A left skew brace is an algebraic structure \((E,+,\circ)\) under which both \((E,+)\) and \((E,\circ)\) are groups and the operations satisfy the relationship \[ a\circ(b + c) = (a\circ b) - a + (a \circ c) \] for all \(a, b, c \in E\). Given an abelian group \((I,+)\), one can extend this to the structure of a left skew brace by simply setting \(\circ = +\), and this will be called a trivial left skew brace. The study of braces has been motivated by their connection with set-theoretic solutions to the Yang-Baxter equation, although no direct applications in that direction are given in this paper. Rather, this paper is focused on studying extensions of skew braces and developing potential cohomology theories.
As for groups, given two left skew braces \(H\) and \(I\), an extension of \(H\) by \(I\) is a short exact sequence \[ 0 \to I \to E \to H \to 0 \] of brace homomorphisms, and one can define a notion of equivalent extensions, leading to the group \(\mathrm{Ext}(H,I)\) of equivalence classes of extensions. The focus here is on the case when \(I\) is an additive abelian group considered as a trivial left skew brace. In this setting, the authors introduce the notion of \(I\) being an \(H\)-module determined by a good triplet \((\nu,\mu,\sigma)\) for a homomorphism \(\nu : (H,\circ) \to \operatorname{Aut}(I,+)\) and anti-homomorphisms \(\mu: (H,+) \to \operatorname{Aut}(I,+)\) and \(\sigma: (H,\circ) \to \operatorname{Aut}(I,+)\) satisfying certain properties. Such a triplet is naturally associated with an extension and the authors define a refined extension group \(\mathrm{Ext}_{(\nu,\mu,\sigma)}(H,I)\), the union of which (over good triplets) gives \(\mathrm{Ext}(H,I)\). When \(I\) is an \(H\)-module, the authors define a degree 2 cohomology group \(H^2_N(H,I)\) via cocycles and coboundaries. In the first main theorem, they show that if the action of \(H\) on \(I\) is associated to the triplet \((\nu,\mu,\sigma)\), then there is a bijection between \(H^2_N(H,I)\) and \(\mathrm{Ext}_{(\nu,\mu,\sigma)}(H,I)\). Some examples of these groups are explicitly computed.
Let \(\mathrm{Autb}(H)\) and \(\mathrm{Autb}(I)\) denote the groups of brace automorphisms. With \(I\) a trivial left skew brace as above, the authors study the action of \(\mathrm{Autb}(H)\times \mathrm{Autb}(I)\) on \(\mathrm{Ext}(H,I)\). Given a triplet \((\nu,\mu,\sigma)\), let \(C_{(\nu,\mu,\sigma)}\) denote the stabilizer of \(\mathrm{Ext}_{(\nu,\mu,\sigma)}(H,I)\) under this action and consider an associated extension of \(H\) by \(I\) as above. A second main result is the construction of a Wells’ like exact sequence (with explicitly given maps): \[ 0 \to Z^1_N(H,I) \to \mathrm{Autb}_{I}(E) \overset{\rho}{\to} C_{(\nu,\mu,\sigma)} \to H^2_N(H,I), \] where \(Z^1_N(H,I)\) are the 1-cocycles and \(\mathrm{Autb}_{I}(E)\) consists of those automorphisms in \(\mathrm{Autb}(E)\) that preserve \(I\). This result is applied to the question of inducibilty of a pair \((\phi,\theta) \in \mathrm{Autb}{H}\times \mathrm{Autb}(I)\), which is defined to mean that the pair arises in the image of the map \(\rho\).
The authors also relate their results to the work of V. Lebed and I. Vendramin [Pac. J. Math. 284, No. 1, 191–212 (2016; Zbl 1357.20009)] on cohomology theories for braces (i.e., non-skew) and the connection with linear cycle sets.