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

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.