Narrow Results

In this paper, we give the definitions of almost cocommutative Hopf coquasigroup and quasitriangular Hopf coquasigroup which are analogous to those in the theory of Hopf algebras. Using the actions of Hopf coquasigroups, we obtain two equivalent conditions for a Hopf coquasigroup to be almost cocommutative. Furthermore, a quasitriangular Hopf coquasigroup determines a class of solutions of the quantum Yang–Baxter equation (QYBE). Finally, we define a 2-cocycle on a Hopf coquasigroup and construct a triangular Hopf coquasigroup by using a Hopf coquasigroup with a 2-cocycle.

Let $G_2$ be a group acting on an abelian group $G_1$ via a homomorphism $\alpha \in \mathrm{\text{Hom}}(G_2;\mathrm{\text{Aut}}(G_1))$ and let a 2-cocycle $𝜀\in Z^2(G_2,G_1,\alpha )$. By Schreier’s theorem, the pair $(\alpha ,𝜀)$ determines a group $G_{(\alpha ,𝜀)}$ which can arise as a non-split extension of $G_1$ by $G_2$, denoted by $G_1\underset{(\alpha ,𝜀)}{\times }{G}_2$ and called the perturbed semidirect product of $G_1$ by $G_2$ under $(\alpha ,𝜀)$. In this paper, we classify the perturbed semidirect products and give some of their properties. Furthermore, we find necessary and sufficient conditions for two perturbed semidirect products to be isomorphic.