Search

We develop the theory of Schur covers of finite skew braces. We prove the existence of at least one Schur cover. We also compute several examples. We prove that different Schur covers are isoclinic. Finally, we prove that Schur covers have the lifting property concerning projective representations of skew braces. [ABSTRACT FROM AUTHOR]

Abstract: We study set-theoretic solutions of the Yang–Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterisation of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid and we show that r extends as a solution on as a set. Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids S of the required type, including iterated products equivalent to a solution, and extensions . Examples include a general ‘double’ construction and some concrete extensions, their actions and graphs based on small sets. [Copyright &y& Elsevier]