Yang–Baxter Cohomology Overview

• Yang–Baxter cohomology is a family of theories that study deformations and extensions of Yang–Baxter operators, with low degrees (2 and 3) controlling infinitesimal changes and obstructions.
• It unifies set-theoretic, birack, and operator-valued models, linking classical knot invariants with Hochschild, Lie, and braided cohomology to address algebraic structure deformations.
• Explicit computations in specific cases reveal that degree 2 governs infinitesimal deformations while degree 3 detects obstructions, enabling practical applications in quantum and state-sum knot invariants.

Yang–Baxter cohomology denotes a family of cohomological theories attached to solutions of the Yang–Baxter equation, ranging from set-theoretic and birack complexes to operator-valued deformation complexes, braided Hochschild-type theories, and extension theories for algebraic structures induced by Yang–Baxter data. Across these formulations, degree $2$ repeatedly governs infinitesimal deformations, extension classes, or cocycle weights, while degree $3$ typically appears as an obstruction space for higher-order lifts (Lebed et al., 2015, Zappala, 2024, Saito et al., 4 Sep 2025).

1. Foundational formulations

At the most basic level, a set-theoretic solution is a map

$R:X\times X\to X\times X$

satisfying

$$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$

Linear theories replace $X$ by a module or vector space $V$ and study operators

$R:V\otimes V\to V\otimes V$

satisfying the corresponding braid relation. The literature uses “Yang–Baxter cohomology” for several nonidentical complexes attached to such data, rather than for a single universal theory (Przytycki et al., 2019, Zappala, 2024).

In operator-valued deformation theory, the low-dimensional cochain complex takes

$$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$

with

$$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$

and

$$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$

In this setting, $3$0 is the first-order Yang–Baxter deformation equation. A distinct formulation due to Saito–Zappalà does not require $3$1 and therefore extends to pre-Yang–Baxter operators; its $3$2 and $3$3 agree with Eisermann’s theory when $3$4 is invertible (Zappala, 2024, Saito et al., 4 Sep 2025).

A recurrent structural theme is that Yang–Baxter cohomology interpolates among several older theories. Set-theoretic theories recover rack- and quandle-type complexes in special cases, while braided and deformation-theoretic theories interact with Hochschild, Lie, and Hopf cohomology. This plurality is not merely terminological: different versions encode different aspects of the same Yang–Baxter datum.

2. Set-theoretic, birack, and normalized theories

For set-theoretic solutions, the Carter–Elhamdadi–Saito chain complex uses

$3$5

with boundary

$3$6

where the left and right faces are built by propagating the $3$7-th entry through repeated applications of $3$8. This construction underlies later normalized and geometric variants for biracks, biquandles, and cycle sets (Przytycki et al., 2019, 2002.04567, Liang et al., 2021).

For left non-degenerate set-theoretic solutions, two a priori different cohomology theories coexist: braided cohomology and birack-style cohomology. An explicit chain isomorphism between them is given by the guitar map, and the same framework associates to every left non-degenerate solution a canonical self-distributive structure, the structure rack. The associated rack reflects major properties of the original solution: involutivity corresponds to triviality of the structure rack, the biquandle condition corresponds to quandle idempotence, and the braid-group actions induced by the original solution and by the structure rack are conjugate via the guitar map. The relation between the cohomology of a solution and the cohomology of its associated self-distributive structure remains an open question in the survey literature (Lebed et al., 2015, Lebed, 2018).

Normalization is subtle and category-dependent. For involutive right non-degenerate solutions presented as cycle sets, the degenerate subgroup is generated by tuples containing a block $3$9, leading to a normalized quotient $R:X\times X\to X\times X$0. For cyclic racks, the homology splits as

$R:X\times X\to X\times X$1

The normalization developed for cycle sets is explicitly not a direct corollary of the earlier Lebed–Vendramin degeneracy framework (Przytycki et al., 2019).

For biquandles, a different normalized theory quotients by chains containing an adjacent fixed pair,

$R:X\times X\to X\times X$2

again producing a normalized quotient $R:X\times X\to X\times X$3. This normalized complex admits a geometric realization $R:X\times X\to X\times X$4, analogous to rack and quandle spaces, and for finite biquandles the second homotopy group $R:X\times X\to X\times X$5 is finitely generated. The normalized viewpoint is also the natural one for cocycle invariants, since degeneracies encode Reidemeister-I contributions that must vanish (2002.04567).

For finite cyclic biquandles $R:X\times X\to X\times X$6, the rational Betti numbers are completely determined: $R:X\times X\to X\times X$7 and the free ranks satisfy

$R:X\times X\to X\times X$8

In low degree,

$R:X\times X\to X\times X$9

and higher torsion is annihilated by $$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$0 for odd $$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$1 and by $$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$2 for even $$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$3 (Liang et al., 2021).

