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

Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation

“…Skew-braces on the other hand are used [23] to describe non-involutive set-theoretic solutions of the YBE, and they may be instrumental in identifying universal R-matrices. This rising research field has been particularly fertile and numerous relevant studies have been produced over the past several years (see for instance [2,3,6,7], [9]- [11], [18]- [20], [21,22], [23,25,26,30], [37]- [39]).…”

Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses

“…Hence the algebra K[A(X, r) Im λ] is representable, and thus its subalgebra K[M (X, r)] is representable as well. (S2) If (X, r) is finite, bijective and non-degenerate then K[M (X, r)] is a prime algebra if and only if it is a domain, and this is equivalent with (X, r) being involutive (see [38,Theorem 2.8] and [40]).…”

“…(N2) If (X, r) is finite and left non-degenerate and, moreover, the diagonal map q : X → X is bijective then K[M (X, r)] is left Noetherian and of finite Gelfand-Kirillov dimension (see [17,Corollary 5.4]). In particular, if additionally (X, r) is bijective then K[M (X, r)] is a left and right Noetherian PI-algebra (see [17,38,40]). To prove this one shows that, under the assumptions, K[M (X, r)] (and also K[A(X, r)]) is a finite left module over a left Noetherian algebra K[B], for some submonoid B of A(X, r) such that λ b = id for each b ∈ B, which ensures that the monoid B may also be treated as a submonoid of M (X, r).…”

“…In the context of set-theoretic solutions (X, r) of the Yang-Baxter equation, these monoids and algebras play a very crucial role and are being intensely studied (see for example [9,11,12,17,26,27,29,31,32,38,40]). The strongest results have been obtained for finite non-degenerate bijective solutions.…”

Set theoretic Yang-Baxter equation, braces and Drinfeld twists

“…Subsequently skewbraces were developed in [18] to describe non-involutive solutions. This emerging research area has been particularly fruitful and numerous relevant studies have been produced over the past few years (see for instance [2,4,5], [15]- [17], [18,20,28,29]).…”