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

Journal Contribution - Journal Article

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)⋊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 |X|. 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,...,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 M_{v_1}(K[G_1])×⋯×M_{v_m}(K[G_m]).
Journal: Transactions of the American Mathematical Society
ISSN: 0002-9947
Issue: 10
Volume: 372
Pages: 7191-7223
Publication year:2019
  • DOI: https://doi.org/10.1090/tran/7837
  • Scopus Id: 85075121202
  • ORCID: /0000-0002-2695-7949/work/70477303
  • ORCID: /0000-0001-7619-6298/work/70477809
  • ORCID: /0000-0002-7848-6405/work/87089696
  • WoS Id: 000514302600014
CSS-citation score:1
Accessibility:Closed