Classifying set-theoretic Yang–Baxter solutions by affine actions beyond the injective case, and the structure-group homomorphism of the semi-direct product of cycle sets

Yang (2016; published 2018) associated to every non-degenerate set-theoretic solution $R$ of the Yang–Baxter equation a regular affine action of its structure group $G_R$ on its derived structure group $A_R$, and proved that injective solutions — those for which the canonical map $X\to G_R$ is injective — are classified up to isomorphism by these affine actions, equivalently by associated C*-dynamical systems $(G_R,M_2(C^*(A_R)),\pi)$. In an appendix on Rump's semi-direct product of cycle sets he constructed a natural homomorphism $\Pi$ from the structure group of the product solution to the semi-direct product of the structure groups, and showed by example that $\Pi$ need not be an isomorphism. Two open questions survive from this work: (1) extend the affine-action and C*-dynamical classification to non-injective non-degenerate solutions, or determine what additional data classifies them; (2) characterize when $\Pi$ is injective, when it is surjective, and when it is an isomorphism. Both questions remain open.

Background

The Yang–Baxter equation (YBE) arose in statistical mechanics and quantum group theory, and Drinfeld proposed studying its set-theoretic solutions: a bijection $R:X\times X\to X\times X$, written $R(x,y)=(\alpha_x(y),\beta_y(x))$, satisfying the braid-type identity $R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23}$ on $X\times X\times X$, where $R_{12}=\mathrm{id}\times R$ and $R_{23}=R\times\mathrm{id}$. A solution is non-degenerate if $\alpha_x$ and $\beta_x$ are bijections of $X$ for every $x$, and symmetric (equivalently, involutive and non-degenerate) if moreover $R^2=\mathrm{id}$. Classifying all such solutions is a major open problem; the best-developed approaches pass through algebra: Rump's braces for involutive solutions and Guarnieri–Vendramin's skew left braces for non-involutive ones — a skew left brace being a set carrying two group structures with a common identity element linked by a mixed distributivity law (Rump's original braces being the case where one structure is abelian).

To a non-degenerate solution $R$ one associates two groups generated by $X$, introduced in the work of Etingof–Schedler–Soloviev, Lu–Yan–Zhu and Soloviev: the structure group $G_R=\langle X\mid xy=\alpha_x(y)\beta_y(x)\rangle$, which linearizes the defining relations, and the derived structure group $A_R=\langle X\mid x\bullet y=y\bullet\alpha_y(\beta_{\alpha_x^{-1}(y)}(x))\rangle$ (the bullet marks the operation of this second group), which is the structure group of the derived solution $R'(x,y)=(y,\alpha_y(\beta_{\alpha_x^{-1}(y)}(x)))$, a second solution on the same set $X$ constructed from $R$ by Soloviev. Soloviev proved that the canonical maps $X\to G_R$ and $X\to A_R$ are injective or not simultaneously; a solution is called injective when $X\to G_R$ is injective. Every symmetric solution is injective, but non-injective non-degenerate solutions exist. Soloviev also showed that the action $\alpha$ extends to an action of $G_R$ on $A_R$; this is the starting point of Yang's paper. An affine action of a group $G$ on a group $A$ is a homomorphism $G\to\mathrm{Aff}(A)=\mathrm{Aut}(A)\ltimes A$, i.e., an action by automorphisms $\pi$ twisted by a 1-cocycle $b:G\to A$ (a map satisfying $b(gh)=b(g)\,\pi_g(b(h))$) via $g\cdot a=b(g)\pi_g(a)$; the action is regular when it is transitive and free, which for such actions is equivalent to $b$ being bijective. Every non-degenerate solution induces a regular affine action $\rho_R$ of $G_R$ on $A_R$ (Yang's Proposition 2.5).

