The Quantum Symmetriser of a Yang–Baxter Solution Need Not Be a Quasi-Isomorphism

For a set-theoretic solution (X, σ) of the Yang–Baxter equation, Farinati and García Galofre showed that a comparison map from braided (co)homology to the Hochschild (co)homology of the structure algebra A = kM(X, σ) can be chosen to be the quantum symmetriser of −σ, and that it factors through a complex A ⊗ B ⊗ A, where B is the Nichols algebra of −σ. They asked whether this complex is a resolution of A, and Lebed asked in a 2019 Oberwolfach report whether the quantum symmetriser is bijective for general solutions; the answer is known to be positive for involutive (characteristic zero) and idempotent solutions. We show that it is negative in general. For the permutation rack X = Z/4, x ◁ y = x + 1, a bijective non-degenerate solution, we give an explicit 2-cycle of weight 3 in A ⊗ B ⊗ A that is not a boundary; it comes from a cubic quantum Serre relation in B. With coefficients in the module on which X acts by 0, the image of the quantum symmetriser misses part of HH₃(A; k) for every quotient of the braided complex. We show that A ⊗ B ⊗ A is a resolution if and only if A is Koszul and B is the Koszul dual coalgebra of A, and that in characteristic zero every finite permutation rack whose permutation has a cycle of length divisible by 4 (resp. 3) fails in weight 3 (resp. 4). For finite non-degenerate solutions that are not involutive and have a finite-dimensional Nichols algebra, such as the dihedral quandle of order 3, a negative answer already follows from a remark of Farinati and García Galofre and a theorem of Jespers, Kubat and Van Antwerpen. By exact computation, 13 of the 29 isomorphism classes of bijective solutions on three points fail. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-17294-017 (ulamai/UnsolvedMath; Oberwolfach Report 51/2019, V. Lebed, Problem 2 on the bijectivity of the quantum symmetriser).