The notion of a bialgebroid. The dual structures of R-rings and corings, i.e. of algebras and coalgebras in a monoidal category of bimodules for an algebra R, will be recalled. Built on them, several equivalent definitions of a bialgebroid will be given. The duality of finitely generated and projective bialgebroids will be studied. The first lecture will be closed by collecting some examples.
Modules for bialgebroids.
The definition of a bialgebroid will be justified by the following
theorem. An
-
ring A is a bialgebroid if and only if the forgetful functor
from the category of A-modules to the category of R-R-
bimodules is strict monoidal. A possible generalisation of a Hopf
algebra, the so called
-Hopf
algebra, will be introduced by the requirement that the forgetful
functor is in addition strong right closed.
Comodules for bialgebroids. The monoidal category of comodules for a bialgebroid, possessing a strict monoidal functor to the category of bimodules for the base algebra, will be studied. Comodule algebras will be defined as algebras in the monoidal category of comodules. It will be shown that a comodule algebra for a bialgebroid determines a Doi-Koppinen datum (over the non-commutative base algebra) hence an entwining structure, hence a coring with a grouplike element. Galois extensions by a bialgebroid will be characterised by the Galois property of the associated coring. A coaction independent description of Galois extensions by finitely generated and projective bialgebroids will be given. The question of (non-) uniqueness of the bialgebroid symmetry for a given Galois extension will be discussed.
Hopf algebroids. Hopf
algebroids -- special kinds of
-Hopf
algebras -- will be defined by introducing an antipode. It will be
explained why does a Hopf algebroid involve two bialgebroid
structures over anti-isomorphic base algebras. The notion of an
integral in a Hopf algebroid will be invented and studied.
Maschke-type theorems will be proven, stating that a Hopf algebroid
is a semi-simple extension of its base algebra if and only if it is
a separable extension and if and only if there exists a normalised
integral in the Hopf algebroid. The dual result relates
co-semi-simplicity and coseparability of the coring, underlying a
Hopf algebroid, to the existence of normalised dual integrals. A
fundamental theorem of Hopf modules is applied to show that a Hopf
algebroid is a Frobenius extension of its base algebra if and only
if it possesses a non-degenerate integral. It will be illustrated by
an example that – in contrast to Hopf algebras -- finitely
generated and projective Hopf algebras are not necessarily
quasi-Frobenius extensions of their base algebras.
Galois extensions by Hopf algebroids. The theory of coring extensions will be used to show that the categories of comodules, for the two constituent bialgebroids in a Hopf algebroid, are monoidally equivalent. This result allows for a definition of a comodule algebra for a Hopf algebroid, i.e. of algebra extensions by Hopf algebroid symmetry. Kreimer-Takeuchi and Schneider type theorems will be proven. These are statements about situations in which the surjectivity of the canonical map implies its bijectivity, that is the Galois property.