Corrigendum and addendum to “The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation”

Abstract:

One of the main results stated in Theorem 4.4 of our article, which appears in Trans. Amer. Math. Soc. 372 (2019), no. 10, 7191–7223, is that the structure algebra $K[M(X,r)]$, over a field $K$, of a finite bijective left non-degenerate solution $(X,r)$ of the Yang–Baxter equation is a module-finite central extension of a commutative affine subalgebra. This is proven by showing that the structure monoid $M(X,r)$ is central-by-finite. This however is not true, even in case $(X,r)$ is a (left and right) non-degenerate involutive solution. The proof contains a subtle mistake. However, it turns out that the monoid $M(X,r)$ is abelian-by-finite and thus the conclusion that $K[M(X,r)]$ is a module-finite normal extension of a commutative affine subalgebra remains valid. In particular, $K[M(X,r)]$ is Noetherian and satisfies a polynomial identity. The aim of this paper is to give a proof of this result. In doing so, we also strengthen Lemma 5.3 (and its consequences, namely Lemma 5.4 and Proposition 5.5) showing that these results on the prime spectrum of the structure monoid hold even if the assumption that the solution $(X,r)$ is square-free is omitted.