, on the set-theoretic rr-matrix, the separate construction of the universal RR-matrix R∈A⊗A\mathcal R\in\mathcal A\otimes\mathcal A being a distinct objective — uniformly for all set-theoretic solutions, in both the involutive and the non-involutive skew-brace cases. This would in particular confirm in full the conjecture of Doikou-Ghionis-Vlaar that the quantum algebra of a set-theoretic solution is a quasi-triangular quasi-bialgebra.

(2) The source paper further notes that the notion of the antipode and of the quasi-Hopf algebra was discussed only briefly, for some special set-theoretic solutions, and that 'further study in this direction is required'. That direction — determining when the resulting quasi-bialgebra admits an antipode, i.e. an anti-homomorphism SS together with elements α,β∈A\alpha,\beta\in\mathcal A satisfying Drinfeld's quasi-Hopf axioms, and how it acts on the generators, and hence the precise quasi-Hopf algebra structure of the quantum algebras of set-theoretic solutions — remains open. The identification of the associator, part (1), is the objective of this problem; the antipode question of part (2) is a further direction flagged by the source.

The further directions also flagged by the source authors — Baxterization of the non-involutive solutions (spectral-parameter deformations), the derivation of a universal RR-matrix R∈A⊗A\mathcal R\in\mathcal A\otimes\mathcal A (listed by the source, with Baxterization, among 'the most challenging questions in this context'), and the RTT/reflection-algebra representation theory of the new zz-deformed solution families — are related but separate objectives.

References

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