Homomorphisms, separable extensions, and Morita maps for weak module algebras. (Q536642)

Let \(k\) be a field, and \(H\) both an algebra and a coalgebra over \(k\). The authors review the definitions of a weak bialgebra, weak Hopf algebra with weak antipode \(S\), weak right comodule algebra, weak left \(H\)-module algebra over a weak bialgebra \(H\), and a weak smash product \(A\#H\) of a weak Hopf algebra \(H\) with bijective antipode \(S\), and a weak left \(H\)-module algebra \(A\). Then some homological properties of weak Hopf algebras and the weak structure theorem in the category of \((H,A)\)-Hopf modules are given. Theorem 1. Let \(H\) be a weak Hopf algebra with bijective weak antipode \(S\). Then \(\Hom_{A\#H}(A,A\#H)\neq 0\) if and only if there exists \(0\neq t\in\int^l\) such that the map \(\varphi\colon A\to A\#H\) given by \(a\to a\#t\) is nontrivial. Theorem 2. Let \(H\) be a cocommutative weak Hopf algebra and \(A\) a weak left \(H\)-module algebra. If \(A\) contains a central element of trace one (i.e., \(a\in A\) such that \(t\cdot a=1_A\) for some left integral \(t\in\int^l\)), then \(A\#H\) is a separable extension of \(A\). Moreover, let \(H\) be a weak left Hopf algebra with weak antipode \(S\), \(A\) a weak left \(H\)-module algebra, and \(_A(_H\mathcal M)\) the category of weak left \((H,A)\)-Hopf modules. The weak structure theorem for \(_A(_H\mathcal M)\) is obtained.