Set theoretic Yang-Baxter equation, braces and Drinfeld twists

Abstract

We consider involutive, non-degenerate, finite set theoretic solutions of the Yang-Baxter equation. Such solutions can be always obtained using certain algebraic structures that generalize nil potent rings called braces.Our main aim here is to express such solutions in terms of admissible Drinfeld twists substantially extending recent preliminary results. We first identify the generic form of the twists associated to set theoretic solutions and we show that these twists are admissible, i.e. they satisfy a certain co-cycle condition.These findings are also valid for Baxterized solutions of the Yang-Baxter equation constructed from the set theoretical ones.

Drinfeld Twists on Skew Braces

“…In [3], it is stated that any involutive YBE solution can be viewed as a quasi-admissible twist of the permutation solution. Although [3] is discussing linear Hopf algebras built on these solutions, the proof of Proposition 3.13 of [3] boils down to the following observation: if r(x, y) = (σ x (y), γ y (x)) satisfies the YBE we can introduce F (x, y) = (x, σ x (y)), Φ(x, y, z) = (x, σ x (y), σ γy (x) (z)) and Ψ(x, y, z) = (x, y, σ x (σ y (z))), so that the triple (F, Φ, Ψ) define a Drinfeld twist on (X, r). Conditions (T1), (T2) and (T3), all follow directly from r being a YBE solution and the identities σ σx(y) (σ γx(y) (z)) = σ x (σ y (z)) and σ γ σy (z) (x) (γ z (y)) = γ σx(y) (σ γy(x) (z)).…”

“…Observe that σ y (γ σ −1 x (y) (x)) appears in the first component of r 2 (x, σ −1 x (y)). Going back to the involutive setting considered in [3], we have that F rF −1 (x, y) = (y, x), since r 2 = id X 2 . Therefore, the permutation solution (X, fl X ) is obtained from a twist on (X, r), and by Theorem 2.2, (X, r) can be obtained by the inverse twist on (X, fl X ).…”

“…Hence, (F, Φ, Ψ) restrict to automorphisms on the subsets of basis elements of Q ⊗2 and Q ⊗3 i.e. R(H) 2 and R(H) 3 . Next we demonstrate that the axioms of Definition 2.2 hold.…”

“…Set-theoretical solutions of the Yang-Baxter equation (YBE) and their classification have attracted a lot of attention in recent years [2,4,3,8,16]. A prominent role in this theory is played by skew braces, which are sets with two compatible group structures [8].…”

“…However, the introduction of twists on skew braces is the truly novel portion of our work. A discussion relating Drinfeld twists and braces also appears [4,3], where the authors study Drinfeld twists on linear FRT-type algebras coming from the linearisation of involutive YBE solutions. We are able to rephrase part of the work done in [3] in terms of our notion of Drinfeld twists in Example 2.1.…”