Equivariant Morita-Takeuchi Theory

TL;DR

This paper develops an equivariant Morita-Takeuchi theory for coalgebras with Hopf algebra symmetries, introducing a cohomology framework to classify coaction lifts and exploring the associated Picard groupoid.

Contribution

It introduces a new equivariant Morita-Takeuchi theory for coalgebras with Hopf algebra symmetries and develops a cohomology classification method.

Findings

Defined a cohomology theory for classifying coaction lifts.

Established a connection between the cohomology and the equivariant Picard groupoid.

Extended Morita-Takeuchi theory to include symmetry considerations.

Abstract

We introduce the notion of H-equivariant Morita-Takeuchi theory for coalgebras with symmetries given by a Hopf algebra H. A cohomology theory is introduced which classifies the possible lifts of coactions on coalgebras to corresponding comodules. An equivariant Picard groupoid is defined and its connection to the developed cohomology theory investigated.