Filomat 2022 Volume 36, Issue 15, Pages: 5179-5220
https://doi.org/10.2298/FIL2215179F
Full text ( 1381 KB)
Braided Galois objects and Sweedler cohomology of certain Radford biproducts
Femić Bojana (Mathematical Institute of Serbian Academy of Sciences and Arts), bfic@mi.sanu.ac.rs
We construct the group of H-Galois objects for a flat and cocommutative Hopf
algebra in a braided monoidal category with equalizers provided that a
certain assumption on the braiding is fulfilled. We show that it is a
subgroup of the group of BiGalois objects of Schauenburg, and prove that the
latter group is isomorphic to the semidirect product of the group of Hopf
automorphisms of H and the group of H-Galois objects. Dropping the
assumption on the braiding, we construct the group of H-Galois objects with
normal basis. For H cocommutative we construct Sweedler cohomology and prove
that the second cohomology group is isomorphic to the group of H-Galois
objects with normal basis. We construct the Picard group of invertible
H-comodules for a flat and cocommutative Hopf algebra H. We show that every
H-Galois object is an invertible H-comodule, yielding a group morphism from
the group of H-Galois objects to the Picard group of H. A short exact
sequence is constructed relating the second cohomology group and the two
latter groups, under the above mentioned assumption on the braiding. We show
how our constructions generalize some results for modules over commutative
rings, and some other known for symmetric monoidal categories. Examples of
Hopf algebras are discussed for which we compute the second cohomology group
and the group of Galois objects.
Keywords: braided monoidal categories, Hopf-Galois objects, Sweedler cohomology, Picard group, quasi-triangular Hopf algebras, Radford biproducts
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