Structure algebras of finite set-theoretic solutions of the Yang--Baxter equation

TL;DR

The article extends the study of structure algebras associated with finite set-theoretic Yang–Baxter solutions beyond the well-studied bijective case. It proves the structure algebra $K[M(X,r)]$ is always left Noetherian for finite left non-degenerate solutions, and develops a robust framework—via left-divisibility, socles, and ideal chains—to characterize right Noetherianity, primeness, cancellative congruences, and GK-dimension (which equals the Krull dimension). It provides explicit decompositions of the monoid and its derived version, connects ring-theoretic properties to properties of the underlying solution, and extends the theory to certain degenerate solutions. These results yield a comprehensive picture linking YB-solution structure to Noetherian, prime, and semiprime behavior, offering new tools for constructing and analyzing finite, non-commutative algebras arising from Yang–Baxter theory.

Abstract

Quadratic algebras related to some classes of finite left non-degenerate solutions (X,r) of the Yang--Baxter equation have been intensively studied since they are the associative ring-theoretical tool to study solutions. These are the monoid algebras K[M(X,r)] and K[A(X,r)], over a field K, of its structure monoid M(X,r) and left derived structure monoid A(X,r). In case r is bijective (and thus also right non-degenerate) it is known that these algebras are representable (hence PI), left and right Noetherian and have finite Gelfand-Kirillov dimension. Moreover, such algebras are domains (or equivalently prime) if and only if they have finite global dimension, which also is equivalent to r being an involutive map. In this paper we deal with structure algebras of arbitrary finite left non-degenerate solutions (X,r), except for the last section. If (X,r) satisfies additional conditions, such as being bijective, idempotent or left derived, it has been shown in a series of papers that K[M(X,r)] is left Noetherian. In the first part of the paper we show that the algebra K[M(X,r)] always is left Noetherian. Via divisibility by generators, we construct an ideal chain in M(X,r) that has very strong algebraic structural properties on its Rees factors. This allows to obtain characterizations of when the algebras K[M(X,r)] and K[A(X,r)] are right Noetherian. Intricate relationships between ring-theoretical and homological properties of these algebras and properties of the solution (X,r) are proven. Furthermore, we describe the cancellative congruences of A(X,r) and M(X,r) as well as the prime spectrum of K[A(X,r)]. This then leads to an explicit formula for the Gelfand-Kirillov dimension of K[M(X,r)] and it equals the classical Krull dimension.

Theorems & Definitions (86)

• Theorem A
• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Proposition F
• Theorem 2.1
• Lemma 2.2
• Proposition 3.1