2019/12/20 by Cedo, Ferran, Jespers, Eric, Verwimp, Charlotte · 1 citation
#16T25 #20M05 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1912.09710
Given a set-theoretic solution (X,r) of the Yang--Baxter equation, we denote by M=M(X,r) the structure monoid and by A=A(X,r), respectively A'=A'(X,r), the left, respectively right, derived structure monoid of (X,r). It is shown that there exist a left action of M on A and a right action of M on A' and 1-cocycles π and π' of M with coefficients in A and in A' with respect to these actions respectively. We investigate when the 1-cocycles are injective, surjective or bijective. In case X is finite, it turns out that π is bijective if and only if (X,r) is left non-degenerate, and π' is bijective if and only if (X,r) is right non-degenerate. In case (X,r) is left non-degenerate, in particular π is bijective, we define a semi-truss structure on M(X,r) and then we show that this naturally induces a set-theoretic solution ( M, r) on the least cancellative image M= M(X,r)/η of M(X,r). In case X is naturally embedded in M(X,r)/η, for example when (X,r) is irretractable, then r is an extension of r. It also is shown that non-degenerate irretractable solutions necessarily are bijective.