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Corrigendum and addendum to "The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang-Baxter equation"
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One of the main results stated in [3, Theorem 4.4] 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 normal 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 \cite[3, Lemma 5.3] (and its consequences, namely [3, Lemma 5.4] and [3, 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.
Tijdschrift: Transactions of the American Mathematical Society
ISSN: 0002-9947
Issue: 6
Volume: 373
Pagina's: 4517-4521
Jaar van publicatie:2020
CSS-citation score:2
Toegankelijkheid:Open