Cohomology and extensions of braces

Victoria Lebed and Leandro Vendramin

Vol. 284 (2016), No. 1, 191–212

DOI: 10.2140/pjm.2016.284.191

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.

Keywords

brace, cycle set, Yang–Baxter equation, extension, cohomology

Mathematical Subject Classification 2010

Primary: 20E22, 20N02, 55N35, 16T25

Milestones

Received: 2 February 2016

Revised: 22 March 2016

Accepted: 4 April 2016

Published: 10 July 2016