Yang's main theorems classify injective solutions by this data: two injective solutions are isomorphic if and only if there is a group isomorphism between their structure groups carrying $X$ onto $Y$ under which the associated regular affine actions are conjugate (Theorem 3.3); equivalently, injective solutions are classified by the C*-dynamical systems $(G_R,M_2(C^*(A_R)),\pi^R)$ built from the affine action (Theorem 3.6), where a C*-algebra is a norm-closed -algebra of bounded operators, $C^*(A_R)$ is the group C-algebra (the universal C*-completion of the complex group algebra of the discrete group $A_R$), and a C*-dynamical system is a group acting by automorphisms on a C*-algebra. Bachiller later observed that Theorem 3.3 is equivalent to saying that two injective non-degenerate solutions are isomorphic exactly when their structure groups are isomorphic as skew left braces carrying $X$ onto $Y$; the injectivity hypothesis remains in force in all known formulations.

For symmetric solutions, Rump introduced cycle sets: a set with a binary operation $(x,y)\mapsto x\cdot y$ satisfying $(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z)$, called non-degenerate when $x\mapsto x\cdot x$ is bijective; non-degenerate cycle sets correspond bijectively to symmetric solutions. Rump also defined a semi-direct product $X\ltimes_\pi S$ of two finite cycle sets along a suitable action $\pi$ of $X$ on $S$ (recorded as Definition A.3 in Yang's appendix), which is again a cycle set and thus yields a symmetric solution $R_{X\ltimes_\pi S}$. Yang's appendix proves a commutation relation for this construction and constructs, in Proposition A.8, a natural group homomorphism $\Pi:G_{R_{X\ltimes_\pi S}}\longrightarrow G_{R_X}\ltimes_\pi G_{R_S},$ where the action of $G_{R_X}$ on $G_{R_S}$ is the extension of $\pi$ (his Lemma A.6). Yang notes that $\Pi$ need not be an isomorphism — for trivial solutions (the solution $R(x,y)=(y,x)$) with trivial action the two sides are the free abelian groups on $X\times S$ and on the disjoint union $X\sqcup S$, which differ whenever $|X|\cdot|S|\neq|X|+|S|$ — and gives no condition under which it is one, just as his classification theorems stop at injective solutions. These two boundaries of the paper are the subject of the present problem.

Problem Statement

Both parts are stated in the conventions of Yang's paper: all set-theoretic solutions are non-degenerate, $R(x,y)=(\alpha_x(y),\beta_y(x))$ is a bijection of $X^2$ satisfying the braid identity, $G_R$ and $A_R$ denote the structure group and derived structure group, $\rho_R$ is the associated regular affine action of $G_R$ on $A_R$, and a solution is injective when the canonical map $X\to G_R$ is injective.

Part 1 (classification beyond the injective case). Yang's Theorem 3.3 states that two injective solutions $R_X$ and $R_Y$ are isomorphic if and only if there exists a group isomorphism $\varphi:G_{R_X}\to G_{R_Y}$ with $\varphi(X)=Y$ such that $\rho_{R_X}$ and $\rho_{R_Y}\circ\varphi$ are conjugate; Theorem 3.6 gives the equivalent classification by the associated C*-dynamical systems. Determine whether these classifications extend to all non-degenerate solutions, i.e., to solutions for which the canonical map $X\to G_R$ is not injective, and, if they do not, what additional data — intrinsic to each solution and computable independently of the isomorphism question, so that tautological choices such as the isomorphism class of a solution itself are excluded — is needed to classify the non-injective non-degenerate solutions.

Part 2 (the semi-direct product homomorphism). Let $X$ and $S$ be finite cycle sets and $\pi:X\times S\to S$ an action in the sense of Rump's semi-direct product (Yang's Definition A.3), so that $X\ltimes_\pi S$ is a cycle set with induced symmetric solution $R_{X\ltimes_\pi S}$, and the action $\pi$ extends to an action of $G_{R_X}$ on $G_{R_S}$ (Yang's Lemma A.6). Yang's Proposition A.8 yields a natural group homomorphism $\Pi:G_{R_{X\ltimes_\pi S}}\longrightarrow G_{R_X}\ltimes_\pi G_{R_S},$ which need not be an isomorphism (for trivial solutions, i.e. $R(x,y)=(y,x)$, with trivial action the two sides are the free abelian groups on $X\times S$ and on $X\sqcup S$). Give necessary and sufficient conditions on $(X,S,\pi)$ under which $\Pi$ is injective, under which it is surjective, and under which it is an isomorphism; each of the three properties is to be characterized (they are logically related, since a group homomorphism is an isomorphism exactly when it is bijective).

