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The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang-Baxter equation


Authors: Eric Jespers, Łukasz Kubat and Arne Van Antwerpen
Journal: Trans. Amer. Math. Soc. 372 (2019), 7191-7223
MSC (2010): Primary 16N60, 16T25; Secondary 16R20, 16S36, 16S37
DOI: https://doi.org/10.1090/tran/7837
Published electronically: June 17, 2019
MathSciNet review: 4024551
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Abstract: For a finite involutive non-degenerate solution $ (X,r)$ of the Yang-Baxter equation it is known that the structure monoid $ M(X,r)$ is a monoid of I-type, and the structure algebra $ K[M(X,r)]$ over a field $ K$ shares many properties with commutative polynomial algebras; in particular, it is a Noetherian PI-domain that has finite Gelfand-Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid $ M(X,r)$ and the algebra $ K[M(X,r)]$ is much more complicated than in the involutive case, we provide some deep insights.

In this general context, using a realization of Lebed and Vendramin of $ M(X,r)$ as a regular submonoid in the semidirect product $ A(X,r)\rtimes \operatorname {Sym} (X)$, where $ A(X,r)$ is the structure monoid of the rack solution associated to $ (X,r)$, we prove that $ K[M(X,r)]$ is a finite module over a central affine subalgebra. In particular, $ K[M(X,r)]$ is a Noetherian PI-algebra of finite Gelfand-Kirillov dimension bounded by $ \vert X\vert$. We also characterize, in ring-theoretical terms of $ K[M(X,r)]$, when $ (X,r)$ is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of $ M(X,r)$.

These results allow us to control the prime spectrum of the algebra
$ K[M(X,r)]$ and to describe the Jacobson radical and prime radical of
$ K[M(X,r)]$. Finally, we give a matrix-type representation of the algebra $ K[M(X,r)]/P$ for each prime ideal $ P$ of $ K[M(X,r)]$. As a consequence, we show that if $ K[M(X,r)]$ is semiprime, then there exist finitely many finitely generated abelian-by-finite groups, $ G_1,\dotsc ,G_m$, each being the group of quotients of a cancellative subsemigroup of $ M(X,r)$ such that the algebra
$ K[M(X,r)]$ embeds into $ \operatorname {M}_{v_1}(K[G_1])\times \dotsb \times \operatorname {M}_{v_m}(K[G_m])$, a direct product of matrix algebras.


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Additional Information

Eric Jespers
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
Email: Eric.Jespers@vub.be

Łukasz Kubat
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
Email: Lukasz.Kubat@vub.be

Arne Van Antwerpen
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
Email: Arne.Van.Antwerpen@vub.be

DOI: https://doi.org/10.1090/tran/7837
Keywords: Yang--Baxter equation, set-theoretic solution, prime ideal, (skew) brace
Received by editor(s): December 5, 2018
Published electronically: June 17, 2019
Additional Notes: The first author was supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Belgium), grant G016117.
The second author was supported by Fonds voor Wetenschappelijk Onderzoek (Flanders), grant G016117.
The third author was supported by Fonds voor Wetenschappelijk Onderzoek (Flanders) via an FWO Aspirant-mandate.
Article copyright: © Copyright 2019 American Mathematical Society