A covering theory for non-involutive set-theoretic solutions to the Yang–Baxter equation

One-generator braces and indecomposable set-theoretic solutions to the Yang–Baxter equation

“…In this preliminary section we collect some basic notation and facts related to the settheoretic Yang-Baxter equation. Recall that a q-cycle set [35] is a set X with binary operations • and : such that the self-map y → x • y of X is invertible and the equations…”

“…With the exponential operations in (4), Equations (5) and (6) state that (G, G) is a matched pairs of groups [30,39]. A group G = (A; •) with a q-affine structure is also called a q-brace with adjoint group A • := G [35]. In accordance with braces [33] and radical rings [28], the unit element is then denoted by 0 and inverses by a .…”

“…hold for x, y, z ∈ X. By [35,Proposition 1], q-cycle sets are equivalent to left * nondegenerate set-theoretic solutions S : X × X → X × X to the Yang-Baxter equation…”

“…While the natural map X → A X is an embedding for involutive solutions, this fails to be true in the non-involutive case. To remedy this shortcoming, the concept of q-brace [35] was introduced.…”

“…35, Corollary 2 of Proposition 8] (see also[32, Proposition 2]), a cycle set X is non-degenerate if and only if the square map x → x • x is bijective. Together with the dual X * , we can form the double X X * , with x * * := x and the extended operation (see[35, Definition 2])x * • y := y x .…”