On groups of $I$-type and involutive Yang-Baxter groups

Counterexample to a conjecture about braces

“…In (2), the bijective correspondence is clear by (1). Then, if B1 and B2 are two left braces with additive group equal to A such that ϕ : B1 → B2 is an isomorphism of left braces, note that ϕ is in particular an automorphism of A.…”

“…Is G the multiplicative group of a left brace? Some results supporting a positive answer to the question can be found in [2,7,9,14]. In [2,7], it is proved that some classes of finite solvable groups are the multiplicative group of a left brace; namely, abelian groups, nilpotent groups of class 2, Hall subgroups of the multiplicative group of a left brace, abelian-by-cyclic groups, and solvable A-groups (i.e.…”

Cohomology and extensions of braces

“…An alternative approach to extensions was suggested earlier by Ben David and Ginosar [5]. Concretely, they studied the lifting problem for bijective 1-cocycleswhich is yet another avatar of braces.…”

Construction of a group of automorphisms for an infinite family of Garside groups

“…Background on set-theoretical solutions of the QYBE. We follow [14] and refer to [14], [20] and [2,3,4,18,19] for more details. Fix a finite dimensional vector space V over the field R. The Quantum Yang-Baxter Equation on V is the equality…”

“…Example 1.2. Let X = {x 1 , x 2 , x 3 , x 4 }, and S : X × X → X × X be defined by S(x i , x j ) = (x g i (j) , x f j (i) ) where g i and f j are permutations on {1, 2, 3, 4} as follows: 3,4). The solution (X, S) is non-degenerate and symmetric and its structure group G(X, S) has the following defining relations:…”