Objectives¶

Since this project is divided into three parts we have three main objectives. The first goal is to formulate a Galois context for commutative H-comodule algebras and then show that it gives rise a categorical Galois theory as defined by G.Janelidze. Having constructed Galois context we shall investigate the Galois theory it brings.

Secondly we want to generalise the Galois correspondence between subalgebras of an H-comodule algebra and quotients of the Hopf algebra H to the case where H is a bialgebroid. Then we want to generalise criteria for closedness of subalgebras and quotients of H the new settings that I proved in my PhD thesis. In the case of cleft extensions we were able to show that the quotient Q of H is closed if and only if the subextension corresponding to Q is Q-Galois, i.e. its canonical map (in the sense of H-comodule algebras) is an isomorphism. I proved a similar theorem for subalgebras: a subalgebra S is closed if a canonical map constructed by Schneider in ‘Normal Basis and Transitivity of Crossed Products for Hopf Algebras’ (Journal of Algebra, 1992, Theorem 1.4) is an isomorphism. We would like to generalise these theorems to the case of cleft extensions of bialgebroids and possibly to a broader case of bialgebroid coactions. In my PhD I generalised Schauenburg’s results on closedness to the case H-comodule algebras over not necessary commutative coinvariants subalgebra (i.e. the extension base). We also want to study a generalisation of Schauenburg results on closedness to the case of bialgebroids.

The third aim is to study categorical Galois theory in model categories (or a suitable generalisations of categories of fibrant objects). The aim is to construct a general categorical setting in which homotopy Galois theory can be formulated and discussed.