The two parts address one scientific question: how completely the structure group, together with its canonical affine action, remembers a set-theoretic solution — for a single solution (Part 1) and under the basic product construction of cycle sets (Part 2).

References

1. Dilian Yang, Affine actions and the Yang–Baxter equation, arXiv:1607.03393 (2016); published as Advances in Operator Theory 3(3) (2018) 710–730, DOI 10.15352/aot.1801-1298, https://arxiv.org/abs/1607.03393.
2. V. G. Drinfeld, On some unsolved problems in quantum group theory, in: Quantum Groups (P. P. Kulish, ed.), Lecture Notes in Mathematics 1510, Springer (1992) 1–8, DOI 10.1007/BFb0101175, https://doi.org/10.1007/BFb0101175.
3. Pavel Etingof, Travis Schedler, Alexandre Soloviev, Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Mathematical Journal 100 (1999) 169–209, DOI 10.1215/s0012-7094-99-10007-x, https://doi.org/10.1215/s0012-7094-99-10007-x.
4. Alexander Soloviev, Non-unitary set-theoretical solutions to the quantum Yang–Baxter equation, Mathematical Research Letters 7 (2000) 577–596, DOI 10.4310/mrl.2000.v7.n5.a4, https://doi.org/10.4310/mrl.2000.v7.n5.a4.
5. Jiang-Hua Lu, Min Yan, Yong-Chang Zhu, On the set-theoretical Yang-Baxter equation, Duke Mathematical Journal 104 (2000) 1–18, DOI 10.1215/s0012-7094-00-10411-5, https://doi.org/10.1215/s0012-7094-00-10411-5.
6. Wolfgang Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation, Advances in Mathematics 193 (2005) 40–55, DOI 10.1016/j.aim.2004.03.019, https://doi.org/10.1016/j.aim.2004.03.019.
7. Wolfgang Rump, Braces, radical rings, and the quantum Yang–Baxter equation, Journal of Algebra 307 (2007) 153–170, DOI 10.1016/j.jalgebra.2006.03.040, https://doi.org/10.1016/j.jalgebra.2006.03.040.
8. Ferran Cedó, Eric Jespers, Jan Okniński, Braces and the Yang–Baxter equation, Communications in Mathematical Physics 327 (2014) 101–116, DOI 10.1007/s00220-014-1935-y, https://doi.org/10.1007/s00220-014-1935-y.
9. L. Guarnieri, L. Vendramin, Skew braces and the Yang–Baxter equation, Mathematics of Computation 86 (2017) 2519–2534, DOI 10.1090/mcom/3161, https://doi.org/10.1090/mcom/3161.
10. David Bachiller, Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks, Journal of Knot Theory and Its Ramifications 27 (2018) 1850055, DOI 10.1142/S0218216518500554, https://arxiv.org/abs/1611.08138.
11. Francesco Catino, Ilaria Colazzo, Paola Stefanelli, The matched product of the solutions to the Yang–Baxter equation of finite order, Mediterranean Journal of Mathematics 17 (2020), article 58, DOI 10.1007/s00009-020-1483-y, https://arxiv.org/abs/1904.07557.
12. F. Cedó, E. Jespers, Ł. Kubat, A. Van Antwerpen, C. Verwimp, On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang–Baxter equation, Journal of Pure and Applied Algebra 227 (2023) 107194, DOI 10.1016/j.jpaa.2022.107194, https://arxiv.org/abs/2011.01724.
13. Hui Li, Dilian Yang, Boundary quotient C*-algebras of products of odometers, Canadian Journal of Mathematics 71 (2019) 183–212, DOI 10.4153/CJM-2017-034-5, https://arxiv.org/abs/1702.05814.
14. Marco Damele, Andrea Loi, Structural and rigidity properties of Lie skew braces, arXiv:2507.06214 (2025), DOI 10.48550/arXiv.2507.06214, https://arxiv.org/abs/2507.06214.