3. Algebraic models and computational frameworks

A major algebraic synthesis is the differential graded bialgebra

$$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$4

attached to a set-theoretic solution $$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$5. It contains the structure algebra

$$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$6

and is built so that

$$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$7

recover the Yang–Baxter homology and cohomology complexes. The bialgebra structure supplies an associative cup product on cohomology, and the quantum symmetrizer yields a comparison map to Hochschild (co)homology of $$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$8 (Farinati et al., 2015).

For idempotent set-theoretic braidings, this comparison becomes especially strong. The braided or Yang–Baxter cohomology of the solution can be reduced to a critical subcomplex, and the explicit quantum symmetrizer identifies that critical complex with the Hochschild complex of the structure monoid up to quasi-isomorphism. In the idempotent case, Yang–Baxter cohomology is therefore an effective model for Hochschild cohomology of the associated monoid, rather than merely an analogous theory (Lebed, 2016).

On the operator side, the normalized Jones $$(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X}) = (\mathrm{Id}_{X}\times R)(R\times \mathrm{Id}_{X})(\mathrm{Id}_{X}\times R).$$9-matrix admits the skein decomposition

$X$0

where $X$1 is a cup pairing and $X$2 is a cap copairing. This turns the Yang–Baxter differential into a sum of horizontal cup–cap concatenations and makes low-dimensional homology and cohomology calculations explicit. The same skein anatomy controls a natural nontrivial $X$3-cocycle, namely the cup pairing $X$4 itself (Elhamdadi et al., 2020).

A different computational framework appears in one-term Yang–Baxter homology for the vector representation $X$5 of $X$6. There the chain complex decomposes as

$X$7

with $X$8 a finite Koszul-type complex and

$X$9

The paper explicitly remarks that one can define Tor and Ext functors on $V$0 induced from the corresponding $V$1-module category. This gives a direct route from homological decompositions to a dual cohomological picture over a polynomial algebra (Tian et al., 6 May 2025).

4. Deformation theory and cohomological control

In deformation-theoretic formulations, Yang–Baxter cohomology behaves in the expected Gerstenhaber–Nijenhuis–Richardson pattern. If

$V$2

then $V$3 satisfies the Yang–Baxter equation modulo $V$4 if and only if $V$5 is a Yang–Baxter $V$6-cocycle. Accordingly,

$V$7

classifies infinitesimal Yang–Baxter deformations up to equivalence, and $V$8 governs obstruction theory for extending such deformations to higher order (Zappala, 2024, Saito et al., 4 Sep 2025).

For Yang–Baxter operators constructed from rack or self-distributive data associated to Lie algebras, a distinguished subclass of deformations is controlled by ordinary Lie cohomology. If

$V$9

comes from Lie $R:V\otimes V\to V\otimes V$0-cochains $R:V\otimes V\to V\otimes V$1, then the obstruction to extending an order-$R:V\otimes V\to V\otimes V$2 Yang–Baxter deformation is governed by the same equation that controls Lie algebra deformations, and the obstruction lies in

$R:V\otimes V\to V\otimes V$3

Vanishing of $R:V\otimes V\to V\otimes V$4 yields full formal integrability for nontrivial infinitesimal $R:V\otimes V\to V\otimes V$5-deformations. At the same time, full Yang–Baxter cohomology is strictly richer than Lie or self-distributive deformation theory: for perfect centerless $R:V\otimes V\to V\otimes V$6,

$R:V\otimes V\to V\otimes V$7

and for $R:V\otimes V\to V\otimes V$8 the associated Yang–Baxter operator has

$R:V\otimes V\to V\otimes V$9

despite Lie-theoretic rigidity. Rigidity of the underlying Lie algebra therefore does not imply rigidity of the associated Yang–Baxter operator (Zappala, 2024).

A parallel refinement appears for braided algebras endowed with both a multiplication $$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$0 and a Yang–Baxter operator $$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$1. Yang–Baxter Hochschild cohomology was extended to a braided-commutative theory by adding the constraint

$$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$2

In that setting,

$$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$3

classifies infinitesimal braided-commutative deformations, and

$$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$4

contains the obstructions to higher-order extensions. For Hopf algebras with the adjoint $$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$5-matrix, this theory is nontrivial and receives a natural map from normalized Hopf algebra second cohomology (Saito et al., 2024).

5. Explicit calculations and low-dimensional patterns

The normalized Jones $$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$6-matrix over $$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$7 provides one of the most explicit low-dimensional calculations presently available. For that operator,

$$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$8

and the cup pairing $$C^1_{\mathrm{YB}}=\operatorname{Hom}(V,V),\qquad C^2_{\mathrm{YB}}=\operatorname{Hom}(V^{\otimes 2},V^{\otimes 2}),$$9 in the skein decomposition $$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$0 defines a concrete nontrivial $$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$1-cocycle. On the homology side,

$$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$2

while over $$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$3,

$$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$4

These results confirm the $$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$5 case of the Przytycki–Wang conjectural pattern for the Jones family (Elhamdadi et al., 2020).

