\(R\)-smash products of Hopf quasigroups. (Q1925581)

A Hopf quasigroup is a coassociative counital coalgebra \((H,\Delta,\varepsilon)\) equipped with a unital multiplication \(\mu_H\) and an antipode \(S_H\) satisfying several compatibility relations. These are non-associative generalizations of Hopf algebras in which the control of associativity is ensured by the antipode. If \(H,A\) are quasigroups and \(R\colon H\otimes A\to A\otimes H\) is a linear map then one can define on \(A\otimes H\) the multiplication \(\mu=(\mu_A\otimes\mu_H)\circ(id_A\otimes R\otimes id_H)\) and the antipode \(S=R\circ(S_H\otimes S_A)\circ\tau\), respectively, where \(\tau\) is the flip map. The main result of the paper gives necessary and sufficient conditions for \(R\) under which the tensor product coalgebra \(A\otimes H\) equipped with the multiplication \(\mu\), unit tensor product and antipode \(S\) is a quasigroup, too. The dual case is treated as well.