Bijectivity of the 1-cocycles between the structure monoid and the derived structure monoids of a set-theoretic solution of the Yang–Baxter equation

For a set-theoretic solution $(X,r)$ of the Yang–Baxter equation, Cedó, Jespers and Verwimp constructed canonical 1-cocycles $\pi: M(X,r) \to A(X,r)$ and $\pi': M(X,r) \to A'(X,r)$ from the structure monoid to the left and right derived structure monoids, and asked (their Question 3.3) when these 1-cocycles are bijective. Surjectivity is already characterized pointwise: $\pi$ is surjective if and only if every $\sigma_x$ is surjective, and dually for $\pi'$. If $X$ is finite, $\pi$ is bijective exactly when $(X,r)$ is left non-degenerate (dually for $\pi'$), but this equivalence fails as a characterization for infinite $X$, and injectivity of $\pi$ and $\pi'$ is known only to follow from, not to be equivalent to, pointwise injectivity of the structure maps. The problem is to give a complete intrinsic characterization of injectivity — equivalently bijectivity — of $\pi$ and $\pi'$ for solutions of arbitrary cardinality, without non-degeneracy or bijectivity hypotheses.

Background

The Yang–Baxter equation (YBE) arises in statistical mechanics and the theory of quantum groups; in its set-theoretic form it asks for a map $r: X \times X \to X \times X$ on a set $X$, called a set-theoretic solution, satisfying the braid relation $(r \times \mathrm{id}) \circ (\mathrm{id} \times r) \circ (r \times \mathrm{id}) = (\mathrm{id} \times r) \circ (r \times \mathrm{id}) \circ (\mathrm{id} \times r)$. Writing $r(x, y) = (\sigma_x(y), \gamma_y(x))$, a solution is left non-degenerate if every map $\sigma_x: X \to X$ is bijective, right non-degenerate if every $\gamma_y$ is bijective, non-degenerate if both, involutive if $r^2 = \mathrm{id}$, and bijective if $r$ is a bijection of $X^2$.

The main algebraic object attached to a solution is its structure monoid $M(X, r) = \langle X \mid x \circ y = \sigma_x(y) \circ \gamma_y(x) \rangle$, the monoid generated by $X$ subject to the rewriting rules determined by $r$; its group completion is the structure group. For a left non-degenerate solution there is a derived solution $r'(x, y) = (y, \sigma_y \gamma_{\sigma_x^{-1}(y)}(x))$, whose structure monoid is the classical derived structure monoid, the prototype of the constructions below. To extend this to arbitrary solutions, Jespers, Kubat and Van Antwerpen, and subsequently Cedó, Jespers and Verwimp, introduced the left and right derived structure monoids $A(X, r) = \langle X \mid x + \sigma_x(y) = \sigma_x(y) + \sigma_{\sigma_x(y)}(\gamma_y(x)) \rangle$ and $A'(X, r) = \langle X \mid \gamma_y(x) \oplus y = \gamma_{\gamma_y(x)}(\sigma_x(y)) \oplus \gamma_y(x) \rangle$ (when $r$ is left non-degenerate, $A(X, r)$ is the structure monoid of the derived solution). Cedó, Jespers and Verwimp proved that $(M, \circ)$ acts on $(A, +)$ on the left and on $(A', \oplus)$ on the right, and that there are unique 1-cocycles with respect to these actions — maps $\pi: M(X, r) \to A(X, r)$ and $\pi': M(X, r) \to A'(X, r)$ satisfying $\pi(a \circ b) = \pi(a) + \lambda'_a(\pi(b))$ and $\pi'(a \circ b) = \rho'_b(\pi'(a)) \oplus \pi'(b)$, where the actions are determined on generators by $\lambda'_x(y) = \sigma_x(y)$ and $\rho'_x(y) = \gamma_x(y)$ — with $\pi(x) = \pi'(x) = x$ for all $x \in X$. Here a 1-cocycle of a monoid acting on another monoid is a map compatible with the action in this multiplicative sense, and its bijectivity identifies the two monoids up to the action.

