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Entwined comodules and contramodules over coalgebras with several objects: Frobenius, separability and Maschke theorems

Abhishek Banerjee, Surjeet Kour

Abstract

We study module like objects over categorical quotients of algebras by the action of coalgebras with several objects. These take the form of ``entwined comodules'' and ``entwined contramodules'' over a triple $(\mathscr C,A,ψ)$, where $A$ is an algebra, $\mathscr C$ is a coalgebra with several objects and $ψ$ is a collection of maps that ``entwines'' $\mathscr C$ with $A$. Our objective is to prove Frobenius, separability and Maschke type theorems for functors between categories of entwined comodules and entwined contramodules.

Entwined comodules and contramodules over coalgebras with several objects: Frobenius, separability and Maschke theorems

Abstract

We study module like objects over categorical quotients of algebras by the action of coalgebras with several objects. These take the form of ``entwined comodules'' and ``entwined contramodules'' over a triple , where is an algebra, is a coalgebra with several objects and is a collection of maps that ``entwines'' with . Our objective is to prove Frobenius, separability and Maschke type theorems for functors between categories of entwined comodules and entwined contramodules.

Paper Structure

This paper contains 9 sections, 43 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Entwined comodules over coalgebras with several objects
  3. Entwined contramodules over coalgebras with several objects
  4. Separability properties for entwined comodules
  5. Separability of the functor $F_\psi$
  6. Separability of the functor $G_\psi$
  7. Separability properties for entwined contramodules
  8. Frobenius properties
  9. Maschke type theorems

Key Result

Theorem 1.1

(see Theorem T4.5hm and Theorem T4.5hm5) Let $\mathscr C$ be a coalgebra with several objects and let $(\mathscr C,A,\psi)$ be an entwining structure. Let $V_1$ be the space such that an element $\sigma\in V_1$ is a collection of maps $\sigma=\{\sigma_X:\mathscr C(X,X)\otimes A\longrightarrow K\}_{X for any $f\in \mathscr C(X,Y)$, $a\in A$, $X$, $Y\in Ob(\mathscr C)$. Then, the following are equiv

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 74 more