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.