(Co)module algebras and their generalizations

TL;DR

This paper explores the structure, stability, and polynomial identities of (co)module algebras over bi- and Hopf algebras, extending previous work with new categorical and grading equivalence insights.

Contribution

It introduces new results on (co)stability of radicals, invariant decompositions, and codimension growth, along with categorical and grading equivalence analyses in (co)module algebras.

Findings

Analysis of (co)stability of radicals

Existence of invariant Levi and Wedderburn decompositions

Growth of polynomial identities in (co)module algebras

Abstract

This manuscript is an extended version of the author's habilitation thesis defended on May 21, 2021 at M.V. Lomonosov Moscow State University. It is devoted to the study of (co)stability of radicals, existence of (co)invariant Levi and Wedderburn decompositions, structure of the corresponding simple algebras and codimension growth of polynomial identities in (co)module algebras over bi- and Hopf algebras and their generalizations. The main difference between this manuscript and the "official" thesis are additional chapters dealing with equivalences of gradings and Hopf algebra (co)actions, V-universal (co)acting bi- and Hopf algebras and related categorical questions. (In Russian.)