Representation theory of skew braces (Q6645106)

A skew (left) brace is a set \(A=(A,\ast,\circ)\) equipped with two group operations \(\ast\) and \(\circ\) such that the so-called brace relation \(a \circ (b \ast c) = (a \circ b) \ast a^{-1} \ast (a \circ c)\) holds for all \(a, b, c \in A\). A brace is a skew brace with \((A, \ast)\) abelian. According to \textit{T. Letourmy} and \textit{L. Vendramin} [J. Algebra 644, 609--654 (2024; Zbl 07804819)], a representation of a skew brace is a pair of representations on the same vector space, one for \((A, \ast)\) and the other for \((A, \circ)\), that satisfies a certain compatibility condition.\N\NIn the paper under review the authors, following the previous definition, explain how some of the results from representation theory of groups, such as Maschke's theorem and Clifford's theorem, extend naturally to that of skew braces. They shall also give some concrete examples to illustrate that skew brace representations are more difficult to classify than group representations.