Cohomology and extensions of braces

Abstract

Braces and linear cycle sets are algebraic structures playing a major role in the classification of involutive set-theoretic solutions to the Yang-Baxter equation. This paper introduces two versions of their (co)homology theories. These theories mix the Harrison (co)homology for the abelian group structure and the (co)homology theory for general cycle sets, developed earlier by the authors. Different classes of brace extensions are completely classified in terms of second cohomology groups.Comment: 16 pages. Final version. Accepted for publication in Pacific Journal of Mathematic

Skew braces of squarefree order

“…Subsequently, Rump showed how set-theoretic solutions arise from cycle sets [Rum05] and radical rings [Rum07a], and in the latter paper he also introduced braces as a generalisation of radical rings. Braces give rise to nondegenerate involutive set-theoretic solutions of QYBE, and have recently been studied intensively [Bac15,CJO16,LV16,CGIS18]. Several generalisations of braces have been investigated, including skew braces [GV17], which give noninvolutive solutions to QYBE, and semi-braces [CCS17], which give degenerate solutions.…”

About a question of Gateva-Ivanova and Cameron on square-free set-theoretic solutions of the Yang-Baxter equation

“…Conversly, if (X, r) is a non-degenerate solution, the binary operation · defined by x · y := λ −1 x (y) for all x, y ∈ X makes X into a left cycle set. The existence of this correspondence allows to move the study of involutive non degenerate solutions to left cycle sets, as recently done in [19,3,4,16,15,18,5,2], and clearly to translate in terms of left cycle set the classical concepts related to the non-degenerate involutive set-theoretic solutions. Therefore, a left cycle set is said to be square-free if the squaring-map q is the identity on X.…”

Extensions and Well's type exact sequence of skew braces

“…Let (β 1 , τ 1 ) and (β 2 , τ 2 ) be the pairs corresponding to s 1 and s 2 , respectively. Then there exists a map θ : H → I satisfying (16) and (17).…”

“…Different homology theories for various structures related to solutions of the Yang-Baxter equations were investigated extensively by Lebed and Vendramin [18]. Cohomology and extensions of linear cycle sets with trivial actions is studied by Lebed and Vendramin [17]. Recently generalized by Jorge A. Guccione, Juan J. Guccione and Christian Valqui [13] to non trivial actions.…”