For cyclic biquandles, explicit normalized cocycles and low-dimensional torsion are also known. The free part of $$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$6 has rank $$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$7, and the low-dimensional groups satisfy

$$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$8

For $$\delta^1_{\mathrm{YB}}(f)=R(f\otimes 1)+R(1\otimes f)-(f\otimes 1)R-(1\otimes f)R,$$9,

$$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$0

The theory also furnishes explicit rational basis cocycles as orbit-indicator functions and explicit torsion-detecting cocycles in degree $$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$1 (Liang et al., 2021).

Alexander biquandles supply another source of concrete cocycles. For

$$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$2

on $$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$3 with $$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$4, the normalized $$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$5-cochain

$$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$6

is an $$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$7-cocycle. The class $$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$8 is explicitly nontrivial for

$$\begin{aligned} \delta^2_{\mathrm{YB}}(\phi) &=(R\otimes 1)(1\otimes R)(\phi\otimes 1) +(R\otimes 1)(1\otimes \phi)(R\otimes 1) +(\phi\otimes 1)(1\otimes R)(R\otimes 1)\ &\quad-(1\otimes R)(R\otimes 1)(1\otimes \phi) -(1\otimes R)(\phi\otimes 1)(1\otimes R) -(1\otimes \phi)(R\otimes 1)(1\otimes R). \end{aligned}$$9

The same paper computes nontrivial normalized homology groups for several finite Alexander biquandles, showing that the normalized quotient retains substantial information (2002.04567).

For the normalized HOMFLYPT operators $3$00, the $3$01-th homology is reduced to $3$02 “final” initial conditions via the decomposition

$3$03

This yields explicit formulas

$3$04

and

$3$05

with

$3$06

$3$07

A plausible implication is that the corresponding low-dimensional cohomology for the HOMFLYPT family should also be controlled by the same finite collection of final complexes, although the paper itself computes homology rather than cohomology (Christiana et al., 28 Feb 2025).

6. Applications, adjacent theories, and open directions

Yang–Baxter cohomology has a direct knot-theoretic role. In normalized set-theoretic theories, $3$08-cocycles and $3$09-cocycles define state-sum invariants of links and knotted surfaces, with normalization eliminating Reidemeister-I degeneracies. In the cycle-set setting, explicit cocycles $3$10 and $3$11 yield link invariants, and in the biquandle setting normalized cocycles evaluate on canonical $3$12- and $3$13-cycles represented by colored diagrams (Przytycki et al., 2019, 2002.04567).

A more recent development interprets operator-valued Yang–Baxter $3$14-cocycles as infinitesimal deformations of quantum knot invariants. If $3$15 is a Yang–Baxter $3$16-cocycle, then

$3$17

can be inserted into trace constructions or pairing/copairing formalisms to produce quantum cocycle invariants of knots. Within this framework, quandle cocycle invariants appear as a special case, while the Yang–Baxter quantum version can be strictly stronger in examples. The same theory interprets both the Jones and Alexander polynomials as invariants arising from higher-order Laurent deformations of Yang–Baxter operators (Saito et al., 4 Sep 2025).

Extension-theoretic variants sit adjacent to, but are not identical with, standard operator-valued Yang–Baxter cohomology. For cycle sets, degree-two cohomology classifies extensions: $3$18 and this extension theory is one of the primary low-degree applications of the cycle-set complex. At the group-theoretic end, groups of $3$19-type and involutive Yang–Baxter groups are organized by bijective $3$20-cocycles, transgression, and Yoneda products, so that the lifting of finite Yang–Baxter data becomes an $3$21-extension problem. For bijective relative Rota–Baxter groups, a distinct degree-two cohomology classifies extensions and is isomorphic to the degree-two cohomology of the associated skew left brace. These theories are Yang–Baxter-adjacent rather than interchangeable with the standard cohomology of a braiding itself (Lebed et al., 2015, David et al., 2014, Belwal et al., 2023).

Several open problems delimit the present state of the subject. The precise relation between the cohomology of a Yang–Baxter solution and the cohomology of its associated self-distributive structure remains open in the survey literature. In the operator-valued setting, the relation between the Saito–Zappalà complex and Eisermann’s complex is clear in degrees $3$22 and $3$23 for invertible operators, but remains unresolved in degree $3$24. For braided-commutative deformation theory, a full higher-dimensional cohomology beyond the low-dimensional obstruction framework is still asked for explicitly. These unresolved comparison problems are characteristic of the field: Yang–Baxter cohomology is by now a family of technically mature theories, but not yet a single fully unified one (Lebed, 2018, Saito et al., 4 Sep 2025, Saito et al., 2024).