Hopf-Galois Theory as a Unifying Framework for Symmetry with a View towards Lifting Problems

Sammanfattning

Hopf–Galois theory provides a unifying framework for the study of symmetries,generalising classical Galois theory, free group actions in geometry,and strongly graded algebras. The first part of this thesis develops the algebraicmachinery underlying Hopf–Galois theory and illustrates how thesethree notions can be understood within a common framework.We then turn our attention to lifting problems. More precisely, motivatedby a classical lifting problem for free group actions in geometry, we use theHopf-Galois framework to formulate an analogous lifting problem for stronglygraded rings. Combining the standard permanence of strong gradings undersubgroup and quotient gradings with our converse result, we obtain thefollowing equivalence: for a G-graded ring S =L_g∈G S_g and a normal subgroup N ⊴ G, the ring S is strongly G-graded if and only if both the inducedN-grading and the induced quotient G/N-grading are strong.The result motivates analogous stability and lifting questions in the broadersetting of Hopf-Galois extensions. We conclude by briefly introducing thesequestions as possible directions for future research.