This identification is the technical backbone of the structure theory of solutions: for left non-degenerate solutions the bijectivity of $\pi$ realizes $M(X, r)$ as a regular submonoid of a semidirect product $A(X, r) \rtimes \mathrm{gr}(\sigma_x \mid x \in X)$, underlies the semi-truss and YB-semitruss frameworks that generate new solutions, and drives the ring-theoretic programme on the structure algebra $K[M(X, r)]$ (Noetherianity, polynomial-identity properties, Gelfand–Kirillov dimension).

Three results delimit what is known about bijectivity of the 1-cocycles, all valid for arbitrary $X$. First, $\pi$ is surjective if and only if every $\sigma_x$ is surjective, and $\pi'$ is surjective if and only if every $\gamma_x$ is surjective — a complete pointwise characterization of surjectivity. Second, if every $\sigma_x$ is injective then $\pi$ is injective (dually for $\pi'$); this sufficient condition is not necessary, since for $r(x, y) = (x, x)$ on any set with $|X| > 1$ the maps $\sigma_x$ are constant yet $\pi$ is injective. Third, if $X$ is finite then $\pi$ is bijective if and only if $(X, r)$ is left non-degenerate, and $\pi'$ is bijective if and only if $(X, r)$ is right non-degenerate. For infinite $X$ the last equivalence collapses as a characterization: on $X = \mathbb{N}$, the solution $r(x, y) = (\xi(y), \xi(x))$ with $\xi(x) = \max\{0, x-1\}$ has every $\sigma_x = \gamma_x = \xi$ surjective but not injective, yet both $\pi$ and $\pi'$ are bijective. It is now also known that every non-degenerate solution is bijective (Jedlička and Pilitowska), so the bijective non-degenerate world is well understood; the 1-cocycle question is sharp precisely for degenerate and one-sided degenerate solutions, where no identification tool is available.

Problem Statement

Give a complete intrinsic characterization of the bijectivity of the canonical 1-cocycles of a set-theoretic solution of the Yang–Baxter equation. Concretely: for a solution $(X, r)$ with $r(x, y) = (\sigma_x(y), \gamma_y(x))$, find necessary and sufficient conditions — expressed in terms of the data of $(X, r)$, such as the maps $\sigma_x, \gamma_y$ or the rewriting behaviour of the defining relations of the associated monoids — under which the 1-cocycle $\pi: M(X, r) \to A(X, r)$ is injective, and dually under which $\pi': M(X, r) \to A'(X, r)$ is injective. Since surjectivity is already characterized pointwise ($\pi$ is surjective if and only if every $\sigma_x$ is surjective, and dually for $\pi'$), this is equivalent to characterizing bijectivity of $\pi$ and $\pi'$. The characterization must be valid for sets $X$ of arbitrary cardinality and must not assume non-degeneracy, one-sided non-degeneracy, bijectivity of $r$, or finiteness; in particular it must correctly classify the known boundary instances: the finite case (bijective if and only if one-sided non-degenerate), the solution $r(x, y) = (x, x)$ with $|X| > 1$ ($\pi$ injective, not surjective; $\pi'$ bijective), the skew-lattice solution on $\{0, 1, 2\}$ given by $r(x, y) = (x \wedge y, y \vee x)$, with the entry in row $x$, column $y$ of the $\wedge$-table equal to $(0,0,0)$, $(0,1,2)$, $(0,1,2)$ for $x = 0, 1, 2$ and of the $\vee$-table equal to $(0,1,2)$, $(1,1,1)$, $(2,2,2)$ (both $\pi$ and $\pi'$ neither injective nor surjective), and the infinite solution on $\mathbb{N}$ given by $r(x, y) = (\xi(y), \xi(x))$ with $\xi(x) = \max\{0, x-1\}$ (both $\pi$ and $\pi'$ bijective although no $\sigma_x$ or $\gamma_x$ is injective). This is Question 3.3 of Cedó–Jespers–Verwimp — 'When are the 1-cocycles $\pi$ and $\pi'$ bijective?' — whose finite case is settled and whose open part is precisely the infinite and degenerate case.

References

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