Lax entwining structures, groupoid algebras and cleft extensions.
An entwining structure consists of an algebra \(A\) and a coalgebra \(C\) together with a map \(\psi\colon C\otimes A\to A\otimes C\) satisfying appropriate axioms. Examples can be constructed starting from a bialgebra \(H\) (playing the role of \(C\)) and an \(H\)-comodule algebra \(A\). Galois theory for entwining structures can be developed (leading to so-called coalgebra Galois extensions), and this is a generalization of Hopf-Galois theory. Generalizations of entwining structures are possible in different directions: weak entwining structures can be constructed from comodule algebras over weak bialgebras; partial entwining structures can be constructed from partial actions of bialgebras on algebras. These two notions unify in lax entwining structures. The aim of this paper is to develop Galois theory of lax entwining structures. The main results are in Sec. 3, where lax \(C\)-Galois extensions with normal basis are characterized. The authors introduce lax \(C\)-cleft extensions and prove that lax \(C\)-cleft extensions are precisely lax \(C\)-Galois extensions with normal basis (Theorem 3.11). This generalizes a classical result of \textit{Y. Doi} and \textit{M. Takeuchi} [Commun. Algebra 14, 801-817 (1986; Zbl 0589.16011)].
scientific article; zbMATH DE number 6307365
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