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.

16 pages. Final version. Accepted for publication in Pacific Journal of Mathematics

References in corpus (3)

• Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation
• Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova
• Retractability of set theoretic solutions of the Yang-Baxter equation

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• Extensions and automorphisms of Rota-Baxter groups
• Relative Rota-Baxter groups and skew left braces
• Schur covers of skew braces
• Extensions and Well's type exact sequence of skew braces
• Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation
• Cohomology and Extensions of Relative Rota-Baxter groups
• Central extensions of some linear cycle sets
• Towards semi-trusses
• Cohomology of idempotent braidings, with applications to factorizable monoids