In this work we study the solutions of the Yang-Baxter equation associated to nineteen vertex models invariant by the parity-time symmetry from the perspective of algebraic geometry. We determine the form of the algebraic ...

Braces were introduced by Rump to study non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. We generalize Rump's braces to the non-commutative setting and use this new structure to study not ...

This paper introduces Hopf braces, a new algebraic structure related to the Yang–Baxter equation, which include Rump’s braces and their non-commutative generalizations as particular cases. Several results of classical ...

We develop a theory of extensions for involutive and nondegenerate solutions of the set-theoretic Yang-Baxter equation and use it to produce new families of solutions. As an application we construct an infinite family of ...

We prove that a finite non-degenerate involutive set-theoretic solution (X, r) of the Yang{Baxter equation is a multipermutation solution if and only if its structure group G(X, r) admits a left ordering or equivalently ...

Using Bieberbach groups, we study multipermutation involutive solutions to the Yang-Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in ...

For a set theoretical solution of the Yang–Baxter equation (X, σ), we define a d.g. bialgebra B = B(X, σ), containing the semigroup algebra A = k{X}/xy = zt : σ(x, y) = (z,t) , such that k ⊗A B ⊗A k and HomA−A(B, k) are ...

We define a knot/link invariant using set theoretical solutions (X, σ) of the Yang-Baxter equation and non commutative 2-cocycles. We also define, for a given (X, σ), a universal group Unc(X) governing all 2-cocycles in ...

We introduce non-degenerate solutions of the Yang–Baxter equation in the setting of symmetric monoidal categories. Our theory includes non-degenerate set-theoretical solutions as basic examples. However, infinite families ...

We define the radical and weight of a skew left brace and provide some basic properties of these notions. In particular, we obtain a Wedderburn type decomposition for Artinian skew left braces. Furthermore, we prove analogues ...